Heuristics, Experimental Subjects, and Treatment
Evaluation in Bigraph Crossing Minimization
Matthias Stallmann, Franc Brglez, and Debabrata Ghosh
North Carolina State University
The bigraph crossing problem, embedding the two node sets of a bipartite graph G = (V0 , V1 , E)
along two parallel lines so that edge crossings are minimized, has application to circuit layout and
graph drawing. Our experimental study looks at various heuristics, input properties, and data
analysis approaches. Among our main findings are the following.
—The most dramatic differences among heuristics show themselves on trees. As graphs, particularly random ones, become more dense, differences are much less clear. The densest graphs in
our study, which have average node degree of 16, produce no noticeable difference in crossing
numbers among leading heuristics.
—Among graphs with roughly the same density, ones that are structured — we consider butterflies,
meshes, hypercubes, graphs motivated by circuits, and minimum spanning trees of randomly
selected points on the plane — tend to benefit most from initial placement (breadth-first or
depth-first search) while ones that are more random respond more to iterative improvement
(for example, repeated application of the barycenter heuristic).
—The best heuristics combine good initial placement with iterative improvement. One such
heuristic, our newly devised tr17, ranks first or second among heuristics of reasonable efficiency
for most classes of graphs.
—The widely-used dot heuristic, ∗ also a combination of initial placement and iterative improvement, performs poorly relative to simpler, more efficient heuristics, especially as graphs become
larger. The best results for dot are obtained on random trees, trees motivated by VLSI circuits,
meshes, and hypercubes.
—Minimum spanning trees on random points within narrow rectangles provide experimental subjects whose minimum number of crossings approach 0 in a controllable fashion
Source code, scripts for generating data, running heuristics, and analyzing results, and random number seeds used to generate all our data are available on the world-wide web at URL
www.cbl.ncsu.edu/software/2001-JEA-Software/.
The research of F. Brglez
tor Research Corporation
EL/DAAH04–94–G–2080),
Corporation.
D. Ghosh is currently with
and D. Ghosh has been supported by contracts from Semiconduc(94–DJ–553), SEMATECH (94–DJ–800), DARPA/ARO (P–3316–
and (DAAG55-97-1-0345), and a grant from Semiconductor Research
Intel Corporation.
Name: Matthias Stallmann, Franc Brglez, and Debrabata Ghosh
Affiliation: Dept. of Computer Science, North Carolina State University
Address: To contact the first author, phone: 919-515-7978, email: Matt [email protected]
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2
·
M. Stallmann, F. Brglez, and D. Ghosh
Categories and Subject Descriptors: D.2.8 [Software]: Software Engineering—metrics, complexity measures, performance measures; F.2 [Theory of Computation]: Analysis of Algorithms
and Problem Complexity; G.2.1 [Mathematics of Computing]: Discrete Mathematics—combinatorial algorithms, permutations and combinations; G.2.2 [Mathematics of Computing]:
Graph Theory—graph algorithms, trees; G.3 [Mathematics of Computing]: Probability and
Statistics—experimental design; G.4 [Mathematics of Computing]: Mathematical Software—
algorithm design and analysis
General Terms: Experimental, Heuristic, NP-Hard, Optimization
Additional Key Words and Phrases: Crossing number, graph drawing, graph embedding, design
of experiments, graph equivalence classes, placement, layout
1. INTRODUCTION
The minimization of the crossing number in a specific graph embedding has often
been motivated by factors such as (1) improving the appearance of a graph drawing
[Di Battista et al. 1999; Eades and Sugiyama 1990; E.R. Gansner, E. Koutsifios,
S.C. North and K.P. Vo 1993; Mutzel 1998; Warfield 1977], (2) reducing the wiring
congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire
length and the layout area [Leighton 1984; Marek-Sadowska and Sarrafzadeh 1995;
Thompson 1979]. This paper is about bigraph crossing defined as follows for any
bipartite graph (bigraph) G = (V0 , V1 , E) [Harary and Schwenk 1972]: Let G be
embedded in the plane so that the nodes in Vi occupy distinct positions on the line
y = i and the edges are straight lines. For a specific embedding f (G), the crossing
number Cf (G) is the number of line intersections induced by f . This depends
only on the permutation of Vi along y = i and not on specific x-coordinates. The
(bigraph) crossing number C(G) = minf Cf (G). Garey and Johnson [Garey and
Johnson 1983] proved that it is NP-hard to compute C(G). Detection of a biplanar
graph, a bigraph G for which C(G) = 0, however, is easy [Harary and Schwenk
1971].
The two node sets V0 and V1 are also called layers of the bigraph. Previous
work on bigraph crossing has focused primarily on the fixed-layer (version of the)
problem, namely computing C(G) subject to the constraint that the permutation
of V0 on y = 0 must stay fixed. Even that is NP-hard [Eades and Wormald 1994].
Work on heuristics for both fixed-layer and general bigraph crossing has mostly
been theoretical [Di Battista et al. 1999; Eades and Whiteside 1994; Eades and
Wormald 1994; Mäkinen 1989; Mäkinen 1990; Shahrokhi et al. 2001; Yamaguchi and
Sugimoto 1999]. Experimental evaluation has focused on dense graphs, for which
good lower bounds are available [Eades and Kelly 1986; Jünger and Mutzel 1997;
Matuszewski et al. 1999], and relatively small sparse random graphs [Schönfeld
et al. 2000; Jünger and Mutzel 1997; Matuszewski et al. 1999]. Graphs arising in
circuit design are large (thousands of nodes, possibly millions), very sparse, and
highly structured.
We have used the dot ∗ package [E.R. Gansner, E. Koutsifios, S.C. North and K.P.
∗ The
dot heuristic is only evaluated for its ability to minimize crossings in bigraph. It also does
layer assignment and drawing of multi-layer graphs.
Bigraph Crossing Minimization
·
3
Vo 1993], intended for graph drawing, to determine placement in circuits [N. Kapur
1998], and found that, after a traditional routing package is applied (e.g [Betz and
Rose 1997; K. Kozminski, (Ed.) 1992]), the wire length and area correlate well with
crossing number [Ghosh et al. 1998a; Ghosh et al. 1998b]. In fact, this correlation
appears to be independent of the quality of the heuristic used to minimize crossings
[Ghosh 2000], so that better crossing minimization implies smaller wire length and
area.
This paper expands and consolidates previous experimental work on bigraph
crossing by: (a) evaluating heuristics based on their performance on general (rather
than fixed-layer) bigraph crossing, (b) presenting new heuristics that perform well
on large instances related to circuit design and random connected graphs of the
same sparsity, (c) doing careful comparison of new heuristics with dot and other
heuristics in the literature, (d) presenting well-defined classes of graph data that
illustrate the strengths and weaknesses of various heuristics, and, (e) mapping out
directions for the development of heuristics and graph data that are most likely to
be significant.
Our work has potential impact on circuit design and beyond in several ways:
(a) new heuristics for bigraph crossing will improve placement, (b) crossing number
can be used as a defining characteristic of a graph or a class of graphs that share
similar behavior under placement and routing heuristics; “benchmark circuits” can
therefore be classes of distinct but similar circuits instead of individual circuits (see
[Ghosh 2000; Ghosh and Brglez 1999; Kapur et al. 1997]), and (c) the methodology
used to generate subjects and evaluate treatments can be emulated in other domains
that call for the solution of NP-hard optimization problems.
The performance of a heuristic can vary widely even on a single input graph,
depending on the order in which the input is presented. This motivates us to define a presentation (of a graph G) as hG, π0 , π1 i, where πi is a permutation of Vi .
As any heuristic implicitly sequences the input when it reads data, the presentation captures essential information about any of the common ways of describing
a bigraph. If described as a list of neighbors for each V0 node, π0 describes the
order of appearance of the V0 nodes while π1 is used to sort the adjacency lists. A
list of edges is sorted using π0 as primary key and π1 as secondary key. Multiple
randomly-generated presentations of a single graph G constitute an isomorphism
class, an important class of experimental subjects — see Section 3.
Quite conveniently, a presentation also yields an embedding of G: use πi to
sequence the Vi nodes along y = i. Let C(hG, π0 , π1 i) denote its crossing number.
The object of the bigraph crossing problem, then, is to compute the crossing number
C(G) = minπ0 ,π1 C(hG, π0 , π1 i).
Pushing this idea further, a heuristic h is a mapping from one graph presentation to another, that is h(hG, π0 , π1 i) = hG, π00 , π10 i and we can study its behavior
statistically by looking at how a distribution on random hG, π0 , π1 i drawn from a
class of presentations imposes a distribution on C(h(hG, π0 , π1 i)). Recent work at
CBL † has focused on using distributions of C(h(hG0 , π0 , π1 i)) where G0 is derived
from G (or is identical to it) to characterize G and synthesize graph equivalence
classes based on G for the evaluation of circuit partitioning and placement heuristics
† Collaborative
Benchmarking Laboratory, North Carolina State University
4
·
M. Stallmann, F. Brglez, and D. Ghosh
[Ghosh 2000; Ghosh and Brglez 1999; Ghosh et al. 1998].
This paper focuses on the heuristics themselves and the experimental methodology for comparing their performance. A direct consequence of the work reported
here is that, whereas the dot heuristic [E.R. Gansner, E. Koutsifios, S.C. North and
K.P. Vo 1993] was used in earlier work characterizing graphs [Ghosh et al. 1998], a
heuristic with better statistical properties (and faster execution time) is now used
in its place [Ghosh 2000].
The paper is organized as follows: Section 2 presents the proposed heuristics,
both existing and new, Section 3 describes the design of our experiments, with
special focus on the data sets used, Section 4 summarizes experimental results,
and Section 5 presents tentative conclusions and suggests further work. There are
four appendices: one discusses validation of our implementations and experimental
design with respect to previous work, another proves a fact relevant to a new type of
experimental subject, a third gives detailed descriptions of some of the experimental
subjects, and the last gives a brief description of software available on the worldwide web.
2. OVERVIEW OF THE HEURISTICS
Our heuristics (experimental treatments) follow the scheme outlined in [Warfield
1977]. There are two phases: an initial ordering and iterative improvement. The
former involves some global computation on the graph to sequence the nodes in each
layer and the latter repeatedly solves a fixed-layer problem on alternating layers.
Various combinations of initial ordering and iterative improvement are considered
in our study. In our notation for heuristics and experimental subjects, n0 = |V0 |,
n1 = |V1 |, and m is the number of edges. The expected number of crossings for
an arbitrary presentation of a bigraph, as reported in [Warfield 1977], is m(m −
1)(n0 − 1)(n1 − 1)/4(n0 n1 − 1).
2.1 Initial Ordering
Two observations suggest that a carefully computed initial ordering can avoid traps
for subsequent attempts at improvement without incurring prohibitive execution
time. The first, already mentioned in the introduction, is that it is easy to detect a
biplanar graph [Harary and Schwenk 1971]. Finding the appropriate embedding is
also easy [Eades et al. 1976] even though the best-known heuristics do poorly with
biplanar graphs as input. Second, a cycle of length 2n has an optimum embedding
with n − 1 crossings [Harary and Schwenk 1972]. That embedding is achieved if the
nodes are sequenced in breadth-first order starting at any node of the cycle.
We use two initial orderings in our experiments in addition to the random ordering for a total of 3 possibilities. One ordering uses a breadth-first search starting
at a random node. This always gets the optimal solution for a simple cycle, and
usually performs better than random ordering.
2.2 Guided Breadth-First Search.
The other initial ordering heuristic is called guided breadth-first search (gbfs). The
main breadth-first search is preceded by another search (also breadth-first) whose
main function is to identify the longest path P in the graph. The main search then
begins at one end of P and, at any branch point, follows all shorter paths before
Bigraph Crossing Minimization
n0 2
2 n2
n1 2
5
n1 1
1
2 c1
2 c0
·
c1 2
c2
watch this
dashed edge!
n0 3
depth in first bfs
5 c0
4 c2
sequence in
2nd bfs
6 n2
Fig. 1.
1
n1
3
n0
6
n2
c1
c2
2
4
c0
5
Guided breadth-first search on a small example.
continuing with P .
For each node v the first search calculates dist[v], the distance from the start
node, and depth[v], the maximum value of dist[w] achieved by any descendant w
of v in the breadth-first search tree. The main search begins at a node s for which
dist[s] is maximized. Adjacency lists are sorted by increasing depth and ties are
broken by visiting nodes w with larger dist[w] first. Sequence numbers are assigned
to nodes based on the order of visitation in the main search and these are used to
sort nodes on each layer of the bigraph.
All of the above can be accomplished easily in linear time: the depth values can
be calculated by visiting nodes in reverse order at the end of the first bfs and letting
depth[v] = maxw a child of v depth[w]. All sorting is based on values that range from 1
to n — bin sorts of all adjacency lists can be combined. Our actual implementation
is not linear time but it runs fast enough in practice. It is easy to show that gbfs
always obtains optimal solutions for biplanar graphs and simple cycles.
Fig. 1 shows the two breadth-first searches and the final ordering of gbfs on a
simple example. The first search begins at n0, a completely arbitrary choice. The
depth[v] number is shown beside each node in the first search. When the dashed
edge is present the crossing induced by the cycle is unavoidable and the order in
which c0 and c2 are visited in the second search does not affect the outcome.
