Collective Tree Spanners and Routing in AT-free Related Graphs F.F. Dragan, C. Yan, D. Corneil Kent State University University of Toronto

G multiplicative tree 4- and additive tree 3- spanner of G Tree t -Spanner Problem Well-known Tree t -Spanner Problem Given unweighted undirected graph G=(V,E) and integers t, s. Does G admit a spanning tree T =(V,E’) such that (a multiplicative tree t-spanner of G) or (an additive tree s-spanner of G)?

G multiplicative 2- and additive 1-spanner of G Sparse t -Spanner Problem Well-known Sparse t -Spanner Problem Given unweighted undirected graph G=(V,E) and integers t, m. Does G admit a spanning graph H =(V,E’) with |E’|  m such that (a multiplicative t-spanner of G) or (an additive s-spanner of G)?

Collective Additive Tree r -Spanners Problem New Collective Additive Tree r -Spanners Problem Given unweighted undirected graph G=(V,E) and integers m, r. T m Does G admit a system of m spanning trees {T 1,T 2,…, T m } such that (a system of m collective additive tree r-spanners of G)? 2 collective additive tree 2-spanners

Applications of Collective Tree Spanners message routing in networks Efficient routing scheme is known for trees but is hard 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. solution for sparse s-spanner problem If a graph admits a system of m collective additive tree s-spanners, then the graph will have an additive graph s-spanner with at most m(n-1) edges, where n is the number of nodes. 2 collective additive tree 2-spanners of G

Some known results for the tree spanner problem 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.) AT-free graphs and their subclasses – 1 additive tree 3-spanner [Pr’99, PKLMW’03] –a permutation graph admits a multiplicative tree 3-spanner [MVP’96] –an interval graph admits an additive tree 2-spanner (mostly multiplicative case)

Some known results for the sparse spanner problem general graphs [PS’89] –t, m  1 is NP-complete 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 k-chordal graphs (additive case) [CDY’03,DYL’04] (G is k-chordal if it has no chordless cycles of length >k) –additive 2  k/2  -spanner with O(n logn) edges –additive (k+1)-spanner with 2n-2 edges

Previous results on the collective tree spanners problem Previous results on the collective tree spanners problem [DYL’2004] n-vertex chordal graphs –log n collective additive tree 2-spanners n-vertex chordal bipartite graphs –log n collective additive tree 2-spanners n-vertex k-chordal graphs –log n collective additive 2  k/2  -spanners n-vertex planar graphs –√n log n collective additive tree 0-spanners n-vertex graphs of bounded tree width tw –O(log n)= tw x log n collective additive tree 0-spanners

New results on the collective tree spanners problem on AT- free related graphs n-vertex AT-free graphs –2 collective additive tree 2-spanners n-vertex permutation graphs –1 additive tree 2-spanner n-vertex DSP-graphs –2 collective additive tree 3-spanners –5 collective additive tree 2-spanners n-vertex graphs of bounded asteroidal number an –an(an-1)/2 collective additive tree 4-spanners –an(an-1) collective additive tree 3-spanners

Permutation, Trapezoid and Co-comparability Graphs Thm: Every permutation graph admits an additive tree 2-spanner, constructable in linear time.  admits a multiplicative tree 3-spanner [MVP’96] Observ: There are bipartite permutation graphs on 2n vertices for which any system of collective additive tree 1-spanners will need to have at least  (n) spanning trees.  the result of the previous Thm cannot be improved Observ: There are trapezoid graphs which do not admit any additive tree 2-spanners.  disprove of the conjecture from [PKLMW’03] that any co- comparability graph admits an additive tree 2-spanner

Trapezoid Graphs Observ: There are trapezoid graphs which do not admit any additive tree 2-spanners

Any AT-free graph G admits an additive tree 3-spanner [PKLMW’03] Thm: Any AT-free graph G admits a system of 2 collective additive tree 2-spanners which can be constructed in linear time. To get +2, one needs at least 2 spanning trees To get +1, one needs at least  (n) spanning trees Collective Tree Spanners For AT-free Graphs an AT-free graph with its backbone

2 collective additive tree 2-spanners of G Collective Tree Spanners For AT-free Graphs caterpillar-tree cactus-tree

Any DSP-graph admits an additive tree 4-spanner [PKLMW’03] Thm: Any DSP-graph admits a system of 2 collective additive tree 3- spanners and a system of 5 collective additive tree 2-spanners. Collective Tree Spanners for DSP Graphs an DSP graph G with its dominating path

2 collective additive tree 3-spanners of G 2 Collective Tree 3- Spanners for DSP Graphs

5 collective additive tree 2-spanners. 5 Collective Tree 2- Spanners for DSP Graphs

Any graph with asteroidal number an(G) admits an additive tree (3an(G) -1)-spanner [PKLMW’03] Thm: Any graph with asteroidal number an(G) admits a system of an(G)(an(G)-1)/2 collective additive tree 4-spanners and a system of an(G)(an(G)-1) collective additive tree 3-spanners. Graphs With Bounded Asteroidal Number A graph with its dominating target

Collective Trees for Graphs with Bounded Asteroidal Number (collective additive tree 4-spanners)

Routing Schemes for AT-free Graphs [FG’01,TZ’01] For family of n-node trees there is a routing labeling scheme with labels of size O(log²n/loglogn)- bits per node and constant time routing decision.  For AT-free graphs, O(log²n/loglogn)-bits per node, constant time routing decision and deviation at most 2. Thm: Every AT-free graph of diameter D=diam(G) and of maximum vertex degree  admits a (3log 2 D+6log 2  +O(1))- bit routing labeling scheme of deviation at most 2. Moreover, the scheme is computable in linear time, and the routing decision is made in constant time per vertex.

Future Plans Find best possible trade-off between number of trees and additive stretch factor for planar graphs (currently: √n log n collective additive tree 0-spanners). Consider the collective additive tree spanners problem for other structured graph families. Complexity of the collective additive tree spanners problem for different m and r on general graphs and special graph classes. More applications of …

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