1. Collective Additive Tree Spanners of Homogeneously Orderable Graphs
F.F. Dragan, C. Yan and Y. Xiang
Kent State University, USA

2. 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
(a multiplicative tree t-spanner of G) or
(an additive tree r-spanner of G)? G
multiplicative tree 4-, additive tree 3-spanner of G T
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

3. 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

4. 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.
(a multiplicative t-spanner of G) or
(an additive r-spanner of G)? H G
multiplicative 2- and additive 1-spanner of G
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

5. 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 2-spanner with n log n edges
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

6. 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
(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

7. 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
(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

8. 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
(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

9. 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
(a system of  collective additive tree r-spanners of G )? , ,
2 collective additive tree 0-spanners or multiplicative tree 1-spanners
2 collective additive tree 2-spanners
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

10. 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.
2 collective tree 2-spanners for G
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

11. 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

12. 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

13. 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

14. 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  r
planar
with genus g
W/o an h-vertex minor
tw(G) ≤ k-1
cw(G) ≤ k 2w
c-chordal
next
slide
to get +0
No constant number of trees guaranties +r for any constant r even for outer-planar graphs
to get +1
w is the length of a longest edge in G
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

15. 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
c-chordal
(c>4)
4-chordal 2w
weakly chordal
No constant number of trees guaranties +r for any constant r even for weakly chordal graphs
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

16. (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.
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

17. Hierarchy of Homogeneously Orderable Graphs (HOGs)
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

18. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs
additive stretch factor
upper bound on number of trees
lower bound on number of trees 3 1 2
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

19. To get +1 one needs trees additive stretch factor
upper bound on number of trees
lower bound on number of trees 3 1 2
trivial
square is a clique chordal
join-split
Take n by n complete bipartite graph
n-1 BFS-trees
trees
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

20. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs
additive stretch factor
upper bound on number of trees
lower bound on number of trees 3 1 2
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

21. 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

22. Additive Tree 3-spanner
Linear Time
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

23. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs
additive stretch factor
upper bound on number of trees
lower bound on number of trees 3 1 2
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

24. Feodor F. Dragan, Kent State University
H and H2
HOG
Chordal 17 15 17 16 13 14 12 11 1 2 3 4 5 6 7 9 8 10 21 20 19 18 22 23 24 25 15 13 16 14 12 25 11 23 1 21 19 3 2 7 8 24 22 20 18 5 4 9 10 6
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

25. H2 (chordal graph) and its balanced decomposition tree17
1, 2, 3, 4, 5, 6, 7, 9, 11, 12 15 13 16 14 12 25 11 23
8, 10
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24 1 21 19 3 2 7 8 24 22 20 18 5 4 9 10 25 6
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

26. 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 13 16 14 12 25 11 23
8, 10
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24 1 21 19 3 2 7 8 24 22 20 18 5 4 9 10 25 6
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

27. Local Additive Tree 2-spanner
Theorem:
must hold
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

28. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs
additive stretch factor
upper bound on number of trees
lower bound on number of trees 3 1 2
One tree cannot give +2
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

29. No constant number d of trees can guarantee additive stretch factor +2
root
gadget
clique
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University

30. 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

31. 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

32. Feodor F. Dragan, Kent State University
Thank You
LATIN 2008, Brazil
Feodor F. Dragan, Kent State University