Without the dashed edge, however, the graph is biplanar, and it is important
to visit c2 first in the second search. Otherwise the edge c2 n0 will cross c0 n2
unnecessarily. The correct order is guaranteed by gbfs because the depth of c2 in
the first search will always be less than that of c0.
2.3 Median and Barycenter Heuristics
A typical way to improve an initial ordering is to apply a heuristic for fixed layer
crossing, alternating repeatedly from layer to layer [E.R. Gansner, E. Koutsifios,
6
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M. Stallmann, F. Brglez, and D. Ghosh
S.C. North and K.P. Vo 1993; Jünger and Mutzel 1997]. Each application of the
fixed-layer heuristic is called a pass. While this is not the only possible paradigm
for iterative improvement, it characterizes all three of the ones reported here, two
popular previously-known heuristics and a new heuristic called adaptive insertion,
described in Section 2.4.
The median heuristic treats the neighbors of each node as a set of integers representing their ordinal numbers on the opposite layer. Nodes are sorted using the
medians of these sets as keys. Implementations of the median heuristic differ in
how the median of a set of even cardinality is computed. Ordinarily, the median
would be defined as the mean of the two middle elements. The median heuristic
introduced in [Eades and Wormald 1994] (see also [Di Battista et al. 1999; Mäkinen
1989]) and evaluated here always uses the smaller of the two candidates, but with
the added condition that, in case of ties, nodes with odd degree always precede
those of even degree. Assuming random initial ordering and a stable sort, ‡ the
probability that the median heuristic embeds a simple path optimally is O(1/n).
It embeds a simple cycle optimally every time.
The barycenter heuristic uses the mean of the set of neighboring positions, rather
than the median, as a key for sorting (the name comes from the fact that it is
a one-dimensional analog of Tutte’s barycenter method for drawing graphs [Tutte
1960; Tutte 1963]). With random initial ordering it performs poorly relative to the
median on paths, cycles, and other very sparse highly-structured graphs (since these
require decisive movement of degree-2 nodes). On graphs that are more random
and/or have several nodes of high degree, barycenter does better than median
(high-degree nodes need to be centrally located with respect to their neighbors).
Jünger and Mutzel [Jünger and Mutzel 1997] observed that the barycenter heuristic
does better than the median on random graphs of various sizes and densities. We
come to the same conclusion. However, as discussed later, our implementation of
the barycenter heuristic and our random graph model differ from theirs (see also
Appendix A).
Our implementations allow median and barycenter to converge, that is, passes
are done until no change occurs. Even though we were unable to prove termination,
the number of passes never exceeded 4000, even on graphs with more than 15,000
edges.
Each pass of the median heuristic can be implemented in linear time (the keys
used for sorting are integers in the range 0, . . . , n − 1), while the barycenter requires
O(m + n log n) per pass (m is the number of edges, n = n0 + n1 , the number of
nodes). For graphs with a constant degree bound, the barycenter heuristic can also
be implemented in linear time per pass. Our experimental implementations of both
median and barycenter used insertion sort, which may be the fastest method in
practice because the number of inversions decreases significantly during each pass
when multiple passes are done.
2.4 Adaptive Insertion Heuristic
Local search is a popular way to improve solutions to optimization problems. A
simple operation on the current presentation is repeated until no instance of the
‡A
stable sort does not change the order of items having the same key.
Bigraph Crossing Minimization
·
7
operation improves cost, i.e. reduces number of crossings. An example of an operation is neighbor swapping, the exchange of nodes (at positions) i and i + 1 on
layer `. Such exchanges would be repeated until no choice of i and ` would yield a
decrease in the number of crossings. The actual difference in number of crossings,
call it d` (i, i + 1), depends only on the relative positions of neighbors of nodes i and
i + 1 on layer 1 − ` and is therefore easy to compute [Di Battista et al. 1999, pp.
281–283].
Adaptive insertion generalizes local search based on neighbor swapping in two
ways: (a) a larger “neighborhood” — each primitive operation inserts a node at any
position among other nodes on its layer of the bipartition; this greatly increases the
chance of finding an improved ordering; and (b) individual operations are batched
into passes as in the Kernighan-Lin graph partitioning heuristic [Kernighan and
Lin 1970] — each node is inserted exactly once during a pass.
Details of adaptive insertion are based on experimental fine tuning. For example,
we discovered that starting each pass where the previous one ended, rather than at
the point of lowest cost in the previous pass, yielded significantly better results. This
appears to be because it forces the heuristic to examine a larger set of presentations.
The primitive operation of adaptive insertion inserts node i in a different position
on its layer P
`. Suppose i is inserted before j, where j < i. The resulting cost change,
D` (i, j), is j≤k<i d` (i, k) — the effect of the insertion is that of a series of swaps
of i with i − 1, . . . , j. The situation is symmetric when j > i and i is inserted after
node j.
One pass of adaptive insertion does a right-to-left sweep of nodes on a layer `.
If the current node i has not already been inserted, the position j, before or after,
with the best D` (i, j) is found. Nodes are not allowed to stay in place even if any
insertion would increase the number of crossings.
A node i is marked if (a) i is inserted in a primitive operation or if (b) i is
immediately to the left of a node k when k is about to be inserted. Once marked,
a node is not inserted (again) during the remainder of the current pass. Part (b)
of the marking rule prevents the immediate undoing of a swap between a node and
its left neighbor when such a swap does not decrease the number of crossings.
To illustrate a pass of adaptive insertion, consider rearranging the V0 nodes of
Fig. 2(a) by a sequence of insertions. Recall that the heuristic works from right to
left, an arbitrary choice. First, node f finds its best position relative to the other
nodes. Ties are broken by choosing the position closest to the current one, except
that staying in place is not an option.
The best position turns out to be an insertion before node e with no change in
number of crossings — Fig. 2(b). Both e and f are marked, f because it has been
inserted — rule (a) above — and e because of its position immediately to the left of
f — rule (b). This makes node d the next one to be inserted into its best position,
before node a. This insertion is equivalent to swapping d with each of c, b, and
a in turn. Each swap can be evaluated independently [Di Battista et al. 1999].
Swapping d with c decreases the crossing number by 3 because edge cD no longer
crosses dA or dB, and edge cB no longer crosses dA. Swaps of d with b and a each
yield a decrease of 1 for a total decrease of 5 — Fig. 2(c). Nodes c and d are now
marked.
The next unmarked node is b, and its best insertion turns out to be to the right,
·
8
M. Stallmann, F. Brglez, and D. Ghosh
A
B
a
b
A
C
D
c
d
e
22 crossings
(a) initial presentation
B
C
f
A
b
c
C
D
D
d
a
B
f
d
22 crossings
a
f
e
*
(+1)
*
*
b
* * *
*
e
c
18 crossings
(d) after the third insertion
marked (+/−0)
position of inserted node is marked also
(b) after the first insertion
A
A
B
C
a
d
*
marked
a
b
c
f
e
*
*
*
(−5)
B
C
D
D
d
c
b
f
* * * * *
(+1)
e
*
19 crossings
(e) after the fourth and final insertion
17 crossings
(c) after the second insertion
Fig. 2.
Adaptive insertion on a simple example.
after c, yielding an increase of 1 — see Fig. 2(d). Finally node a is forced to move
and finds its best position before d. The position after one pass of adaptive insertion
is shown in Fig. 2(e).
A single pass of adaptive insertion takes O(m2 ) time. If all adjacency lists are
initially sorted using layer 1 − ` positions as keys (this can be done in linear time),
the calculation of d` (i, k), and hence D` (i, k), for all k 6= i takes time O(mk) where
k = deg(i). Summing over all nodes i gives the O(m2 ) bound. § For denser graphs
the time per node can be reduced to O(m log k) using a simple data structure, for
an overall bound of O(mn log(m/n)) per pass.
Adaptive insertion has many similarities with sifting, as reported recently in
[Matuszewski et al. 1999; Schönfeld et al. 2000]. The two main differences are (a) we
do not allow a node to stay in its current position, and (b) we consider nodes in
their current right-to-left order rather than based on their degrees. Restriction (a)
introduces an element of unpredictability which appears to enhance the potential
§ Our
implementation is slower than the analysis suggests — we use insertion sort to sort adjacency
lists and, since we identify nodes by their position on a layer, extra costs are introduced when the
position of a node changes.
Bigraph Crossing Minimization
·
9
solution quality produced by adaptive insertion (it does not get stuck in the manner
of a local search). But the restriction also takes away any natural stopping criterion,
leading to potentially long series of iterations in which no improvement takes place.
2.5 A New Heuristic
Adaptive insertion appears to perform at least as well as the other iterative improvement heuristics and often considerably better. Unfortunately, the number of
passes required to outperform median or barycenter is large and appears to grow
nonlinearly with the number of edges (see [Stallmann et al. 1999a]). After much
experimentation we engineered a combined iterative improvement heuristic that
repeats the following two-part step until it fails to find a presentation with fewer
crossings: (a) run the barycenter heuristic for enough passes to achieve convergence,
then (b) run adaptive insertion for a fixed number of passes, alternating layers —
we chose 2048 passes.
The new heuristic, when combined with gbfs initial ordering, outperforms all the
other we considered, but has prohibitive execution times on the larger graphs. We
use it primarily as an indicator of the existence of better solutions, and perhaps as
a precursor of a more efficient heuristic that performs equally well.
2.6 The Dot Heuristic
The dot heuristic used in our experiments is a modified version of the graph drawing package described in [E.R. Gansner, E. Koutsifios, S.C. North and K.P. Vo
1993] — the final phase that computes the geometry of the drawing has been eliminated. The revised main program only outputs the number of crossings. ¶ Also
eliminated is a phase that decomposes the graph into layers — in our experiments,
the decomposition into two layers is given.
Dot begins with an initial ordering based on depth-first, instead of breadth-first,
search. This is followed by iterative improvement based on repeated application
of the fixed-layer median heuristic. The median as implemented in dot uses the
mean when there are exactly two elements, and a biased mean — biased toward
the side on which neighbors are closer together in the current ordering — for even
cardinalities > 2. Each time dot applies the median heuristic to layer `, a local
search performs swaps of neighbors on layer ` until no swap is able to decrease cost.
Dot alternates application of the median/swapping passes until a fixed number of
passes (10 in the implementation we used) has resulted in no cost reduction.
2.7 Experimental Treatments
Our software uses the treatment number nomenclature of an earlier conference paper [Stallmann et al. 1999b]. Table 1 gives a brief overview of the combinations
discussed here. Treatment numbers are non-consecutive — some treatments were
omitted because they failed to contribute anything of interest to the results reported here. The output presentation of treatment tr00 is identical to the input
presentation — it therefore plays the role of a control in our experiments.
We recently added three more treatments to have a basis for comparison with
¶ An
additional version that outputs the permutation on each layer was developed so that we could
verify the results, but it runs significantly slower.
·
10
M. Stallmann, F. Brglez, and D. Ghosh
Treatment Initial Ordering
Iterative Improvement
Time ∗
tr00
random †
none
O(n)
tr01
random †
median
O(n)
tr03
random †
barycenter
O(n lg n)
tr06
bfs
none
O(n)
tr07
bfs
median
O(n)
tr09
bfs
barycenter
O(n lg n)
tr12/dot ‡
dfs
median/swap §
O(n2 )
tr14
gbfs
none
O(n)
tr15
gbfs
median
O(n)
tr17
gbfs
barycenter
O(n lg n)
tr23
gbfs
adaptive-insertion/barycenter ¶
O(n2 )
tr24
random
median k
O(n2 )
tr25
random
barycenter k
O(n2 )
tr30
random
median k∗∗
O(n2 )
∗
Asymptotic time of an optimal implementation based on number of nodes n — we
assume that graphs have O(n) edges; time per iteration is given if iterative improvement
dominates.
† No modification takes place — input presentation is assumed to be random.
‡ We refer to this as dot rather than tr12 in what follows.
§ A pass consists of one application of the median heuristic followed by repeated swapping of neighbors on the same layer until there is no further improvement; passes are
done on alternating layers until there is no improvement.
¶ Barycenter alternated with 2048 passes of adaptive insertion; the number of passes
could have been greatly decreased for all but the largest graphs without any change in
the results.
k Stops when there is no reduction in crossings.
∗∗ Averages two middle elements when taking the median of an even-cardinality set.
Table 1. Various heuristics (treatments) used in our experiments. Numbering is based partially
on an earlier conference paper and partially on continued experimentation.
the recent experimental work of others [Jünger and Mutzel 1997; Matuszewski
et al. 1999]. When the median and barycenter heuristics are used iteratively on
alternating layers, experiments by previous authors iterate until there is no decrease
in the number of crossings. Our iterative versions of median (tr01) and barycenter
(tr03) keep iterating until there is no change in the order of nodes on either side
and then use the final presentation as output even if solutions with fewer crossings
were encountered earlier. This has the advantage that there is no need to update
the number of crossings after each iteration.
