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Additive Spanners for k -Chordal Graphs

Additive Spanners for k -Chordal Graphs. V. D. Chepoi, F.F. Dragan , C. Yan. University Aix-Marseille II, France Kent State University, Ohio, USA. Sparse t -Spanner Problem. Given unweighted undirected graph G=(V,E) and integers t, m.

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Additive Spanners for k -Chordal Graphs

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  1. Additive Spanners for k-Chordal Graphs V. D. Chepoi,F.F. Dragan, C. Yan University Aix-Marseille II, France Kent State University, Ohio, USA

  2. Sparse t -SpannerProblem 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 t-spanner of G)? Gmultiplicative 2- and additive 1-spanner of G

  3. Sparse t -SpannerProblem 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 t-spanner of G)? Gmultiplicative 2- and additive 1-spanner of G

  4. Applications in distributed systems and communication networks • synchronizers in parallel systems • Close relationship were established between the quality of spanners for a given undirected graph (in terms of the stretch factort and the number of edges |E’|), and the time and communication complexities of any synchronizer for the network based on this graph • topology for message routing • efficient routing schemes can use only the edges of the spanner G 2-spanner for G

  5. Applications in distributed systems and communication networks • synchronizers in parallel systems • Close relationship were established between the quality of spanners for a given undirected graph (in terms of the stretch factort and the number of edges |E’|), and the time and communication complexities of any synchronizer for the network based on this graph • topology for message routing • efficient routing schemes can use only the edges of the spanner G 2-spanner for G

  6. Some Known Results (multiplicative case) • general graphs [Peleg&Schaffer’89] • given a graph G=(V, E) and two integers t, m1, whether G has a t-spanner with m or fewer edges, is NP-complete • chordal graphs [Peleg&Schaffer’89] G is chordal if it has no chordless cycles of length >3 • every n-vertex chordal graph G=(V, E) admits a 2-spanner with O(n1.5)edges • there exist (infinitely many) n-vertex chordal graphs G=(V, E) for which every 2-spanner requires (n1.5) edges • every n-vertex chordal graph G=(V, E) admits a 3-spanner with O(n logn) edges • every n-vertex chordal graph G=(V, E) admits a 5-spanner with at most 2n-2edges

  7. Some Known Results (multiplicative case) • general graphs [Peleg&Schaffer’89] • given a graph G=(V, E) and two integers t, m1, whether G has a t-spanner with m or fewer edges, is NP-complete • chordal graphs [Peleg&Schaffer’89] G is chordal if it has no chordless cycles of length >3 • every n-vertex chordal graph G=(V, E) admits a 3-spanner with O(n logn) edges • every n-vertex chordal graph G=(V, E) admits a 5-spanner with at most 2n-2edges • tree spanner [BDLL’2002] • given a chordal graph G=(V, E) and an integer t>3, whether G has a t-spanner with n-1 edges (tree t-spanner), is NP-complete

  8. Some Known Results (multiplicative case) • general graphs [Peleg&Schaffer’89] • given a graph G=(V, E) and two integers t, m1, whether G has a t-spanner with m or fewer edges, is NP-complete • chordal graphs [Peleg&Schaffer’89] G is chordal if it has no chordless cycles of length >3 • every n-vertex chordal graph G=(V, E) admits a 3-spanner with O(n logn) edges • every n-vertex chordal graph G=(V, E) admits a 5-spanner with at most 2n-2edges 2-appr. algorithm for any t  5 • tree spanner [BDLL’2002] • given a chordal graph G=(V, E) and an integer t>3, whether G has a t-spanner with n-1 edges (tree t-spanner), is NP-complete

  9. This Talk • From multiplicative to additive • every chordal graph admits an additive 4-spanner with at most 2n-2 edges which can be constructed in linear time • every chordal graph admits an additive 3-spanner with O(n logn) edges which can be constructed in polynomial time • Extension to k-chordal graphs G is k-chordal if it has no chordless cycle of length >k • Every k-chordal graph admits an additive (k+1)-spanner with at most 2n-2 edges which can be constructed in O(nk+m) • Better bounds for subclasses of 4-chordal graphs • Every HH-free graph (or chordal bipartite graph) admits an additive 4-spanner with at most 2n-2 edges which can be constructed in linear time • Note that any additive t-spanner is a multiplicative (t+1)-spanner

  10. Method:Constructing Additive 4-Spanner • Given a chordal graphG=(V, E) and an arbitrary vertex u u

  11. BFS-Ordering and BFS-Tree up-phase • We start from u and construct a BFS tree. The red edges are tree edges. • First layer. 16 17 15 u 18

  12. BFS-Ordering and BFS-Tree up-phase • Second layer 13 14 11 12 16 17 15 u 18

  13. BFS-Ordering and BFS-Tree up-phase • Third layer 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  14. BFS-Ordering and BFS-Tree up-phase • Fourth Layer 4 3 2 1 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  15. Constructing Spanner down-phase • Start from the last layer. For vertices of each connected component in the layer create a star for the fathers. 4 3 2 1 connected components 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  16. Constructing Spanner down-phase • Third Layer 4 3 2 1 connected components 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  17. Constructing Spanner down-phase • Second layer 4 3 2 1 10 9 8 7 6 5 13 14 11 12 connected components 16 17 15 u 18

  18. Final Spanner • The final spanner is showed in red 4 3 2 1 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  19. Final Spanner • The final spanner is showed in red 4 vs 5 4 3 2 1 1 vs 3 10 9 8 7 6 5 13 14 11 12 16 17 15 u 18

