Collective Additive
Tree Spanners of
Homogeneously Orderable
Graphs
F.F. Dragan, C. Yan and Y. Xiang
Kent State University, USA
Well-known
Tree t -Spanner Problem
Given unweighted undirected graph G=(V,E) and integers t,r.
Does G admit a spanning tree T =(V,E’) such that
 u, v V , dist T (v, u)  t  dist G (v, u) (a multiplicative tree t-spanner of G)
or
u, v V , dist T (u, v)  dist G (u, v)  r (an additive tree r-spanner of G)?
G
LATIN 2008, Brazil
T
multiplicative tree 4-,
additive tree 3-spanner
of G
Feodor F. Dragan, Kent State University
Some known results for the tree
spanner problem
(mostly multiplicative case)
•
general graphs [CC’95]
– t  4 is NP-complete. (t=3 is still open, t  2 is P)
•
approximation algorithm for general graphs [EP’04]
– O(logn) approximation algorithm
• chordal graphs [BDLL’02]
– t  4 is NP-complete. (t=3 is still open.)
• planar graphs [FK’01]
– t 4 is NP-complete. (t=3 is polynomial time solvable.)
• easy to construct for some special families of graphs.
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Well-known
Sparse t -Spanner Problem
Given unweighted undirected graph G=(V,E) and integers t,m,r.
Does G admit a spanning graph H =(V,E’) with |E’|  m s.t.
 u, v  V , dist H (v, u)  t  dist G (v, u) (a multiplicative t-spanner of G)
or
u, v V , dist H (u, v)  dist G (u, v)  r
G
(an additive r-spanner of G)?
H
multiplicative 2- and additive 1-spanner of G
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Some known results for sparse
spanner problems
•
general graphs
– t, m1 is NP-complete [PS’89]
– multiplicative (2k-1)-spanner with n1+1/k edges [TZ’01, BS’03]
• n-vertex chordal graphs (multiplicative case) [PS’89]
(G is chordal if it has no chordless cycles of length >3)
– multiplicative 3-spanner with O(n logn) edges
– multiplicative 5-spanner with 2n-2 edges
• n-vertex c-chordal graphs (additive case) [CDY’03, DYL’04]
(G is c-chordal if it has no chordless cycles of length >c)
– additive (c+1)-spanner with 2n-2 edges
– additive (2 c/2 )-spanner with n log n edges
 For chordal graphs: additive 4-spanner with 2n-2 edges, additive 2spanner with n log n edges
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Collective Additive Tree
r -Spanners Problem (a middle way)
Given unweighted undirected graph G=(V,E) and integers , r.
Does G admit a system of  collective additive tree r-spanners
{T1, T2…, T}
such that
 u, v  V and 0  i   , dist Ti (v, u )  dist G (v, u )  r
(a system of  collective additive tree r-spanners of G )?
surplus
,
collective multiplicative
tree t-spanners
can be defined similarly
2 collective additive tree 2-spanners
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Collective Additive Tree
r -Spanners Problem
Given unweighted undirected graph G=(V,E) and integers , r.
Does G admit a system of  collective additive tree r-spanners
{T1, T2…, T}
such that
 u, v  V and 0  i   , dist Ti (v, u )  dist G (v, u )  r
(a system of  collective additive tree r-spanners of G )?
,
2 collective additive tree 2-spanners
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Collective Additive Tree
r -Spanners Problem
Given unweighted undirected graph G=(V,E) and integers , r.
Does G admit a system of  collective additive tree r-spanners
{T1, T2…, T}
such that
 u, v  V and 0  i   , dist Ti (v, u )  dist G (v, u )  r