The treatments tr24 and tr25 use the same median and barycenter heuristics
for each iteration as tr01 and tr03, respectively. However, tr24 and tr25 stop
iterating when the solutions produced during the last two passes (one on each layer)
have no fewer crossings than the best previous solution. And, while the sorting
algorithm used in tr01 and tr03 is stable (to ensure convergence), tr24 and tr25
Bigraph Crossing Minimization
Heuristic
tr25 ∗
crossings
time
iterations
tr03 †
crossings
time
iterations
tr17 ‡
crossings
time §
iterations
dot
crossings
time
iterations k
tr23 ∗∗
crossings
time
iterations
·
11
Results for random trees
Graph Size (m, n0 , n1 )
(230,115,116) (470,235,236) (950,475,476) (1910,955,956)
[1040,1140]
[0.08,0.25]
[8,38]
[3420,3810]
[0.25,1.13]
[13,72]
[9880,10700]
[1.45,3.55]
[35,104]
[28800,31200]
[6.12,15.3]
[55,169]
[965,1040]
[0.02,0.05]
[14,66]
[2970,3150]
[0.13,0.22]
[50,142]
[8350,8890]
[0.53,0.7]
[92,240]
[22900,24000]
[2.38,3.35]
[196,536]
[420,474]
[0.02,0.05]
[5,29]
[1300,1430]
[0.08,0.15]
[9,34]
[3800,4220]
[0.37,0.48]
[16,85]
[10900,11800]
[1.55,1.95]
[32,165]
[645,743]
—¶
—
[1940,2250]
[1,2]
—
[6110,6980]
[7,9]
—
[17800,20700]
[36,69]
—
[231,296]
[49,50]
[3,3]
[708,978]
[205,207]
[3,3]
[2170,2930]
[834,3236]
[3,10]
[5730,7390]
[3434,15909]
[3,14]
∗
barycenter, iterated until there is no improvement
barycenter, iterated until there is no change on either side
‡ barycenter, same as tr03, with a good initial placement
§ does not count time for initial placement
¶ too small to be measured
k not able to count iterations for the dot heuristic
∗∗ good initial placement, adaptive insertion, and barycenter; results based
on 8 subjects, not 64
†
Table 2. Number of crossings versus execution time for various iterative improvement heuristics.
Each heuristic was executed on 64 random graphs with the specified number of nodes and edges.
A 95% confidence interval about the mean is shown for number of crossings. The intervals for
time and iterations represent ranges from minimum to maximum. Execution time is in seconds on
an UltraSparc IIi, 360 MHz, with 256 MB memory. Each iteration of a heuristic other than tr23
is one application of the heuristic to each layer. An iteration of tr23 is 2048 passes of adaptive
insertion followed by iterations of barycenter until there is no change (tr03).
use an unstable sort since it gives slightly better results. Finally, tr30 is introduced
because, among several attempted implementations of an iterated median heuristic,
it comes closest to approximating the results reported for the median heuristic in
[Jünger and Mutzel 1997] (see Appendix A for more details). The key feature that
distinguishes tr30 from tr24 is that it takes the mean of two middle elements when
a node has an even-cardinality adjacency list instead of using the subtler approach
of [Eades and Wormald 1994].
Table 2 shows number of crossings and execution times reported for the leading
iterative improvement heuristics on random trees of increasing sizes. Execution
times for initial placement, even with insertion sort being used during placement,
were negligible in comparison with those for iterative improvement. A comparison
12
·
M. Stallmann, F. Brglez, and D. Ghosh
of tr25 with tr03 shows that our absolute convergence strategy pays off — tr03
reports significantly fewer crossings and runs in less time. It is able to do more
iterations in less time because there is no need to update crossings. The results
shown are typical and hold for other types of graphs as well.
The combined heuristic tr17 obtains solutions with many fewer crossings than
tr03 using fewer iterations, the key being an excellent initial placement. This phenomenon is not universal — there are even graphs, for example, hypercubes, on
which tr03 outperforms tr17. The dot heuristic achieves less with longer execution
times than tr17 on most classes of graphs — it appears to spend a lot of extra effort
swapping neighbors for only a small reduction in the number of crossings. On the
other hand, tr23 sometimes gets dramatic reductions in the number of crossings
and is used to indicate how far from optimal the other solutions are. Because of
the prohibitive execution times of tr23, we usually ran it on only 8 random subjects
from a class of 64. Results are not affected significantly because the variance of
tr23 is relatively small.
3. EXPERIMENTAL SUBJECTS
A challenging aspect of this work is devising a collection of experimental subjects
that leads to well-reasoned conclusions about the quality of the various heuristics
and that can also serve as benchmarks for future heuristics. Ultimately we want to
explore ways of generating graph equivalence classes that have the same characteristics as specific reference graphs. This, however, is an ongoing effort [Ghosh 2000],
beyond the scope of this paper. For now we believe our proposed classes energetically address the issue of differentiating among present and future heuristics.
A distinguishing feature of our work is the use of both random and isomorphism classes of graphs instead of isolated benchmarks or generic random graphs
to examine the behavior of heuristics. A random class contains graphs (64 in our
experiments) generated randomly given n0 , n1 , m, and possibly other information.
An isomorphism class consists of random presentations of the same graph G — in
some cases G is generated randomly, in others it is designed deterministically. Every
isomorphism class for a graph G has a corresponding random class so that we can
answer the question: how typical is G of graphs having the same number of nodes
and edges? Conversely, every random class has an isomorphism class constructed
from one of its members — thus, we can see if the observed behavior of heuristics
on the random class is an average that smoothes out widely varying behavior, or
whether behavior of any individual instance typifies behavior for the class.
The current work strikes a balance between parameterized random graph classes
and isomorphism classes based on specific graphs occurring ‘naturally’ in network
design or contrived to have specific characteristics. We describe the design of these
equivalence classes of experimental subjects here.
The size of a graph is always measured by its number of edges m. However, a
bigraph type, consisting of similar bigraph classes of various sizes can be characterized by two measures that allow n0 and n1 to be derived from m. The density
factor a = m/(n0 + n1 − 1) is the ratio between the size of the graph and that of
one of its spanning trees †† . We can also specify the balance b = n0 /n1 . Now let
†† [West
1996, page 362] defines edge density as
m
,
n0 +n1
essentially the same as our density factor.
·
Bigraph Crossing Minimization
name
n0
13
n1
m
a
bm
c + 1 − n0
a
15 · 2k − 10
—
b m+1
c
2
m + 1 − n0
15 · 2k − 10
1
Gr,k (even k)
2k−1
2k−1
2 · (2k − 2 2 )
Gr,k (odd k)
2k−1
2k−1
random graphs, 1 ≤ k ≤ 8
R(a, 15 · 2k − 10), a =
R(1, 15 ·
2k
d = 1,
33 17 9 5
, , , , 2,4, 8
32 16 8 4
b
bm
c+1
a
c
2
− 10, d),
1001 101 11
,
, ,2, ∞
1000 100 10
grid graphs, 1 ≤ k ≤ 10
k
2k+1 − 3 · 2
k−1
2
≈2
≈2
butterfly graphs, 1 ≤ k ≤ 7
(k/2 + 1)2k
Gb,k (even k)
(k + 1)/2 ·
Gb,k (odd k)
2k
(k/2) · 2k
(k + 1)/2 ·
2k
2k · 2k
≈2
2k
≈2
2k ·
hypercubes, 1 ≤ k ≤ 9
Gh,k
2k−1
2k−1
k · 2k−1
k/2
3 · 2k − 2
12 · 2k − 7
15 · 2k − 10
1
VLSI graphs, 2 ≤ k ≤ 7
G00,k , G01,k
G10,k , G11,k
G20,k , G21,k
Table 3.
3·
2k
3·
2k
−2
−2
9·
2k−1
3·
2k−2
−1
+4
15 ·
2k
− 10
≈2
15 ·
2k
− 10
≈4
Parameters that characterize the proposed graph classes.
n = bm/ac + 1, n0 = bn/(b + 1)c, and n1 = n − n0 . The choice of b has little impact
on the number of crossings reported by any heuristic except in extreme cases (small
graphs and/or extreme values of b).
Table 3 summarizes our experimental subject classes. Within each bigraph type,
the size is controlled by the parameter k, called the rank. With the exception of
butterfly graphs and hypercubes, an increment in rank corresponds roughly to a
doubling in size. We discuss the table in more detail in Section 3.4 after describing
the bigraph types.
3.1 Random Graph Classes
Not all random graphs are created equal — with respect to suitability for crossing
minimization experiments, that is. This is illustrated plainly in Table 4. Our random graph generator creates connected random graphs by first running a minimumspanning-tree algorithm with random distances between nodes to create a random
spanning tree. Such a tree is later called a dimension-∞ tree. Additional edges, if
any are needed, are added by randomly selecting them from the edges not used in
the spanning tree. The not connected graphs are generated using our own implementation of the random bipartite graph generator from GraphBase [Knuth 1993].
The contrast between the rows marked [1] and [2] speaks for itself. The crossing
numbers of random placements reported by tr00 are close to the expected ones of
Warfield’s formula [Warfield 1977] and differ little among graphs of the same size.
However, the minimum crossing number — and that produced by good heuristics
14
·
M. Stallmann, F. Brglez, and D. Ghosh
Size
(m, n0 , n1 )
(230,115,116)
a=1
[1]
connected
[2]
not connected
(230,97,97)
a = 1.1875
[3]
connected
(230,58,58)
a=2
[4]
connected
[5]
not connected
(470,235,236)
a=1
[1]
connected
[2]
not connected
(470,198,198) a = 1.1875
[3]
connected
(470,118,118)
a=2
[4]
connected
[5]
not connected
(950,475,476)
a=1
[1]
connected
[2]
not connected
(950,400,401) a = 1.1875
[3]
connected
(950,238,238)
a=2
[4]
connected
[5]
not connected
(1910,955,956)
a=1
[1]
connected
[2]
not connected
(1910,804,805) a = 1.1875
[3]
connected
(1910,478,478)
a=2
[4]
connected
[5]
not connected
tr00
tr07
tr17
tr23
Heuristic
tr00
tr07
tr17
tr23
[12800,13100]
[12800,13100]
[749,880]
[2540,2860]
[420,474]
[1710,1820]
[232,296]
[1320,1660]
[12800,13000]
[2560,2670]
[1870,1950]
[1500,1690]
[12600,12900]
[12700,12900]
[5520,5640]
[5560,5680]
[4670,4770]
[4760,4840]
[4360,4750]
[4440,4680]
[54300,55200]
[54200,55000]
[2490,2810]
[11000,12300]
[1300,1430]
[6790,7070]
[708,978]
[5590,6590]
[54500,55100]
[10600,11000]
[7330,7540]
[5960,6500]
[53300,54000]
[53800,54500]
[23300,23700]
[23500,24000]
[19300,19500]
[19600,19800]
[18500,19100]
[18400,19000]
[223000,225000]
[223000,226000]
[7760,8840]
[44200,48900]
[3800,4220]
[27300,28100]
[2170,2930]
[22500,24700]
[224000,226000]
[43900,44800]
[28700,29200]
[22900,24300]
[223000,225000]
[223000,225000]
[95500,96700]
[97600,98800]
[78200,78700]
[79600,80300]
[74500,76100]
[75800,78300]
[908000,915000] [24800,28400]
[10900,11800]
[911000,918000] [173000,191000] [107000,110000]
[5730,7390]
[90600,99400]
[906000,911000] [179000,182000] [113000,114000]
[96200,99400]
[906000,912000] [387000,391000] [313000,316000] [300000,305000]
[904000,910000] [394000,398000] [322000,323000] [308000,314000]
Legend
random order
breadth-first search + median; best linear-time heuristic
gbfs + barycenter; best sub-quadratic time heuristic
best solution quality; long execution time; results are based on a
sample of 8 instead of 64
Table 4. Number of crossings for connected versus not necessarily connected random graphs.
The intervals are 95% confidence intervals with respect to the mean number of crossings for 64
random graphs of each category.
Bigraph Crossing Minimization
·
15
— differs widely between trees, row [1], and random graphs with the same number of edges as trees, row [2]. The latter have many isolated nodes, some small
components, and usually one larger component that is denser than a tree (see, e.g.
[Palmer 1985], for theoretical background). We have included row [3] to show that
the behavior of random not-necessarily-connected graphs with density factor a = 1
3
, graphs with the same
closely resembles that of connected graphs with a = 1 16
number of edges but fewer nodes.
Rows [4] and [5] repeat the comparison of rows [1] and [2], this time with
a = 2, a larger density factor. The difference here is much less pronounced, as can
be expected — denser random graphs are more likely to be connected and their
largest connected component is likely to be larger if they are not. The differences
in performance among the better heuristics are also less striking.
The decision to use only connected random bigraphs in our experiments was
clear from the beginning. We felt that (a) our random graphs should behave like
our deterministically generated graphs with the same number of nodes and edges,
(b) heuristics should not be inadvertently judged on the basis of how well they solve
the problem of separating connected components, (c) our measures of graph density
should not be skewed by the fact that the largest component in a multicomponent
graph is denser than the graph itself, and (d) finding a close-to-optimal embedding
of a connected graph is likely to be harder than finding one in a graph with multiple,
mostly small, components.
Our test data includes random graph classes with a = 1, 1.03125, 1.0625, 1.125,
1.25, 2, 4, 8. Sizes of random graphs range from 20 edges (rank 1) to 3830 edges
(rank 8), each rank having twice as many as the previous rank, plus 10. The choice
of sizes has historical reasons — early experiments were done on specific circuitbased graphs and we wanted random graphs with the same sizes for comparison.
