Graphs on Grids:
area and volume tradeoffs
in circuit layouts
By
Dan Archdeacon
University of Vermont
Abstract:
Finding a layout for a computer circuit can be
considered as drawing a graph on an n x n grid
or mesh. There are several measures of how
efficient the layout is: the first strives to minimize
the area, a second examines the maximum wire
run, a third bounds the number of bends in the
wires. Sometimes a circuit that is good in one
sense is bad in another. We examine these
tradeoffs and how to balance conflicting goals.
We also examine what happens when we layout
circuits in 3-dimensional n x n x n grids.
Goal: represent a graph in the plane
Assumptions:




Vertices are points with integer
coordinates (i,j), e.g., pixels
Avoids scaling to make pictures
arbitrarily small
Can use approximation by rational
coordinates for other drawings
Typical measures:
1.
2.
3.
Length of longest line (wire run)
Length of shortest line
Ratio max/min of lines
Examples





Polyline drawings (a)
Straight-line drawings (b)
Orthogonal drawings (c)
Upward digraph drawings (d)
Upward rooted trees
Section 1: Straight-line drawings



Want to draw vertices on integer points,
edges as straight lines
Must edges cross?
If edges don’t have to cross, is there any
advantage to having them cross?
Kuratowski and Fary

Theorem (K): A graph is
planar if and only if it has
no K5 or K3,3 as a
topological subgraph

Theorem (F): If a graph is
planar, then it has a
crossing free drawing
with all edges are
straight-line segments
Measures of straight-line drawings





Minimize area (width times height)
Aspect ratio close to 1 (ratio of width
verses height)
Angular resolution (smallest angle
between edges at a vertex)
Minimum number of crossings
Maximum number of symmetries
Trading crossings and symmetries



Picture on left has no
crossings
Picture on right better
reveals symmetries
There is no picture
revealing all
symmetries but
without edge
crossings
Trading crossings and upwardness



Picture on left has
all arrows upward
Picture on right has
no crossings
There is no picture
with all arrows
upwards but without
edge crossings
Minimizing crossings
• Crossing # of a planar graph = minimum #
crossings over all drawings.
• Rectilinear crossing # = same, but only
allowing straight-line drawings
Drawings with small area for
arbitrary graphs
Theorem: There exist constants c1 and c2 such
that every graph on n vertices has a straight-line
drawing with
c1 n < Area < c2 n log n
Note edges may cross in these drawings
Open problem: narrow this gap?
Balancing area and
aspect ratio in rooted trees
Theorem: There exists constants c1 and c2
such that every rooted tree on n vertices
has an upward straight-line drawing with
area < c1 n log n, and
aspect ratio < c2 n / log n
Balancing area and angular
resolution in all graphs

If we require that no angles are small, the area
of the layout may grow large very rapidly

Theorem: If we require all angles to be of size at
least r, then there are examples of graphs on n
vertices where the area needed is at least
c1 (c2)rn
Balancing area and angular resolution
in planar graphs

Theorem: There are c1 and c2 such that
every planar graph on n vertices with
degree d has a straight-line drawing with
area < c1 d3 n and
angular resolution > c2 / d
Open problem




Consider straight-line drawings of general
(nonplanar) graphs, so crossings are allowed
The best known lower bound applicable to all
graphs is
angular resolution > c1 / d2
There are graphs for which
angular resolution < c2 (log d) / d2
Which bound is the “right” answer?
Section 2: Polyline drawings

Now allow lines to have bends
Measures of good drawings
1.
2.
3.
All of the previous (area, aspect ratio, etc.)
Total number of bends
Maximum number of bends per edge
Area of polyline drawings

There are constants c1 and c2 such that
any tree of order n has a polyline drawing
with
c1 n < area < c2 n

There are constants c3 and c4 such that
any graph of order n has a polyline
drawing with
c3 n2 < area < c4 n2
Balancing area and angular resolution
in polyline drawings
Theorem: There are c3 and c4 such that
every graph on n vertices with degree d
has a polyline non-crossing drawing with
area < c3 n2 and
angular resolution > c4 / d
Section 3: Orthogonal drawings


Now require all line segments to be horizontal or vertical.
Note the maximum degree of a vertex is at most 4, or
else there is no hope of such drawings
Edges may cross (non-planar graphs) or not cross. If G is
planar, allowing edges to cross may make a difference
Area in orthogonal drawings
Theorem: any planar graph with maximum
degree 4 has a drawing without crossings
with
c1 n2 < area < c2 n2
(i.e., even if we require lines to always go
horizontal or vertical, the bounds on the
area remain the same)

Orthogonal drawings with crossings

Let G be a planar graphs, but let the
orthogonal drawings have crossings. Then
c_1 n log n <= area <= c_2 n (log n)^2
1.
If allow crossings, area needed for planar
graphs drops dramatically
Gap between existential lower bound and
universal upper bound
2.
Aspect ratio and orthogonal drawings