(a system of  collective additive tree r-spanners of G )?
,
2 collective additive tree 2-spanners
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Collective Additive Tree
r -Spanners Problem
Given unweighted undirected graph G=(V,E) and integers , r.
Does G admit a system of  collective additive tree r-spanners
{T1, T2…, T}
such that
 u, v  V and 0  i   , dist Ti (v, u )  dist G (v, u )  r
(a system of  collective additive tree r-spanners of G )?
,
2 collective additive tree 2-spanners
LATIN 2008, Brazil
,
2 collective additive
tree 0-spanners or
multiplicative tree
1-spanners
Feodor F. Dragan, Kent State University
Applications of Collective Tree
Spanners
representing complicated graph-distances
with few tree-distances
• message routing in networks
Efficient routing schemes are known for trees
but not for general graphs. For any two nodes,
we can route the message between them in one
of the trees which approximates the distance
between them.
- ( log2n/ log log n)-bit labels,
- O( ) initiation, O(1) decision
• solution for sparse t-spanner
problem
If a graph admits a system of  collective additive
tree r-spanners, then the graph admits a sparse
additive r-spanner with at most (n-1) edges,
where n is the number of nodes.
LATIN 2008, Brazil
2 collective tree 2spanners for G
Feodor F. Dragan, Kent State University
Previous results on the collective
tree spanners problem
(Dragan, Yan, Lomonosov [SWAT’04])
(Corneil, Dragan, Köhler, Yan [WG’05])
• chordal graphs, chordal bipartite graphs
– log n collective additive tree 2-spanners in polynomial time
– Ώ(n1/2) or Ώ(n) trees necessary to get +1
– no constant number of trees guaranties +2 (+3)
• circular-arc graphs
– 2 collective additive tree 2-spanners in polynomial time
• c-chordal graphs
– log n collective additive tree 2 c/2 -spanners in polynomial time
• interval graphs
– log n collective additive tree 1-spanners in polynomial time
– no constant number of trees guaranties +1
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Previous results on the collective
tree spanners problem
(Dragan, Yan, Corneil [WG’04])
• AT-free graphs
– include: interval, permutation, trapezoid, co-comparability
– 2 collective additive tree 2-spanners in linear time
– an additive tree 3-spanner in linear time (before)
• graphs with a dominating shortest path
– an additive tree 4-spanner in polynomial time (before)
– 2 collective additive tree 3-spanners in polynomial time
– 5 collective additive tree 2-spanners in polynomial time
• graphs with asteroidal number an(G)=k
– k(k-1)/2 collective additive tree 4-spanners in polynomial time
– k(k-1) collective additive tree 3-spanners in polynomial time
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Previous results on the collective
tree spanners problem
(Gupta, Kumar,Rastogi [SICOMP’05])
• the only paper (before) on collective multiplicative tree
spanners in weighted planar graphs
• any weighted planar graph admits a system of O(log n)
collective multiplicative tree 3-spanners
• they are called there the tree-covers.
• it follows from (Corneil, Dragan, Köhler, Yan [WG’05])
that
– no constant number of trees guaranties +c (for any
constant c)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Some results on collective additive tree
spanners of weighted graphs with
bounded parameters
(Dragan, Yan [ISAAC’04])
Graph class
planar