  20. Analysis Of The Algorithm • Given a chordal graph G=(V, E), we produce a spanning graph H=(V,E’) such that • H is an additive 4-spanner of G • H contains at most 2n-2edges • H can be constructed in O(n+m) time

  21. Analysis Of The Algorithm • Given a chordal graph G=(V, E), we produce a spanning graph H=(V,E’) such that • H is an additive 4-spanner of G • H contains at most 2n-2edges • H can be constructed in O(n+m) time y x Layer i Layer i-1 c u

  22. Constructing Additive 3-Spanner • Gis a chordal graph with n vertices and with a BFS ordering (started at u) • Take all the edges of the additive 4-spanner • in each connected component S induced by layer r, we run the algorithm presented in [Peleg&Schaffer’89], to construct a multiplicative 3-spanner for S 4 3 2 1 10 9 8 7 6 5 13 14 11 12 16 17 15 18

  23. Constructing Additive 3-Spanner • Gis a chordal graph with n vertices and with a BFS ordering (started at u) • Take all the edges of the additive 4-spanner • in each connected component S induced by layer r, we run the algorithm presented in [Peleg&Schaffer’89], to construct a multiplicative 3-spanner for S 4 3 2 1 10 9 8 7 6 5 13 14 11 12 16 17 15 18

  24. Analysis Of The Algorithm • Given a chordal graph G=(V, E)with n vertices and medges, we produce a spanning graph H=(V,E’) such that • H is an additive 3-spanner of G • H contains O(n logn) edges • H can be constructed in polynomial time

  25. Method:Constructing Additive (k+1)-Spanner Given a k-chordal graphG=(V, E) and an arbitrary vertex u u

  26. BFS-Ordering and BFS-Tree up-phase • We start from u and construct a BFS tree. The red edges are tree edges. 2 1 6 3 4 5 7 9 8 10 11 12 u

  27. Constructing Spanner down-phase • Start from the last layer. For vertices of each component,choose • the smallest one. Then try to connect others to it or its ancestor. 2 1 6 3 3 4 5 a component on layer 3 7 9 8 10 11 12 u

  28. Constructing Spanner down-phase • Start from the last layer. For vertices of each component,choose • the smallest one. Then try to connect others to it or its ancestor. 2 1 6 3 3 4 5 5 a component on layer 3 7 7 9 9 8 edge used to connect 3 and 5 10 11 12 u

  29. Constructing Spanner down-phase • Start from the last layer. For vertices of each component,choose • the smallest one. Then try to connect others to it or its ancestor. 2 1 6 3 4 5 a component on layer 3 7 9 8 edge used to connect 3 and 5 10 11 12 u

  30. Final Spanner • Final spanner is shown in red. 2 1 6 3 4 5 7 9 8 10 11 12 u

  31. Analysis Of The Algorithm • G is k-chordal if it has no chordless cycles of length >k • The spanner constructed by the above algorithm has the following properties • It is an additive (k+1)-spanner • It contains at most 2n-2edges • It can be constructed in O(k·n+m)time

  32. Open questions and future directions • Can these ideas be applied to other graph families to obtain good sparse additive spanners? • Can one get a constant approximation for the additive 3-spanner problem on chordal graphs? • so far, • only a log-approximation for t=3 • 2-approximation for t>3 • What aboutt=2 (additive)? • so far, (from Peleg&Schaffer’89) • a log-approximation for multiplicative3-spanner • for t=1, the lower bound is (n1.5) edges (as multiplicative 2-spanner)

  33. Thank You

  34. Layering • Given a graph G=(V, E) and an arbitrary vertex uV, the sphere of u is defined as • The ball of radius centered at u is defined as • A layeringof G with respect to some vertex u is a partition • of V into the spheres

  35. BFS Ordering • G=(V, E)is a graph with n vertices • InBreadth-First-Search (BFS), started at vertex u, we number the vertices fromn to 1 as follows • uis numbered by nand is put on an initially empty queue • a vertex v is repeatedly removed from the head of the queue and the neighbors of vwhich are still unnumbered are consequently numbered and placed onto the queue • we call v the father of those vertices which are placed onto the queue when v is removed from the queue. We use f(v) to denote the father of v • Anordering generated by BFS is calledBFS-ordering

  36. Layering and BFS-ordering, an example • The vertices are numbered in BFS-ordering and the BFS tree is shown in red Layer 3 1 2 6 Layer 2 5 4 3 7 Layer 1 8 9 Layer 0 10

  37. Constructing Additive 4-Spanner • Method of constructing spanner H=(V, E’) • let be arbitrary vertex of G and , we use Breadth-First-Search (BFS) rooted at to label all the vertices of G. • start from the layer of , for each vertex we add into • start from the layer and for each connected component induced by we find its projection on layer and make it star and put all the edges in the star into

  38. Example • The following is an example. The think red lines consists • the spanners for the chordal graph 4 3 2 1 10 9 8 7 6 5 13 14 11 12 16 17 15 18

  39. Definitions and Symbols for k-chordal graph • G is k-chordal if it has no chordless cycles of length>k • Let u be an arbitrary vertex of G. We define a graph with the lth sphere as a vertex set. Two vertices are adjacent in if and only if they can be connected by a path outside the ball . We use to denote all the connected component of • Also we define

  40. ConstructingSpanners • G is k-chordal if it has no chordless cycles of length>k • Method of constructing a spanner H=(V, E’) • for each vertex , we add into E’ • for each connected component we identify a vertex such that is the minimum in BFS-ordering among all vertices in • check if then we add to • check if then we add to • If none of the above is true, we let and repeat 3 and 4

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