3.2 Random Trees
When creating a random tree we already know n0 and n1 and hence must assign each
node to one of V0 or V1 before creating a random spanning tree. For the dimension∞ trees used to ensure connectivity, we choose cost(v0 , v1 ), where v0 ∈ V0 and
v1 ∈ V1 , at random. The bipartition is preserved by letting cost(x, y) = ∞ when x
and y are on the same layer.
The cost of a potential tree edge can also be based on a distance between points
in a metric space of a given dimension. For example, random dimension-2 trees are
created by mapping each node to a random point in the unit square and letting
cost(v0 , v1 ) be the ∞-norm distance between v0 and v1 (equivalently, we could
use a unit circle and the Euclidean norm). These behave quite differently from
dimension-∞ trees with respect to the heuristics.
Dimension-(1 + α) trees, where 0 ≤ α < 1 use random points in a 1 × α rectangle
instead of a unit square. When α = 0 the coordinates are points on a line segment.
The set of possible dimension-1 trees turns out to be exactly the set of biplanar
graphs — see Appendix B for proof of this. As α approaches 0 and tree size
stays fixed, the minimum number of crossings found by the better heuristics in
dimension-(1 + α) trees also approaches 0.
Our tree classes — classes with a = 1 — include random trees with dimension
d = 1, 1.001, 1.01, 1.1, 2, and ∞.
·
16
0
1
2
a5
3
M. Stallmann, F. Brglez, and D. Ghosh
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
a4
a8
a1
a3
a0
a9
a6
a2
a7
(a) The biplanar tree G00,2 (easy, meaning few crossings)
21 22 23 24 7 17 18 19 20 5
a5
a4
6
8
1 33 34 35 36 13 14 16 3 15 0
a1
a8
a3
a0
4
2 37 38 39 40 25 26 27 28 9 10 12 11 29 30 31 32
a9
a6
a2
a7
(b) The tree G01,2 (hard, meaning many crossings)
Fig. 3.
Two VLSI trees (density factor a = 1).
3.3 Isomorphism Classes
For our isomorphism classes we use bigraphs that are well-known in the literature
and some that were specifically designed to reflect the behaviors we encountered in
circuit-design applications. For the latter, we develop our own graph types, mainly
to explore issues that were not addressed by the other graph types.
3.3.1 Communication Graphs. Our collection of isomorphism classes includes some
well-known bipartite communication graphs — grids, butterflies, and hypercubes
(see, e.g., [Leighton 1992]). The grid Gr,k has 2k nodes, forming a square when
k is even and a rectangle with 2:1 aspect ratio when k is odd. In either case, a
node at position (i, j) is connected to nodes at (i − 1, j), (i, j − 1), (i + 1, j), and
(i, j + 1). One or two of these edges may be missing for nodes on a boundary or in
a corner, respectively. Nodes with even i + j are in V0 , those with odd i + j in V1 .
The butterfly graph Gb,k has nodes defined by ordered pairs (i, w), where i ranges
from 0 to k and w is a string of k bits. Edges are of the form (i, w)(i + 1, w0 ), where
w0 is either the same as w or differs from w only at the i-th bit (i = 0, . . . , k − 1).
Hypercubes, the only graphs in our study whose density factor a increases with
size, have 2k nodes. Each node is denoted by a string of k bits. The set of edges is
{xy | x, y differ in exactly one bit }. A 2k -node hypercube is denoted by Gh,k .
3.3.2 Graphs Motivated by Circuit-Design Applications. We consider isomorphism
classes of graphs specifically designed to test the suitability of various treatments
for the circuit layout applications that motivate this work. In circuit applications,
V0 typically represents a set of cell nodes and these often have fixed degree (5 in
our examples). The nodes in V1 are net nodes; an alternate abstraction for a net
node is a hyper-edge consisting of all the cell nodes to which it is adjacent.
The circuit, or VLSI, graphs are designated Gxy,k where x = 0, 1, or 2 (meaning
a = 1, 2, or 4), y = 0 or 1 (meaning a small or large number of crossings), and k
ranges from 1 to 7 (the rank — number of edges is 15 · 2k − 10 as in the random
graphs). What follows is a brief description of the ideas behind the VLSI graphs.
A more detailed account is given in Appendix C.
The G00,k graphs are biplanar — Fig. 3(a) shows G00,2 . A “hard” tree is created
Bigraph Crossing Minimization
b0
b1
b2
b3
b4
b5
b6
b7
a1
10 crossings
a2
4 crossings
a3
10 crossings
(a) The graph G10,1 (easy, meaning relatively
few crossings)
b0
b1
b6
a0
b2
a1
b3
b7
a2
b4
b5
a3
(b) The graph G11,1 (hard, meaning many
crossings)
Fig. 4.
17
In G10,1 , the ‘easy’ VLSI graph with a = 2,
the 5 neighbors of µ
a2i ¶
and a2i+1 are the
same, resulting in
a0
·
5
2
= 10 crossings
among their edges.
Furthermore, the nodes a2i+1 and a2i+2
have 2 neighbors in common. Since these
neighbors are also common to a2i and a2i+3 ,
an additional 4 crossings result from this
overlap.
In G11,1 , the ‘hard’ VLSI graph with a = 2,
the nodes a2i and a2i+1 share only 4 neighbors. The 5th edge is used to connect to a
bj that, in turn, connects to a ‘far away’ ak .
As in G10,1 , each pair shares 2 neighbors
with the next pair.
If the ‘far away’ connections — via the
circled
µ
¶ bi nodes — are ignored, there are
4
= 6 crossings for each pair and
2
4 crossings due to overlap of neighboring
pairs.
Two VLSI graphs with density factor 2.
by taking a complete binary tree and adding extra leaves to give the cell nodes
a degree of 5. Fig. 3(b) shows G01,2 in an embedding that has 28 crossings, the
minimum any of our treatments were able to find.
The number of edges and therefore the number of cell nodes is the same for all
VLSI graphs of any given rank. To increase density while keeping rank the same,
the number of net nodes is decreased. For example, to achieve a density of 2, a
VLSI graph has roughly 3n0 /2 net nodes. A density of 4 is achieved with roughly
n0 /4 net nodes.
Easy, or low-crossing, graphs are obtained by simply making each cell node adjacent to the 5 “closest” net nodes based on an a priori ordering of nodes of each
kind. Fig. 4(a) shows G10,1 .
To increase the number of crossings and confound the heuristics, the hard, or
high-crossing, graphs set aside a fraction of the nodes for far-away adjacencies —
each net node is adjacent to at least two cell nodes that are far apart in the original
ordering. See Fig. 4(b) for a good embedding of G11,1 — the circled net nodes are
the ones used for far-away adjacencies.
3.4 Summary
We are now ready to take a more detailed look at Table 3. Random classes are
denoted R(a, m, d), where d defaults to ∞ if it is omitted (random spanning trees
for graphs with a > 1 are always generated using d = ∞, for example). Each
isomorphism class is denoted by the name of the subject (also called reference
graph) used to generate it; for example, G20,5 denotes a class consisting of 64
random presentations of G20,5 .
18
·
M. Stallmann, F. Brglez, and D. Ghosh
We also use two additional pieces of notation: rnd (X), where X is an isomorphism class, is a random class R(a, m), where a and m are the density factor and
number of edges of X; and iso(R), where R is a random class, is an isomorphism
class generated from one randomly-chosen member of R.
Recall that a bigraph type is a collection of classes that differ only in size (number
of edges). Table 3 shows 23 basic bigraph types — 7 different a > 1 and 6 different d
with a = 1 for random graphs; even (square) and odd (rectangular) grids (counted
as two distinct types), butterflies, hypercubes ∗ , and 6 types of VLSI graphs. For
each type T of 7 isomorphism types, there is a type rnd (T ) derived by creating a
random graph class with the same m, n0 and n1 as each class in T (there are really
10 isomorphism types, but G00,k has the same parameters as G01,k , etc.). Now for
each type T of the 20 random types discussed so far, there is a type iso(T ) derived
by selecting a random member of each class in T and creating an isomorphism class
from it.
The total number of classes is 358 (meaning 22, 912 experimental subjects were
treated, 64 per class) — 8 sizes for each of 26 classes, the random and their derived
isomorphism classes, 10 sizes of grid graphs (times 3 because of the derived random
and its derived isomorphism class), 7 of butterflies (times 3), 9 hypercubes (times
3), and 6 size classes for each of 12 VLSI types (6 original, 3 derived random, and
3 derived isomorphism from the random).
For all classes listed in Table 3, the average number of crossings reported by tr00,
the random placement, is within 0.2% of the expected crossing number reported in
[Warfield 1977]: m(m − 1)(n0 − 1)(n1 − 1)/4(n0 n1 − 1).
To see the relationship between isomorphism classes and random classes, consider the histograms in Fig. 5. The top row shows number of crossings obtained by
three treatments (tr01, tr09, and dot) on a 64-member isomorphism class of G01,7 .
In the second row are results for the class rnd (G01,7 ) — 64 randomly generated
trees with dimension d = ∞ and the same n0 , n1 , and m as G01,7 . The third row
shows iso(rnd (G01,7 )), the isomorphism class of a random member of rnd (G01,7 ).
A weaker treatment, such as tr01, only takes slight advantage of the special structure of G01,7 , as seen in the first column of Fig. 5. The other two columns show
treatments that perform equally well on an average run with G01,7 . Dot (rightmost column) has a large variance, even on the isomorphism classes. For G01,7
this is an advantage — dot is able to find solutions with fewer crossings than tr09.
For random graphs, however, it appears to be a disadvantage — tr09 consistently
outperforms dot on these.
4. TREATMENT EVALUATION
Treatments can be evaluated in many ways. Our emphases in this section are (a) to
illustrate performance of treatments relative to each other, (b) to establish which
experimental subjects are most useful for distinguishing treatments from each other,
and (c) to suggest the most fruitful avenues for development of future heuristics.
We begin in Section 4.1 by looking at the effect of the density factor on relative
performance. Section 4.2 considers the variations that result from different types
∗ Technically,
hypercubes are not a bigraph type because different size hypercubes have different
density factors.
Bigraph Crossing Minimization
50
·
19
Number of Crossings (TR09)
40
20
Number of Crossings (DOT)
Number of Crossings (TR01)
20
30
15
15
20
10
10
10
5
5
0
20
0
0
[140000, 200000],
µ = 167380, σ = 7402.3
Number of Crossings (TR01)
[5000, 30000],
µ = 9900.97, σ = 176.749
20
Number of Crossings (TR09)
[5000, 30000],
µ = 10165.2, σ = 2036.02
20
15
15
15
10
10
10
5
5
5
0
0
0
[140000, 200000],
µ = 178076, σ = 6978.82
20
Number of Crossings (TR01)
[5000, 30000],
µ = 15975.2, σ = 3014.38
20
Number of Crossings (TR09)
[5000, 30000],
µ = 19000.9, σ = 4417.65
20
15
15
15
10
10
10
5
5
5
0
0
0
[140000, 200000],
µ = 176808, σ = 6199.56
[5000, 30000],
µ = 14073.2, σ = 1547.85
Number of Crossings (DOT)
Number of Crossings (DOT)
[5000, 30000],
µ = 19492, σ = 4104.72
Each of three treatments is applied to 64 subjects of each of 3 classes. The
columns represent tr01 (median), tr09 (bfs+barycenter), and dot, respectively. The rows represent G01,7 (64 presentations of the “hard” VLSI tree),
rnd(G01,7 ) (64 random graphs having same size and density as G01,7 ), and
iso(rnd(G01,7 )) (64 presentations of one of the random graphs).
Fig. 5.
A comparison of isomorphism and random classes.
·
M. Stallmann, F. Brglez, and D. Ghosh
Average number of crossings reported
1000000
1000000
00
01
100000
06
14
03
07
•
15
dot
09
10000
Average number of crossings reported
20
17
+ 23
5000
01
03
07
• dot
+
09
14
15
17,23
100000
A random graph (a=2, m=1910)
(c)
Average number of crossings reported
Average number of crossings reported
00
06
01
06
dot
•
07
100000
03
09
14
15
+17
23
10000
5000
A random graph (a=1.125, m=1910)
(b)
A random graph (a=1, m=1910)
(a)
1000000
00
1000000
00
01
03
09
07
dot •
17,23
+
06
14
15
100000
A random graph (a=4, m=1910)
(d)
Fig. 6. A comparison among the various treatments for random graphs with increasing density.
The averages are based on 64 random presentations of each graph. Similar results are obtained if
64 random graphs from each class are chosen.
of experimental subjects. In Section 4.3 we turn to the asymptotic relative performance of treatments on both random and isomorphism classes. We conclude with
other interesting observations in Section 4.4.
We use decomposition diagrams like those in Fig. 6 to show not only the number
of crossings reported by all treatments but also the ingredients of each treatment —
see Table 1 on page 10. A solid line from t1 to t2 with t2 to the left of t1 indicates
that t2 is t1 followed by iterative improvement, either in the form of the median
(one column to the left) or barycenter (two columns). A dashed line from t1 to t2
with t2 to the right of t1 indicates that t2 is t1 preceded by initial placement, either
bfs (one column) or gbfs (two columns). The choice of left/right and solid/dashed
is arbitrary. Treatments dot and tr23 are shown separately in the center column.