r
O( n )
0
with genus g
O( gn )
0
W/o an h-vertex minor
O( h 3 n )
k log 2 n
0
tw(G) ≤ k-1
0
cw(G) ≤ k
k log 3 / 2 n
2w
c-chordal
next
slide

( n log log n / log 2 n)
to get +0
No constant number of
trees guaranties +r for
any constant r even for
outer-planar graphs

 (n)
to get +1
• w is the length of a longest edge in G
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Some results on collective additive tree
spanners of weighted c-chordal graphs
(Dragan, Yan [ISAAC’04])
Graph class

r
4-chordal
6 log 2 n
c
2  w
2
c 
2(    1) w
3
c  2
2
w

 3 
2w
weakly chordal
4 log 2 n
2w
log 2 n
c-chordal
(c>4)
4 log 2 n
5 log 2 n
LATIN 2008, Brazil
No constant number of
trees guaranties +r for
any constant r even for
weakly chordal graphs
Feodor F. Dragan, Kent State University
(This paper)
Homogeneously orderable Graphs
• A graph G is homogeneously orderable if G has an h-extremal
ordering [Brandstädt et.al.’95].
• Equivalently: A graph G is homogeneously orderable if and only if
the graph L(D(G)) of G is chordal and each maximal two-set of G is
join-split.
D(G)  {N k [v] : v V , k  Ν}
– L(D(G)) is the intersection graph of D(G).
– Two-set is a set of vertices at pair-wise distance ≤ 2.
join-split
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Hierarchy of Homogeneously
Orderable Graphs (HOGs)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Our results on Collective additive tree
spanners of n-vertex homogeneously
orderable graphs
additive stretch
factor
3
upper bound on
number of trees
1
lower bound on
number of trees
1
2
log 2 n
 c
1
n 1
(n)
0
n 1
(n)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
To get +1 one needs (n) trees
additive stretch
factor
upper bound on
number of trees
lower bound on
number of trees
3
1
1
2
log 2 n
 c
1
n 1
(n)
0
n 1
(n)
trivial
n-1
BFStrees
LATIN 2008, Brazil
Take n by n complete bipartite graph

nn
trees
n 1
• square is a clique chordal
• join-split
Feodor F. Dragan, Kent State University
Our results on Collective additive tree
spanners of n-vertex homogeneously
orderable graphs
additive stretch
factor
3
upper bound on
number of trees
1
lower bound on
number of trees
1
2
log 2 n
 c
1
n 1
(n)
0
n 1
(n)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Layering and Clustering
• The projection of each
cluster is a two-set.
• The connected
components of
projections are two-sets
and have a common
neighbor down.
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Additive Tree 3-spanner
Linear Time
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Our results on Collective additive tree
spanners of n-vertex homogeneously
orderable graphs
additive stretch
factor
3
upper bound on
number of trees
1
lower bound on
number of trees
1
2
log 2 n
 c
1
n 1
(n)
0
n 1
(n)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
H and H2
HOG
Chordal
17
15
16
13
16
13
14
14
12
12
25
25
11
23
11
23
1
21
1
21
19
2
3
24
17
15
7
8
24
22
20
18
5
4
6
LATIN 2008, Brazil
9
19
10
3
2
7
8
5
4
9
10
22
20
18
6
Feodor F. Dragan, Kent State University
H2 (chordal graph) and its
balanced decomposition tree
17
15
1, 2, 3, 4, 5, 6, 7, 9, 11, 12
16
13
14
12
25
11
23
8, 10
1
21
24
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24
19
3
2
7
8
5
4
9
10
22
20
18
25
6
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Constructing Local Spanning Trees
for H
• For each layer of the decomposition tree, construct local spanning trees
of H (shortest path trees in the subgraph).
•Here, we use the second layer for illustration.
17
15
1, 2, 3, 4, 5, 6, 7, 9, 11, 12
16
13
14
12
25
11
23
8, 10
1
21
24
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24
19
3
2
7
8
5
4
9
10
22
20
18
25
6
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Local Additive Tree 2-spanner
Theorem: d T ( x, y )  d H ( x, y )  2 must hold
i
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Our results on Collective additive tree
spanners of n-vertex homogeneously
orderable graphs
additive stretch
factor
3
upper bound on
number of trees
1
lower bound on
number of trees
1
2
log 2 n
 c
1
n 1
(n)
0
n 1
(n)
One tree cannot
give +2
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
No constant number d of trees can
guarantee additive stretch factor +2
gadget
root
k  f (d )
2k
2k
2k
clique
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
No constant number d of trees can
guarantee additive stretch factor +2
Tree of
gadgets
…
…
…
The depth is a
function of d
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Open questions and future plans
• Given a graph G=(V, E) and two integers  and r, what is
the complexity of finding a system of  collective additive
(multiplicative) tree r-spanner for G? (Clearly, for most 
and r, it is an NP-complete problem.)
• Find better trade-offs between  and r for planar graphs,
genus g graphs and graphs w/o an h-minor.
• We may consider some other graph classes. What’s the
optimal  for each r?
• More applications of collective tree spanner…
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University
Thank You
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University