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Collective Additive Tree Spanners of Homogeneously Orderable Graphs

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.

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Collective Additive Tree Spanners of Homogeneously Orderable Graphs

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

  2. Well-knownTree 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)? multiplicative tree 4-, additive tree 3-spanner of G G T 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. Feodor F. Dragan, Kent State University

  4. Well-knownSparse 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 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/kedges [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-2edges • 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-2edges • additive (2 c/2)-spanner with n log nedges  For chordal graphs: additive 4-spanner with 2n-2edges, additive 2-spanner with n log nedges 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 2collective additive tree 2-spanners 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)? , 2collective additive tree 2-spanners 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)? , 2collective additive tree 2-spanners 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)? , , 2collective additive tree 0-spanners or multiplicative tree 1-spanners 2collective additive tree 2-spanners Feodor F. Dragan, Kent State University

  10. Applications of Collective Tree Spannersrepresenting 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 Feodor F. Dragan, Kent State University

  11. 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 ncollective 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 • 2collective additive tree2-spanners in polynomial time • c-chordal graphs • log ncollective additive tree2 c/2 -spanners in polynomial time • interval graphs • log ncollective additive tree 1-spanners in polynomial time • no constant number of trees guaranties +1 Feodor F. Dragan, Kent State University

  12. Previous results on the collective tree spanners problem(Dragan, Yan, Corneil [WG’04]) • AT-free graphs • include:interval, permutation, trapezoid, co-comparability • 2collective additive tree2-spanners in linear time • an additive tree3-spanner in linear time (before) • graphs with a dominating shortest path • anadditive tree4-spanner in polynomial time (before) • 2collective additive tree3-spanners in polynomial time • 5collective additive tree2-spanners in polynomial time • graphs with asteroidal number an(G)=k • k(k-1)/2collective additive tree 4-spanners in polynomial time • k(k-1)collective additive tree 3-spanners in polynomial time Feodor F. Dragan, Kent State University

  13. 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 ofO(log n)collective multiplicative tree 3-spanners • they are called therethetree-covers. • it followsfrom(Corneil, Dragan, Köhler, Yan [WG’05]) that • no constant number of trees guaranties +c (for any constant c) Feodor F. Dragan, Kent State University

  14. Some results on collective additive tree spanners of weighted graphs with bounded parameters(Dragan, Yan [ISAAC’04]) 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 Feodor F. Dragan, Kent State University

  15. Some results on collective additive tree spanners of weighted c-chordal graphs(Dragan, Yan [ISAAC’04]) No constant number of trees guaranties +r for any constant r even for weakly chordal graphs 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 orderableif 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 Feodor F. Dragan, Kent State University

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

  18. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs Feodor F. Dragan, Kent State University

  19. Take n by n complete bipartite graph • square is a clique chordal • join-split To get +1 one needs trees trivial n-1 BFS-trees trees Feodor F. Dragan, Kent State University

  20. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs 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. Feodor F. Dragan, Kent State University

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

  23. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs Feodor F. Dragan, Kent State University

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

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

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

  28. Our results on Collective additive tree spanners of n-vertex homogeneously orderable graphs One tree cannot give +2 Feodor F. Dragan, Kent State University

  29. No constant numberd of trees can guarantee additive stretch factor +2 root gadget clique Feodor F. Dragan, Kent State University

  30. No constant numberd of trees can guarantee additive stretch factor +2 Tree of gadgets … … … The depth is a function of d Feodor F. Dragan, Kent State University

  31. Open questions and future plans • Given a graph G=(V, E) and two integers andr,what is the complexity of finding a system of collective additive (multiplicative) tree r-spanner for G? (Clearly, for most andr,it is an NP-complete problem.) • Find better trade-offs between andrfor 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… Feodor F. Dragan, Kent State University

  32. Thank You Feodor F. Dragan, Kent State University

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