Bigraph Crossing Minimization
·
21
For example, tr07 in the diagram can be reached from tr00 by following either
the dashed line to tr06 and the solid line to tr07 or the solid line to tr01 followed by
the dashed line to tr07. These two paths illustrate two different ways to isolate the
ingredients of tr07. In either case, we can see how much is contributed by initial
placement tr06 (bfs) — the dashed lines to the right — and how much by iterative
improvement tr01 (median) — the solid lines to the left. The relative importance
of the two ingredients of a hybrid heuristic such as tr07 on a particular graph is
immediately evident from the decomposition diagram.
The y-axis in the decomposition diagrams is logarithmic scale so that vertical
distance between treatments is proportional to the ratio between their reported
crossing numbers.
4.1 The Effect of Density
Jünger and Mutzel [Jünger and Mutzel 1997] report that among random graphs
only sparse ones exhibit significant differences in the performance of treatments.
Fig. 6 corroborates this point — an increase in density factor leads to significant
decrease in ratio between treatments. When a = 4 even the best of our treatments
yields only about a 44% improvement over random placement — 4.9 × 104 versus
9 × 104 crossings. When a = 8 this decreases to 29% — 6.2 versus 9.0 (not shown).
The best improvement we get over tr07, a simple linear-time heuristic, when a = 4
is 9% — 4.9 versus 5.5. When a = 8, it is only 5.5% — 6.2 versus 6.7. Even
if such small differences are sought after, the ranking of the treatments does not
appear to change beyond a = 2. Furthermore, the 5 best treatments have mutually
overlapping 95% confidence intervals with respect to the mean number of crossings
for a > 2.
Ranking of treatments does, however, change in subtle ways as we progress from
trees to denser graphs. Initial placement is at least as important as iterative improvement for trees but diminishes drastically in importance even for very sparse
non-trees. Note the positions of treatments tr06, tr07, tr14, and tr15 relative to
tr03 in Fig. 6(a) and (b). Initial placement, yielding a difference as dramatic as
iterative improvement on the sparser graphs, is much less important for the denser
ones. This can also be seen by the position of tr03 relative to tr09 and tr17 —
a t-test here is unable to establish any improvement in performance when better
initial placement is added to the barycenter heuristic.
The ranking of dot is best for trees but worst for the sparsest non-trees. This
phenomenon appears to be solely the result of dot’s initial placement strategy,
depth-first rather than breadth-first search. Dot’s iterative improvement performs
as expected: better than the median heuristic but not quite as good as barycenter.
The relative position of tr23 versus tr17 suggests less about any particular feature
of tr23 than it does about the potential for improved heuristics. With increasing
density it becomes less likely that any enhancement will yield dramatically lower
crossing numbers.
The discontinuity in our results, both in terms of actual number of crossings
reported and other trends, between trees and extremely sparse other graphs suggests
that arboricity (the maximum density factor of any node-induced subgraph) may
have an influence on crossing number, something already suggested by the upper
bound of [Shahrokhi et al. 2001]. The graphs in Fig. 6 all have the same number of
22
·
M. Stallmann, F. Brglez, and D. Ghosh
edges. A small decrease in the percentage of spanning-tree edges represents only a
small increase in overall density factor, but the density factor of some node-induced
subgraphs can increase dramatically.
4.2 Which Subjects are Interesting
We now look at features other than density factor and how they affect our results.
Fig. 7 shows decomposition diagrams for 4 graphs, all of which have density factors of approximately 2 and roughly the same number of edges. The ranking of
treatments for the butterfly (Fig. 7(a)) is the same as for the random graph with
a = 1.125 (not a = 2), but differences in crossings are not as great (yet still greater
than for the random graph with a = 2).
Square grids (Fig. 7(b)) are very sensitive to initial placement. Applying the
median heuristic after a breadth-first search increases rather than decreases the
number of crossings! Ironically, dot’s improved position in the rankings is not the
result of initial placement — in fact, the number of crossings after initial placement
is higher even than that of tr01. Depth-first search is a particularly bad way to
sequence the nodes of a square grid. Here it is the weighted median combined with
swapping of neighbors that yields a dramatic reduction.
Even more dramatic cases where initial placement trumps iterative improvement
occur in results for the two VLSI graphs — Fig. 7(c) and (d). The hard VLSI
graph G11,7 favors bfs slightly over gbfs. The median applied after either search
leads to an increase in the number of crossings. The easy graph G10,7 renders our
iterative improvement heuristics almost completely superfluous — neither median
nor barycenter can “deduce” the simple linear structure of G10,k on their own, nor
can they do much to improve an initial placement that does.
Even more extreme differences can be seen among trees. The hard VLSI tree
G01,7 seems particularly amenable to good iterative improvement via barycenter
— see Fig. 8(a). These decomposition diagrams cover a much larger range on the
y-axis, so the differences among heuristics are actually much larger than they might
appear relative to those of Figs. 6 and 7. Here, the good relative performance of dot
is, again, based on initial placement. Depth-first search is better than breadth-first
search for trees. In fact, for several of the 64 presentations in the isomorphism
class, dot does as well as tr23 and its initial placement outperforms all treatments
but tr23.
For the easy VLSI tree — G00,7 , Fig. 8(b) — treatments that use gbfs all report
0 crossings and are not shown. The differences are as large as any we’ve seen, and
initial placement determines the quality of solutions — except in the case of dot
(a depth-first search on a biplanar tree yields a good solution only if the starting
point is at the end of one of the longest paths). Dot has a large variance, reporting
solutions with as few as 20 crossings or as many as 1900.
Figs. 8(c) and (d) show two random trees with different dimension. The one with
d = ∞ is the same as the one in Fig. 6(a) (but the scale is different). The d = 1.1
tree represents a large step in the direction of biplanar trees. The main changes are
the increasing advantage of gbfs over bfs and the decreasing effectiveness of (the
initial placement phase of) dot.
·
Bigraph Crossing Minimization
1000000
1000000
23
00
01
06
dot
03
09
•
+17,23
07
100000
14
15
Average number of crossings reported
Average number of crossings reported
00
01
100000
15
+
09
06
14
17,23
10000
10000
A square grid (a=2, m=1984)
(b)
A butterfly graph (a=2, m=1784)
(a)
1000000
1000000
00
01
100000
03
07
• dot
+
17,23
09
15
14
06
10000
Average number of crossings reported
Average number of crossings reported
07
dot •
03
00
01
100000
03
10000
• dot
09
07
+
17,23
A "hard" VLSI graph (a=2, m=1910)
(c)
06
14
15
An "easy" VLSI graph (a=2, m=1910)
(d)
Fig. 7. A comparison among the various treatments for specific graphs with a density factor of
roughly 2.
·
M. Stallmann, F. Brglez, and D. Ghosh
Average number of crossings reported
1000000
1000000
00
01
100000
07
06
15
14
03
dot
•
10000
09
17
+
23
Average number of crossings reported
24
00
01
100000
03
10000
1000
1000
500
500
09
A "hard" VLSI tree (a=1, m=1910)
(b)
1000000
00
01
100000
06
14
07
• dot
15
09
+
23
17
Average number of crossings reported
Average number of crossings reported
1000000
10000
07
An "easy" VLSI tree (a=1, m=1910)
(a)
03
06
dot
•
00
01
100000
• dot
03
10000
06
07
09
17
+
14
15
23
1000
A random tree (a=1, d=infinity, m=1910)
(c)
Fig. 8.
1000
A random tree (a=1, d=1.1, m=1910)
(d)
A comparison among the various treatments for various trees.
Bigraph Crossing Minimization
·
25
4.3 Asymptotic Relative Performance
The relative performance of various treatments on specific graphs presented up to
now leads us to wonder how these differences behave as graphs grow larger. For
the circuit layout applications, where millions of nodes are typical, it is important
to know whether a given treatment is likely to produce competitive results with
increasing size. Naturally we can only guess these results by extrapolation, but
many of the trends are clear-cut and compelling. We turn our attention now to
charts that show ratios between crossings numbers of two given treatments as a
function of number of edges. Each curve on a chart represents a specific graph
type and each point represents a single class. The y value is obtained by taking
the ratio of two averages, the average crossing number reported by each treatment
for the class. A key difference from earlier charts for random classes is that instead
of using the isomorphism class of a single random graph (to be consistent with the
isomorphism classes in the same figure), we use the 64 distinct graphs of the original
random class (to make the asymptotic curves smoother). As suggested earlier —
Section 3.4 — this should not matter much.
The differences in performance illustrated in the previous section are supported
by asymptotic trends — large differences in the performance of two treatments on
a specific graph from a single class imply a steady increase in the ratio between
these two treatments for the corresponding graph type, small differences remain
the same or decrease with increasing size.
A typical example is the ratio between tr03 and tr06 (pure barycenter versus
pure bfs). Fig. 9(a) supplements the decomposition diagrams of Fig. 6. In all cases,
tr03 reports fewer crossings and the ratios become smaller with increasing size. As
density factor increases, the rate of decrease in the ratio slows † . A slowing down
of rate of decrease or increase with increasing density factor can be observed in
general for various ratios on random graphs and, of course, for the ratio tr00/tr17.
This suggests that with larger sizes of random graphs, the differences in height
among the decomposition diagrams in Fig. 6 would be more extreme.
Ratios for communication graphs, Fig. 9(b), all favor tr03. The upward trend for
grids, especially square ones, suggests, however, that larger subjects might favor
tr06. Fig. 9(c) shows the ratio tr03/tr06 for various random trees. All those with
d < ∞ (i.e. based on geometry) favor tr06 with the rate of increase getting bigger
as d approaches 1. The non-geometric trees (d = ∞) favor tr03, as already seen in
Fig. 9(a).
Fig. 9(d) shows ratios for VLSI graphs. The easy ones (G00,k , G10,k , and G20,k )
all favor tr06 with dramatically increasing ratios. The hard trees exhibit almost the
same behavior as random trees (this is true for most of the other asymptotic ratios
between treatments), but the hard graphs with density factor 2 (G11,k ) appear to
have enough structure to favor tr06. It is not clear what the hard graphs with
density factor 4 will do.
The ratio tr03/tr06 can be regarded as a measure of the amount of structure in
a graph (and its asymptotic behavior an indication of the amount of structure in
the graph type). Graphs that are more random favor iterative improvement, par† The
a = 1.0 curve is for a = 1.0 and d = ∞.
Ratio of CN averages: tr03/tr06
1.1
1
M. Stallmann, F. Brglez, and D. Ghosh
a = 1.0
a = 1.125
a = 1.03125
a = 1.25
a = 1.0625
a = 2.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
20
100
1000
number of edges
4000
Ratio of CN averages: tr03/tr06
·
26
1.1
1
⊗♥
0.9
0.8
0.7
0.6
0.5 =
d = 1.001
d = 2.0
d = 1.01
d = infinity
30
10
1
0.3
20
100
1000
number of edges
(c) Random trees.
Fig. 9.
⊗
butterflies
⊗
hypercubes
=
rectangular
grids
♦
square grids
♥
♦
⊗
=
=⊗ ♦⊗
⊗
♥
♦
100
edges
♥
♥
1000
4000
(b) Communication graphs.
4000
Ratio of CN averages: tr03/tr06
Ratio of CN averages: tr03/tr06
d = 1.1
=
♦
0.4
0.3
10
(a) Random graphs.
d = 1.0
⊗
♥
♥
G_00
G_10
G_20
G_01
G_11
G_21
30
10
1
0.3
50
100
1000 2000
number of edges
(d) VLSI graphs.
An asymptotic comparison between pure barycenter and pure bfs.
Bigraph Crossing Minimization
Ratio of CN averages: tr06/tr14
d = 1.001
1000
50
tr06/tr14 (d=1.01)
d = 1.1
100
27
tr06/tr14 (d=1.001)
d = 1.01
d = 2.0
·
tr06/tr14 (d=1.1)
tr06/tr14 (d=2)
10
10
1
20
1
100
1000
4000
number of edges
(a) bfs vs. gbfs as a function of size.
Fig. 10.
view.
1
10
100
1000 4000
"Width" of the tree
(b) bfs vs. gbfs as a function of “width”.
How initial placement based on gbfs improves on breadth-first search — asymptotic
ticularly barycenter, while more structured ones favor initial placement, especially
breadth-first search.
4.3.1 Asymptotic Behavior of Gbfs. One difference that is not reflected directly
in asymptotic behavior is the advantage of gbfs over bfs for random trees with
dimension ranging from 2 down to 1. Fig. 10(a) shows the curves of the ratio
tr06/tr14 for various dimensions of trees. For any fixed dimension, the advantage
of gbfs dwindles rapidly as size increases.
The gbfs heuristic on a class of random trees of dimension 1 + α is very sensitive
to what we call the width, the number of edges multiplied by α. Roughly speaking,
the width of an individual tree represents the average number of edges for all paths
that branch away from the longest path. Fig. 10(b) shows the relationship between
width and the tr06/tr14 ratio. The same symbols as in Fig. 10(a) are used to show
which tree dimension contributed each point along the curve. To get the largest
two widths for d = 1.001 we generated additional tree classes with up to 15350
edges.