Theorem: Every binary tree on n vertices
has an upward planar orthogonal grid with
area <= c_1 n log (log n) and
aspect ratio <= c_2 n log (log n) / (log n)^2
(As they say in England, mind the gap)
Section 4: Planar orthogonal drawings
(subdivided subgraphs of the grid)




Now restrict attention to
graphs whose vertices lie
on the grid, and whose
edges are (disjoint) paths
through the grid
Again, any such graph
has maximum degree at
most 4
Allow bends, but may
bound these as part of a
tradeoff
Now very close to VLSI
concerns
The general problems are tough

Theorem: It is NP-hard to determine the
minimum area needed in a grid that contains G
as a subdivided subgraph

Theorem: It is NP-hard to determine the
minimum number of bends in G represented as
a subdivided subgraph of the grid
Both results in J.A. Storer, Networks 14-2 (1984)181-212
Bounding the number of bends
Theorem: Let G be a planar graph of order n
and maximum degree at most 4. Suppose
that G has at most 2 bends per edge and
that crossings are allowed. Then the
maximum number of bends is between n
and 2n+4
Trading total number of bends
and bends per edge
Given an embedding of a graph, how large of a
grid do we need to represent it?




If 3 bends per edge
2n-2 <= total # bends <= (12/5)n + 2
If 3 bends per edge and 2-connected
2n-2 <= total # bends <= 2n + 2
If 2 bends per edge and 3-connected
(4/3)(n-1)+2 <= total # bends <= (3/2)n + 4
If 1 bend per edge and cubic
n/2+1 <= total # bends <= (12/5)n + 2
Finding grid subgraphs
with few bends or with small area
There are a lot of papers about how to find grid
embeddings with a few number of bends

Can find grid embedding with at most 4 bends
per edge in linear time

Can find grid embedding with W+H = n and
aspect ratio 3/2 in linear time
Graphs of larger degree
If the degree of G is 5 or more, no
subdivision is a subgraph of a grid.
Instead allow vertices of G to be
represented by large connected
sets of grid vertices.
Equivalent to G being a minor of a
planar grid
Theorem: Every planar embedding is
a minor of some n x n grid
Section 5: 3-dimensional Grids
Application to circuit layouts


The rapid rate of progress in VLSI
technology suggests that multi-layered
chips and packages will be commonplace
in the not too distant future
President of TI predicts the production of
3D chips by the end of the decade
(unfortunately, quote was from 1982)
More practical applications





In “real life” circuits are constructed in layers
Each layer contains either horizontal or vertical
lines, but not both
Connections between layers are through cuts
called vias
The number of layers is limited to about 6 or so
Equivalent to finding a subdivision as a
subgraph of a 3-D grid n x m x 3
Where are the processors?




“one active layer” model requires all vertices to
be on the bottom layer
“many active layer” model allows vertices on any
level
Latter is more flexible, but has higher production
costs
“[Surprisingly…] many-active-layer layouts are
little or no more efficient than one-active-layer
layouts when the number of layers is relatively
small” (Leighton and Rosenberg)
Basics for the 3D problem
Any subgraph of a 3-D grids must have
maximum degree 6. Let n be the number of
vertices in G. Any graph embeds in 3-space,
so it is a subgraph of some 3D-grid.
Minimize volume? Optimize other parameters
such as bends?
Existential results




Any graph embeds in a 3-D grid of volume
c1 n3/2 with at most 6 bends per edge
(volume is known to be optimal)
Volume up to n5/2 if 5 bends per edge
Volume up to n3 if 4 or 3 bends per edge
Volume n2 if maximum degree 4
Balancing volume and wire run
Theorem: if the maximum degree of the
graph G is at most 6, then there are
constants c1 and c2 such that G is a minor
of a 3D-grid with
Volume < c1 n3/2
Max wire run < c2 n1/2
Gains over 2D layouts

Suppose that we have a 2D-layout of a
circuit (subgraph of a 2D-grid) with area A
and maximum wire run R

Theorem: There exists a 3D-layout of
height H with
Vol = A/H
Max wire run = R/H
Section 6: minors of 3D grids

Theorem: Any graph on n vertices is a
minor of an n x n x 2 grid (slightly more
subtle argument shows (n-1) x (n-1) x 2
grid)

Define the grid width of G, gw(G), as the
smallest n such that G is a minor of the
nxnx2
Determining the grid-width


Lower bounds: show that G contains a subgraph
that requires a large grid-width. Candidates are
“brambles” that in a sense are highly connected.
These are related to “cross-bar switches” that
annoy real-life VLSI designs
Upper bounds: show that G has a structure
much like a tree, which allows recursive
constructions of grid embeddings = layouts
Future topics



Determine the grid-width of classes of
graph
Find efficient algorithms for VLSI layouts:
use bifurcation (recursive) algorithms and
results from eigenspaces
Relate to isoparametric properties
The End