Thus we see that the diminishing effectiveness of gbfs is really due to increasing
width and width increases as we increase size while keeping dimension fixed.
4.3.2 Asymptotic Behavior of Dot. Another interesting comparison is that between dot and tr07. Both have roughly equal emphasis on initial placement versus
iterative improvement, but the former uses dfs while the latter uses bfs.
The random trees with d = ∞ clearly favor dot — see Figs. 11(a) and (c) —
but all other random graphs give the edge to tr07. This advantage is only slight
for graphs with a > 1 and it only shows itself for the largest graphs. For the
geometrically-generated trees it is dramatic and the gap grows larger as d approaches 1. For example, the tr07/dot ratio goes from about 0.18 to 0.08 when
d = 1.1 and the number of edges goes from 1910 to 3830 — thus the inverse
dot/tr07 ratio would increase from 5.5 to 12.5.
Among the isomorphism graph types only the hard VLSI trees favor dot and even
more dramatically so than the random trees — see Fig. 11(d). Hypercubes also
Ratio of CN averages: tr07/dot
1.8
M. Stallmann, F. Brglez, and D. Ghosh
a = 1.0
a = 1.125
a = 1.03125
a = 1.25
a = 1.0625
a = 2.0
Ratio of CN averages: tr07/dot
·
28
1.6
1.4
1.2
1
0.8
0.6
20
100
1000
number of edges
4000
♥
butterflies
⊗
hypercubes
1.8
=
rectangular
grids
1.6
♦
square grids
1.4
1.2
1 =⊗
0.6
10
d = 1.1
d = 2.0
d = infinity
1.5
1
0.5
0
20
100
1000
number of edges
(c) Random trees.
Fig. 11.
⊗
⊗♥
♦
⊗
⊗
♥
♥
=
⊗
♥
⊗
♦
♥
100
1000 4000
edges
(b) Communication graphs.
Ratio of CN averages: tr07/dot
Ratio of CN averages: tr07/dot
2
♥♦
=
0.8
(a) Random graphs.
d = 1.0
d=1.001 and
d=1.01 not
shown
= ♦
4000
G_00
G_10
G_20
G_01
G_11
G_21
2
1.5
1
0.5
0
50
100
1000 2000
number of edges
(d) VLSI graphs.
An asymptotic comparison between bfs plus median and dot (dfs plus enhanced median).
Bigraph Crossing Minimization
·
29
show a slight trend in favor of dot as their dimension increases — see Fig. 11(b).
Since the density factor also increases, the increase in ratio here is contrary to what
one would expect for random graphs with the same a and m.
In all other cases, the difference between tr07 and dot is hardly noticeable and
the asymptotic trends are far from clear.
The lesson here appears to be that depth-first search does reasonably well on trees
that are more balanced while breadth-first search does better when most subtrees
at a given node are small. When embedding a biplanar or almost biplanar tree, it
is important to embed any small subtrees emanating from a node close to the node
itself, while letting long paths stretch from one end to the other. Conversely, with
the number of nodes divided more evenly among subtrees (the extreme case of this
are the hard VLSI trees G01,k — see Fig. 16, Appendix C) the shuffling of subtrees
that occurs with breadth-first search produces many unnecessary crossings.
4.4 Other Observations
At the end of Section 3 we showed that sometimes the large variance of dot could
be advantageous, allowing low-crossing presentations to be found in one or more
of the 64 runs in an isomorphism class. Fig. 12 illustrates this with G01,7 and
iso(rnd (G01,7 )), whose histograms appear in the first and last rows of Fig. 5. The
main change is in the ranking (Fig. 12(a) and (b)) or relative position (Fig. 12(c)
and (d)) of dot. There is also a slight decrease in the advantage of gbfs over bfs,
suggesting that at least 1 out of 64 times bfs finds a presentation almost as good
as that of gbfs. For some graphs bfs finds presentations better than those of gbfs.
We now take a look at how the treatments we studied compare with known upper
bounds. Fig. 13 illustrates the asymptotic behavior of several treatments and the
known upper bounds for four graph types, two hard VLSI graph types, hypercubes,
and grids (both rectangular and square). The upper bounds for the VLSI graphs
come from Appendix C and the ones for grids and hypercubes come from a recent
paper [Shahrokhi et al. 2000].
Only tr23 manages to keep up with the upper bound for the trees. However,
the minimum number of crossings reported by dot matches our upper bound in
every case (our recursive derivation is based on a depth-first search from the root
of the tree). The upper bound for the VLSI graphs with a = 2 is not good at all
and even some of the less powerful treatments manage to do better rather easily.
The minimum number of crossings certainly grows at a much slower rate than the
quadratic one suggested by our upper bound. In other words, our attempt to create
a type of graphs with a relatively large crossing number was not successful.
The Shahrokhi et al. bound for hypercubes is excellent in relation to our treatments, which is not surprising given that it is based on the recursive structure of
the hypercube. Dot is the closest competitor. In fact, for all but the two largest
cubes, the minimum crossing number reported by dot matches the upper bound.
Their bound for grids appears to take better advantage of the structure of grids that
are not square (square grids are marked with # in front of the number of edges)
than most of our treatments. However, tr23 has the same pattern (with respect to
square and rectangular grids) and does better still. Since [Shahrokhi et al. 2000]
gives a lower bound that is within a constant factor of the upper bound, our results
suggest that the better treatments have the same asymptotic behavior as the actual
·
M. Stallmann, F. Brglez, and D. Ghosh
Average number of crossings reported
1000000
00
1000000
01
100000
06
15
07
14
03
dot
•
10000
09
17
+
23
Minimum number of crossings reported
30
01
100000
03
10000
09
500
500
A "hard" VLSI tree (a=1, m=1910)
01
10000
06
07
15
14
09
17
+
23
1000
A random tree (m=1910, n0=382, n1=1529)
(c) iso(rnd(G01,7 )), average
Fig. 12.
trees.
Minimum number of crossings reported
Average number of crossings reported
1000000
00
•dot
+•dot
A "hard" VLSI tree (a=1, m=1910)
(b) G01,7 , minimum
(a) G01,7 , average
03
17
23
1000
100000
06,15 14
07
1000
1000000
00
00
01
100000
03
10000
06
07
09
• dot
15
14
17
+
23
1000
A random tree (m=1910, n0=382, n1=1529)
(d) iso(rnd(G01,7 )), minimum
A comparison between minimum value reported and average value reported for two
4
3.5
3
2.5
2
tr03/tr17
Bigraph Crossing Minimization
tr06/tr17
UB/tr17
tr23/tr17
dot/tr17
tr03/tr17
2.5
2
tr06/tr17
dot/tr17
tr23/tr17
UB/tr17
1.5
# 112 edges
1.5
230 edges
1
0.5
110 edges
950 edges
·
UB/tr17
tr23/tr17
dot/tr17
tr06/tr17
tr03/tr17
(b) Hard graphs with a = 2 (G11,k ).
2.2
2
1.8
1.6
1.4
1.2
1
0.8
52 edges
(d) Grids (meshes).
# 24 edges
1
ratio of mean crossing numbers
ratio of mean crossing numbers
0.5
(a) Hard trees (G01,k ).
tr03/tr17
tr06/tr17
dot/tr17
tr23/tr17
UB/tr17
470 edges
232 edges
470 edges
448 edges
0
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
0.92
0.9
192 edges
50 edges
10 edges
110 edges
32 edges
31
1910 edges
# 1984 edges
1910 edges
2304 edges
50 edges
12 edges
(c) Hypercubes.
976 edges
20 edges
4 edges
Fig. 13. An asymptotic comparison between known upper bounds and average number of crossings reported by several treatments.
# 480 edges
950 edges
1024 edges
230 edges
80 edges
ratio of mean crossing numbers
ratio of mean crossing numbers
32
·
M. Stallmann, F. Brglez, and D. Ghosh
minimum and approximate it reasonably well.
We learned much more from the behavior of treatments on our carefully-designed
classes of subjects than we would have learned examining only the results on random graphs. In a sense, the observed “behavior” of a graph type with respect to
our treatments became at least as important as the behavior of the treatments. By
identifying some extremes in the range of this latter behavior we hope to have reduced the effort required in judging the efficacy of a new treatment or the usefulness
of a new graph type.
5. CONCLUSIONS AND FURTHER WORK
This study is only the beginning of what we hope is a new approach to experimental study of bigraph crossing and other intractable problems. Input data
and treated output can be shared and verified so that different groups working on
the same problem can conduct repeatable experiments. Better heuristics are often developed through a detailed understanding of why specific instances present
difficulties, and such understanding is made more likely when careful thought is
given to the design of types of test subjects. We hope to facilitate future development of graph type design and algorithm engineering by making our experimental
infrastructure available on the world-wide web — see Appendix D.
The work we have done is extensive in volume and scope. Certainly there are
many hypotheses beyond those we presented that can be inferred and partially or
fully verified from the data already gathered. But there are also important gaps in
our data and the scope of our algorithm engineering, as well as unproven conjectures
about the bigraph crossing problem.
The most pressing issue from a practical perspective is generalization to k > 2
layers. ∗ There are obvious generalizations for all heuristics discussed here. Initial
ordering heuristics depend only on sorting each layer by sequence numbers assigned
during a search. Iterative improvement heuristics repeatedly solve fixed-layer problems one layer at a time — their generalization can repeat forward and backward
passes through the layers, where a forward pass sequences layer i so as to minimize
crossings among edges between it and layer i − 1 and a backward one minimizes
crossings among edges between layers i + 1 and i. The dot heuristic is already generalized in this fashion [E.R. Gansner, E. Koutsifios, S.C. North and K.P. Vo 1993].
Theoretical questions related to k-layer generalization include (a) is there a simple
characterization of a k-planar graph, a k-layer graph that can be embedded on k
horizontal lines without edge crossings; † and (b) are there types of graphs, e.g.
trees, for which k-layer crossing minimization can be solved in polynomial time?
Future work can be divided into three main categories: theory, algorithm engineering, and experimental design. Some ideas are listed below.
Theory
—Better lower bounds for specific graphs and for general sparse graphs.
∗ A k-layer graph is a k-partite graph in which all edges are incident to nodes on adjacent layers;
dot converts the former into the latter by adding “virtual nodes” to edges between non-adjacent
layers.
† This was recently settled by [Healy et al. 2000]
Bigraph Crossing Minimization
·
33
—Algorithms with provable approximation bounds for special graph types (as in
[Shahrokhi et al. 2000]) or general graphs.
—Faster exact algorithms for trees — the results of [Shahrokhi et al. 2001] imply an O(n1.6 ) algorithm [Chung 1984; Shiloach 1979], or faster approximation
algorithms — the minimum number of edges whose removal will eliminate all
crossings can be found in linear time for trees [Shahrokhi et al. 2001].
—More general types of graphs, beyond trees, for which the crossing number can
be found in polynomial time. ‡
—Proof of convergence for the median and barycenter heuristics (or a counterexample) — neither one is necessarily monotone with respect to crossing number.
Algorithm Engineering
—Better initial ordering — for example, a decomposition into biconnected components could be combined with a good heuristic or exact algorithm for trees.
—Better iterative improvement — tr23 suggests that this is possible but efficiency
is an important issue.
—Treatments that, like simulated annealing, use randomization.
—Novel treatments that stray from the standard paradigm of initial ordering and
iterative improvement, or iterative improvement strategies that are not based on
fixed-layer heuristics.
Experimental Design
—Fill in some gaps in the graph types introduced here: (a) trees with dimension
d > 2, both integer and non-integer values — what is the maximum value for
d before a class is indistinguishable from d = ∞? and (b) general graphs with
d 6= ∞ — give nodes a random location in a metric space and choose all edges
based on increasing distance, first constructing a minimum spanning tree and
then adding edges to it.
—Equivalence classes based on specific circuits or different schemes than the ones
presented here (see, for example, [Ghosh 2000]).
—Infrastructure to support participation of other researchers who wish to contribute new graph classes, new treatments, or new approaches to analyzing the
data (see, for example, [Brglez et al. 2000; Brglez and Lavana 2001])
ACKNOWLEDGMENTS
Some of the initial crossing number experiments with 2-layer graphs posted on the
Web in 1997 were organized and executed by Nevin Kapur. He also contributed the
software utilities that support gathering and tabulating data about the experiments
and the statistical summaries. Hemang Lavana contributed utilities to dynamically
generate web-pages of directories that link the data related to the reported experiments. We appreciate their support.
Also appreciated are the extensive comments of Cathy McGeogh, our editor, and
helpful suggestions by the referees and by Imrich Vrt́o.
‡ One
simple case is that of bipartite permutation graphs [Spinrad et al. 1987].
34
·
M. Stallmann, F. Brglez, and D. Ghosh
APPENDIX
A. COMPARISONS WITH PREVIOUSLY PUBLISHED RESULTS
There are four main differences between the way we conducted our experiments
and previous results of [Jünger and Mutzel 1997] or the experiments on the sifting
technique [Matuszewski et al. 1999; Schönfeld et al. 2000]. We focus our discussion
on the Jünger-Mutzel results since these are used as a benchmark for the others.
(1) Our random graphs are forced to be connected (we generate a random spanning
tree before distributing the remaining edges at random). As already pointed
out in Section 3 — see Table 4 — graphs generated using the standard random
graph model behave, with respect to crossing number, as if they were denser
than they really are.
(2) Since there are no published random number seeds for previous experiments,
the results cannot be duplicated exactly — for example, as a test of the accuracy
of the implementations. In most cases our own results, including our attempted
recreation of the relevant experiments in the Jünger-Mutzel paper, were only
slightly sensitive different seeds. The only exception were graphs with 10 nodes
on each layer and 10% density [Jünger and Mutzel 1997, Table 4], where most
of the statistics appear to be meaningless. Most heuristics reported 0 crossings
more than half the time, so the value of the mean was a matter of chance more
than anything else.
(3) Our implementations of both median and barycenter use stable sorts and we
keep iterating until there is no change at all in the positions of nodes. We also
implemented versions of both median and barycenter that stop iterating when
there is no improvement after the heuristic is applied to both sides (tr24 and
tr25 — see Section 2). In these cases there is no need for the sort to be stable
— we used an unstable sort because it yielded better results. Table 2 shows a
comparison of our implementations with the ones used in other experiments.
(4) It is not clear whether the median heuristic used in the Jünger-Mutzel paper
is the one from [Eades and Wormald 1994] or uses the standard definition of
the median of an even-cardinality set — the mean of the two middle items.
Our attempts to recreate the experiments — see Tables 5 and 6 suggest the
latter (tr30 uses the mean of two middle items) even though the original paper
suggests the former [Jünger and Mutzel 1997; Mutzel 2001]. The exact nature
of the sorting algorithm might also have an effect. We used insertion sort in
all cases, which meant that our unstable sort reversed the order of nodes with
equal keys. Jünger and Mutzel, on the other hand, used priority queues as
implemented in LEDA [Mehlhorn and Näher 1999]. These lead to an unstable
sort, but not necessarily to the reversal of items with equal keys.
Tables 5 and 6 show our results as compared to those of Tables 4 and 6, respectively in [Jünger and Mutzel 1997].
∗ Data
for this number of edges is extremely sensitive to the random number seed. 100 instances
generated with a different seed yielded intervals [0.14,0.39] for tr23, [0.81,1.43] for tr25, and
[1.40,2.00] for tr30.
Bigraph Crossing Minimization
Heuristic
minimum
tr23
barycenter
tr25
median
tr30
tr24
m = 10 ∗
0.29
[0.04,0.16]
1.52
[0.60,1.14]
1.53
[1.26,1.86]
[1.03,1.75]
Heuristic
minimum
tr23
barycenter
tr25
median
tr30
tr24
m = 20
11.62
[10.6,12.1]
18.78
[14.8,17.2]
24.08
[19.7,22.3]
[19.1,22.3]
m = 60
463.17
[456,465]
475.52
[462,475]
539.59
[487,502]
[488,504]
m = 30
56.60
[58.1,61.6]
65.30
[63.5,68.5]
81.78
[71.4,76.6]
[74.1,80.4]
m = 70
698.35
[699,707]
710.88
[704,715]
782.33
[739,756]
[727,744]
m = 40
146.89
[145,149]
157.70
[152,158]
189.55
[165,174]
[167,176]
m = 80
1008.38
[1000,1010]
1021.44
[1000,1020]
1110.39
[1050,1070]
[1038,1053]
·
35
m = 50
276.78
[277,283]
288.15
[284,292]
333.25
[306,318]
[303,315]
m = 90
1405.57
[1410,1420]
1420.68
[1420,1430]
1524.18
[1440,1460]
[1427,1440]
The graphs yielding the above results all have n0 = n1 = 10. 100 instances for each m are generated randomly using standard techniques
for random bipartite graphs, like those of, for example, Knuth’s GraphBase [Knuth 1993]. Numbers reported for the minimum, median, and
barycenter are the means from Table 4 of [Jünger and Mutzel 1997]. Our
data for tr23 (the best results obtained), tr24, tr25, and tr30 are 95%
confidence intervals of the mean rounded to 3 digits of accuracy.
Table 5. Comparison of crossing number mean values reported by Jünger and Mutzel for random
graphs of various densities with our results.
B. PROOF THAT DIMENSION-1 TREES ARE BIPLANAR TREES
Recall that a tree has dimension d = 1 if it is generated in the following way. Assign
each node to either V0 or V1 . Assign a real number xi ∈ (0, 1] to each node i. Let
the distance between nodes i and j be ∞ if {i, j} ⊆ V` for ` = 0 or 1 and |xi − xj |
otherwise. Construct a minimum spanning tree using the distance just defined.
Definition 1. A comb is a tree that becomes a path if all the leaves are removed.
Fig. 14 shows a comb; the thicker path in the diagram is the spine, the path that
remains when leaves have been deleted. A connected bigraph is biplanar if and only
if it is a comb [Harary and Schwenk 1971].
z2
z4
z6
s
s
s
×
×
A
A
¢A
¢@
¢ A
¢ A@
A
¢
¢
A @
A
A
¢
A @
A ¢
A
s¢ × A× @s A¢s
As
×
×
z1
Fig. 14.
z3
z5
z7
An example of a comb — the spine is along the thicker line, z1 . . . z7 .
36
·
M. Stallmann, F. Brglez, and D. Ghosh
Heuristic
tr23
barycenter
tr25
median
tr30
tr24
m = 20
[11.7,13.3]
15.70
[17.1,19.8]
25.70
[21.6,24.8]
[19.8,23.0]
m = 40
[45.4,50.4]
72.50
[67.5,75.1]
79.60
[85.8,94.2]
[94.4,104]
m = 60
[103,112]
147.90
[152,165]
188.50
[189,203]
[233,249]
m = 80
[179,193]
273.30
[259,281]
374.20
[334,359]
[425,449]
m = 100
[279,297]
392.30
[400,429]
561.90
[518,552]
[658,705]
Heuristic
m = 120
m = 140
m = 160
m = 180
m = 200
tr23
[409,436]
[548,581]
[697,736]
[904,953] [1100,1160]
barycenter
567.00
764.60
1080.70
1272.40
1555.10
tr25
[572,612]
[765,819]
[957,1020] [1250,1320] [1520,1600]
median
811.20
1146.20
1481.30
1848.00
2084.10
tr30
[757,802] [1020,1070] [1310,1380] [1690,1770] [2050,2140]
tr24
[1000,1050] [1320,1380] [1750,1840] [2240,2340] [2750,2870]
The graphs yielding the above results all have n0 = n1 = m/2. 100 instances for each m are generated randomly using standard techniques for
random bipartite graphs. Numbers reported for the median and barycenter are the means from Table 7 of [Jünger and Mutzel 1997]. Our data
for tr23 (the best results obtained), tr25, and tr30 are 95% confidence
intervals of the mean rounded to 3 digits of accuracy.
Table 6. Comparison of crossing number mean values reported by Jünger and Mutzel for sparse
random graphs of various sizes with our results.
Lemma 1. A tree has dimension d = 1 if and only if it is biplanar (a comb).
Proof. (⇐=) Suppose that T is a comb with nodes z1 , . . . , zk along the spine
(as in Fig. 14). Assign the real number i/k to zi and numbers in the open interval
i+0.5
( i−0.5
k , k ) to the non-spine neighbors of zi . It is easy to see that the minimum
spanning tree, restricted to obey the bipartition, will be T .
(=⇒) Suppose T is dimension-1 and place all nodes of T on the x-axis according
to the real number assigned to them (assume wlog that no two nodes occupy the
same spot). Move the nodes of V1 vertically to y = 1 and draw the edges of T as
straight lines. Suppose the edges a1 b1 and a2 b2 , both in T , cross (the a’s are in V0 )
and assume wlog that a1 is the leftmost node (in either V0 or V1 ) on any edge that
crosses — otherwise the proof is symmetric with b2 playing the role of a1 .
Adding a1 b2 to T creates a cycle C. Let a1 b0 be the other edges of C incident
with a1 . Depending on the position of b0 we contradict either the fact that T is a
restricted minimum spanning tree or the assumption about a1 . See Fig. 15.
b0 is to the left of b2 . In order for C to reconnect with b2 there must be another
crossing involving an edge with an endpoint to the left of a1 , contradicting our
assumption about a1 .
b0 is to the right of b2 . Since b2 is to the right of a1 by assumption, b0 is farther
from a1 than b2 . This means T ∪{a1 b2 }−{a1 b0 } has less weight than T , contradicting
the fact that T is a minimum spanning tree.
b0 is b2 , i.e. a1 b2 is an edge of T . Add the edge a2 b1 to T (it cannot be in
T already) to form the cycle a1 b1 a2 b2 . Depending on the positions of a2 and b1 ,
Bigraph Crossing Minimization
b0
s· ·
··
b2
b1
s
·s
···
· ·´
A · · · · · · · · ··¢·A·· · · · · · · ·´
¢ A ´
A
¢ ´
A
´A
A
A ¢´
s¢
As
A´
a1
a2
(a) b0 to the left of b2 .
Fig. 15.
b2
b0
s
s
¢@ ¡
¢ ¡
@
¢ ¡ @
¢¡
@
s¢¡
@s
a1
a2
b0 = b 2
·
37
b1
s ¾ s¢@ ¡A
¢ ¡
@ A
¢ ¡ @A
¢¡
@A
s¢¡
@
As
a1
(b) b0 to the right of b2 .
a2
(c) b0 is b2 .
Cases in the proof that a dimension-1 tree is biplanar.
either T ∪ {a2 b1 } − {a2 b2 } or T ∪ {a2 b1 } − {a1 b1 } will have less weight than T ,
contradicting the fact that T is a minimum spanning tree.
The proof suggests that, with an appropriate definition of what it means for
all biplanar trees to be equally likely, it may be possible to argue that a random
dimension-1 tree with a given n0 , n1 , and m is equally likely to be any of the biplanar
trees with that n0 , n1 , and m. The structure of the tree is more dependent on the
assignment of nodes to V0 or V1 than to the positions of nodes on the line. Without
that a priori assignment, the tree will always be a path.
C. DETAILS ON THE CONSTRUCTION OF VLSI GRAPHS
Our goal: to construct bigraph types (i.e. collections of classes having common
characteristics other than size and whose sizes grow exponentially) similar to VLSI
circuits with varying densities and difficulty.
Recall that the set V0 represents cell nodes. In many applications these all have
the same degree — we chose 5 to allow for density factors up to 4. The set V1
consists of net nodes. Each net node is adjacent to a group of cell nodes that are
to be connected in a circuit.
The six bigraph types have three density factors (1, 2, and 4), and two “degrees
of difficulty” (easy, meaning relatively few crossings, and hard, a larger number of
crossings, but also with the intent of stumping our heuristics).
Easy instances of trees are trivial to create. Just make them biplanar — we
connect each cell node to 5 net nodes and share a net node between each two
neighboring cell nodes, as illustrated in Fig. 3(a), Section 3.3. The result has no
crossings.
To create hard instances of trees, the G01,k graphs, we let n = n0 = 3 · 2k − 3,
V0 = {a0 , . . . , an−1 }, V1 = {b0 , . . . , b4n }, and E = {ai b4i+j | 0 ≤ i ≤ n − 1, 1 ≤
j ≤ 4} ∪ {a0 b0 } ∪ {ai bi | 1 ≤ i ≤ 3} ∪ {ai b2i−3 | 4 ≤ i ≤ n − 1}. Fig. 16 shows
how G01,2 looks when made to look like a binary tree with extra leaves. Fig. 3(b),
Section 3.3, shows an embedding with 28 crossings.
Using a straightforward recursive construction, we know the number of crossings
k
k
for G01,k to be no more than 15
2 k · 2 − 10 · 2 + 10, hence O(n log n). This exact
bound is clearly not optimal, given the embedding with 28 crossings for k = 2;
whether n log n is the right asymptotic bound is unclear.
Denser graphs are created by decreasing the number of net (V1 ) nodes. The easier
variants generalize the structure of a comb with a sequence of overlapping complete
bipartite graphs, called clusters: a k0 ×k1 cluster has k` layer-` nodes and all possible
38
·
M. Stallmann, F. Brglez, and D. Ghosh
a0
b0
b2
b1
a1
b5
a4
b6
b7
a5
b3
a2
b8
b9
a3
b10
a6
b11
a7
b17 18 19 20 21 22 23 24 25 26 27 28
Fig. 16.
b4
b12
b13
a8
b14
b15
b16
a9
29 30 31 32 33 34 35 36 37 38 39 40
The tree G01,2 .
edges
between
µ ¶µ
¶ them — the minimum number of crossings for such a cluster is
k0
k1
. In G10,k graphs, 2 × 5 clusters (forcing 10 crossings each) overlap at
2
2
two net nodes on each end; the overlap forms a 4 × 2 cluster that yields 4 additional
crossings (1 crossing on each end only involves edges belonging to the same 2 × 5
cluster — otherwise the contribution of a 4 × 2 cluster would be 6 crossings). If
c = n0 , the number of cell nodes, there are 3c
2 + 2 net nodes. Numbering the node
sets a0 , . . . and b0 , . . . as before, the edges are {ai b3i/2+j | 0 ≤ i ≤ c−1, 0 ≤ j ≤ 4}.
The minimum number of crossings is easily calculated: 7c−4. Fig. 4(a), Section 3.3,
shows the smallest instance.
To increase the density again, we reduce the number of net nodes to c−2
4 + 5. A
similar overlap strategy to the one used for G10,k has a sequence of 4 × 5 clusters
overlapping at 4 net nodes. Counting the number of crossings is complicated by the
fact that a cluster overlaps at 3 net nodes with the second cluster after it and at 2
net nodes with the third one (overlapping at one node induces no new crossings).
In general, the minimum number of crossings is 55c − 260, but k = 2 (c = 10) is
a special case because some potential overlaps are missing: 298 is the optimum in
that case.
To increase difficulty — and number of crossings — for each of the two higher
densities, only four of the five edges incident on each cell node are used in an
overlapping cluster construction. The fifth edge is used to connect to a net node
that also connects to another net node “as far away as possible” in the cluster
sequence. For G11,k a sequence of 2 × 4 clusters overlap at 2 net nodes forming
4 × 2 clusters at the overlap. Roughly one third of the net nodes are reserved
for the far-away connections. The edges are {ai bi+j | 0 ≤ i ≤ c − 1, 0 ≤ j ≤
2} ∪ {ai bc+2+i mod c/2 | 0 ≤ i ≤ c − 1} ∪ {ai bi+3 | i is even } ∪ {ai bi−1 | i is odd }.
See Fig. 4(b), Section 3.3. If the cell nodes are put in numerical sequence and the
net nodes sequenced according to a barycenter pass (as shown in the figure), the
number of crossings is m2 /20 + 7m/10 − 4.
Finally the G21,k graphs have overlapping 8 × 4 clusters (3 × 8 then 2 × 8 at
the overlap). Half of the roughly c/4 net nodes are used for clusters, the other
Bigraph Crossing Minimization
·
39
half for long-distance connections. Let c = n0 as before and let h = (c − 6)/8 + 2
(when k ≥ 2, the number of cell nodes = 6 mod 8). The edges are then defined as
{ai bi/8+j | 0 ≤ i ≤ c, 0 ≤ j ≤ 3} ∪ {ai bh+2+(i/4) mod h | 0 ≤ i ≤ c}.
D. DATA AND PROGRAMS AVAILABLE ON THE WEB
Data and programs used in our experiments are available on the Web at URL
www.cbl.ncsu.edu/software/2001-JEA-Software/. A unique feature of our experimental design is the separation of the experimental software into three parts:
(1) The generation of experimental data: Programs and scripts are provided for
generating both random and isomorphic graph classes, those reported in the
paper and custom-designed ones for which density factor, dimension, and sizes
are specified.
(2) The execution of heuristics: There is a program that selectively runs any of the
treatments reported in the paper and scripts for running one or more treatments
on a set of graph classes specified by Unix wild-card conventions.
(3) The evaluation of the cost function and compilation of results: The separation
of execution and evaluation makes it possible for other researchers to run their
own heuristics on our data, submit new data to our heuristics, and evaluate any
bigraph presentation independent of the heuristic that generated it. Various
scripts are available for gathering summary statistics, calculating confidence
intervals, and putting comparative results in tabular form.
All software is clearly documented and a tutorial example is provided to illustrate
the use of the main scripts.
REFERENCES
Betz, V. and Rose, J. 1997. VPR: A New Packing, Placement and Routing Tool for FPGA
Research. In Proceedings of the 7th International Workshop on Field-Programmable Logic
(August 1997), pp. 213–222. Software and postscript of paper can be downloaded from
http://www.eecg.toronto.edu/~vaughn/vpr/vpr.html .
Brglez, F. and Lavana, H. 2001. A Universal Client for Distributed Networked Design
and Computing. In Proceedings of the 38th Design Automation Conference (June 2001).
Also available at http://www.cbl.ncsu.edu/publications/#2001-DAC-Brglez.
Brglez, F., Lavana, H., Ghosh, D., Allen, B., Casstevens, R., III, J. H., Kurve, R.,
Page, S., and Stallmann, M. 2000. OpenExperiment: A Configurable Environment
for Collaborative Experimental Design and Execution on the Internet . Technical Report
2000-TR@CBL-02-Brglez (March), CBL, CS Dept., NCSU, Box 8206, Raleigh, NC 27695.
Chung, F. R. K. 1984. On optimal linear arrangement of trees. Comp. and Maths. with
Appls. 10, 1, 43–60.
Di Battista, G., Eades, P., Tamassia, R., and Tollis, I. G. 1999.
Graph Drawing:
Algorithms for the Visualization of Graphs. Prentice Hall.
Eades, P. and Kelly, D. 1986. Heuristics for reducing crossings in 2-layered networks. Ars
Combinatoria 21-A, 89–98.
Eades, P., McKay, B. D., and Wormald, N. C. 1976. On an Edge Crossing Problem.
Technical report, Department of Computer Science, University of Newcastle, New South
Wales 2308, Australia.
Eades, P. and Sugiyama, K. 1990. How to Draw a Directed Graph. Journal of Information
Processing 13, 424–437.
Eades, P. and Whiteside, S. 1994. Drawing graphs in two layers. Theoretical Computer
Science 131, 361–374.
40
·
M. Stallmann, F. Brglez, and D. Ghosh
Eades, P. and Wormald, N. C. 1994. Edge Crossings in Drawings of Bipartite Graphs.
Algorithmica 11, 379–403.
E.R. Gansner, E. Koutsifios, S.C. North and K.P. Vo. 1993. A Technique for Drawing
Directed Graphs. IEEE Trans. Software Engg. 19, 214–230. The drawing package dot is
available from http://www.research.att.com/sw/tools/graphviz/.
Garey, M. R. and Johnson, D. S. 1983.
Crossing Number is NP-complete. SIAM J.
Algebraic Discrete Methods 4, 312–316.
Ghosh, D. 2000. Generation of Tightly Controlled Equivalence Classes for Experimental
Design of Heuristics for Graph–Based NP-hard Problems. Ph. D. thesis, Electrical and
Computer Engineering, North Carolina State University, Raleigh, N.C. Also available at
http://www.cbl.ncsu.edu/publications/#2000-Thesis-PhD-Ghosh.
Ghosh, D. and Brglez, F. 1999. Equivalence Classes of Circuit Mutants for Experimental
Design. In IEEE 1999 International Symposium on Circuits and Systems – ISCAS’99
(May 1999). A reprint also accessible from http://www.cbl.ncsu.edu/experiments/DoEArchives/I999-ISCAS.
Ghosh, D., Brglez, F., and Stallmann, M. 1998a.
First steps towards experimental design in evaluating layout algorithms: Wire length versus wire crossing in linear
placement optimization. Technical Report 1998-TR@CBL-11-Ghosh (October), CBL, CS
Dept., NCSU, Box 7550, Raleigh, NC 27695. Also available at http://www.cbl.ncsu.edu/publications/#1998-TR@CBL-11-Ghosh.
Ghosh, D., Brglez, F., and Stallmann, M. 1998b. Hypercrossing Number: A New and
Effective Cost Function for Cell Placement Optimization. Technical Report 1998-TR@CBL12-Ghosh (December), CBL, CS Dept., NCSU, Box 7550, Raleigh, NC 27695. Also available
at http://www.cbl.ncsu.edu/publications/#1998-TR@CBL-12-Ghosh.
Ghosh, D., Kapur, N., Harlow, J. E., and Brglez, F. 1998.
Synthesis of Wiring
Signature-Invariant Equivalence Class Circuit Mutants and Applications to Benchmarking. In Proceedings, Design Automation and Test in Europe (Feb 1998), pp. 656–663. Also
available at http://www.cbl.ncsu.edu/publications/#1998-DATE-Ghosh.
Harary, F. and Schwenk, A. J. 1971. Trees with Hamiltonian Square. Mathematika 18,
138–140.
Harary, F. and Schwenk, A. J. 1972.
A New Crossing Number for Bipartite Graphs.
Utilitas Mathematica 1, 203–209.
Healy, P., Kuusik, A., and Leipert, S. 2000. Characterization of level non-planar graphs
by minimal patterns. In COCOON , Number 1858 in LNCS (2000), pp. 74–84.
Jünger, M. and Mutzel, P. 1997.
2–Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms. Journal of Graph Algorithms and Applications
(JGAA) 1, 1, 1–25.
K. Kozminski, (Ed.). 1992. OASIS2.0 User’s Guide. MCNC, Research Triangle Park, N.C.
27709. (Over 600 pages, distributed to over 60 teaching and research universities worldwide).
Kapur, N., Ghosh, D., and Brglez, F. 1997. Towards A New Benchmarking Paradigm in
EDA: Analysis of Equivalence Class Mutant Circuit Distributions. In ACM International
Symposium on Physical Design (April 1997).
Kernighan, B. W. and Lin, S. 1970.
An efficient heuristic procedure for partitioning
graphs. Bell System Technical Journal, 291 – 307.
Knuth, D. E. 1993. The Stanford Graphbase. Addison Wesley.
Leighton, F. T. 1984. New Lower Bound Techniques for VLSI. Math. Systems Theory 17,
47–70.
Leighton, F. T. 1992.
Introduction to Parallel Algorithms and Architectures: Arrays,
Trees, Hypercubes. Morgan Kaufmann.
Mäkinen, E. 1989. A Note on the Median Heuristic for Drawing Bipartite Graphs. Fundamenta Informaticae 12, 563–570.
Mäkinen, E. 1990. Experiments on drawing 2-level hierarchical graphs. International Journal of Computer Mathematics 37, 129–135.
Bigraph Crossing Minimization
·
41
Marek-Sadowska, M. and Sarrafzadeh, M. 1995. The Crossing Distribution Problem.
IEEE Transactions on Computer–Aided Design of Integrated Circuits and Systems 14, 4
(April), 423–433.
Matuszewski, C., Schönfeld, R., and Molitor, P. 1999.
Using sifting for k-layer
straightline crossing minimization. In Proc. 7th Graph Drawing Conference, Number 1731
in LNCS (1999), pp. 217–224.
Mehlhorn, K. and Näher, S. 1999. LEDA: A Platform for Combinatorial and Geometric
Computing. Cambridge University Press.
Mutzel, P. 1998.
The AGD-Library: Algorithms for Graph Drawing. Available from
http://www.mpi-sb.mpg.de/~
mutzel/dfgdraw/agdlib.html.
Mutzel, P. 2001. Personal communication.
N. Kapur. 1998. Cell Placement and Minimization of Crossing Numbers. Master’s thesis,
Electrical and Computer Engineering, North Carolina State University, Raleigh, N.C. Also
available at http://www.cbl.ncsu.edu/publications/#1998-Thesis-MS-Kapur.
Palmer, E. M. 1985.
Graphical Evolution: An Introduction to the Theory of Random
Graphs. John Wiley and Sons.
Schönfeld, R., Günter, W., Becker, B., and Molitor, P. 2000.
K-layer straightline
crossing minimization by soeeding up sifting. In Proc. 8th Graph Drawing Conference,
Number 1984 in LNCS (2000), pp. 253–258.
Shahrokhi, F., Sýkora, O., Székely, L. A., and Vrt́o, I. 2000. A new lower bound for
bipartite crossing number with applications. Theoretical Computer Science. To appear.
Shahrokhi, F., Sýkora, O., Székely, L. A., and Vrt́o, I. 2001. On bipartite drawing
and the linear arrangement problem. SIAM J. Computing. To appear. Preliminary version
was published in WADS’97.
Shiloach, Y. 1979. A minimum linear arrangement algorithm for undirected trees. SIAM
J. Computing 8, 1, 15–32.
Spinrad, J., Brandstädt, A., and Stewart, L. 1987. Bipartite permutation graphs. Discrete Applied Mathematics 19, 279–292.
Stallmann, M., Brglez, F., and Ghosh, D. 1999a.
Evaluating Iterative Improvement Heuristics for Bigraph Crossing Minimization. In IEEE 1999 International Symposium on Circuits and Systems – ISCAS’99 (May 1999). A reprint also accessible from
http://www.cbl.ncsu.edu/publications/#I999-ISCAS-Stallmann.
Stallmann, M., Brglez, F., and Ghosh, D. 1999b.
Heuristics and Experimental Design for Bigraph Crossing Number Minimization. In Algorithm Engineering and Experimentation (ALENEX’99), Number 1619 in Lecture Notes in Computer Science (1999),
pp. 74–93. Springer Verlag. Also available at http://www.cbl.ncsu.edu/publications/#1999-ALENEX-Stallmann.
Thompson, C. D. 1979. Area–Time complexity for VLSI. In Proceedings, 11th Annual ACM
Symposium on Theory of Computing (May 1979), pp. 81–88.
Tutte, W. T. 1960. Convex Representations of Graphs. Proc. London Math. Soc. 10, 304–
320.
Tutte, W. T. 1963. How to Draw a Graph. Proc. London Math. Soc. 13, 743–768.
Warfield, J. N. 1977.
Crossing Theory and Hierarchy Mapping. IEEE Transactions on
Systems, Man, and Cybernetics SMC–7, 7 (July), 505–523.
West, D. B. 1996. Introduction to Graph Theory. Prentice Hall.
Yamaguchi, A. and Sugimoto, A. 1999. An approximation algorithm for the two-layered
graph drawing problem. In COCOON , Number 1627 in LNCS (1999), pp. 81–91.