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Collective Tree Spanners of Graphs with Bounded Parameters F.F. Dragan and C. Yan Kent State University, USA.

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1 Collective Tree Spanners of Graphs with Bounded Parameters F.F. Dragan and C. Yan Kent State University, USA

2 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University multiplicative tree 4-, 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,r. Does G admit a spanning tree T =(V,E’) such that (a multiplicative tree t-spanner of G) (an additive tree r-spanner of G)? or G T

3 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 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.) easy to construct for some special families of graphs. (mostly multiplicative case)

4 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 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,r. Does G admit a spanning graph H =(V,E’) with |E’|  m s.t. (a multiplicative t-spanner of G) (an additive r-spanner of G)? G H or

5 ISAAC 2005, SanyaFeodor 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 n 1+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

6 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Collective Additive Tree r -Spanners Problem New 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 {T 1, T 2 …, T  } such that (a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners collective multiplicative tree t-spanners can be defined similarly, surplus

7 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Collective Additive Tree r -Spanners Problem New 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 {T 1, T 2 …, T  } such that (a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners,

8 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Collective Additive Tree r -Spanners Problem New 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 {T 1, T 2 …, T  } such that (a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners,

9 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Collective Additive Tree r -Spanners Problem New 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 {T 1, T 2 …, 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,,

10 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Applications of Collective Tree Spanners 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. - (  log 2 n/ 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

11 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University chordal graphs, chordal bipartite graphs –log n collective additive tree 2-spanners in polynomial time –Ώ(n 1/2 ) 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 Previous results on the collective tree spanners problem (Dragan, Yan, Lomonosov [SWAT’04]) (Corneil, Dragan, Köhler, Yan [WG’05])

12 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 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 Previous results on the collective tree spanners problem (Dragan, Yan, Corneil [WG’04])

13 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 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. (Corneil, Dragan, Köhler, Yan [WG’05])it follows from (Corneil, Dragan, Köhler, Yan [WG’05]) that –no constant number of trees guaranties +c (for any constant c) Previous results on the collective tree spanners problem (Gupta, Kumar,Rastogi [SICOMP’05])

14 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University New results on collective additive tree spanners of weighted graphs with bounded parameters Graph class  r planar0 with genus g 0 W/o an h -vertex minor 0 tw(G) ≤ k-1 0 cw(G) ≤ k 2w2w c- chordal nextslide No constant number of trees guaranties +r for any constant r even for outer-planar graphs to get +0 to get +1 w is the length of a longest edge in G

15 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University New results on collective additive tree spanners of weighted c-chordal graphs Graph class  r c-chordal (c>4) 4-chordal2w weakly chordal2w No constant number of trees guaranties +r for any constant r even for weakly chordal graphs

16 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University ( , γ, r)-Decomposable Graphs A graph G=(V, E) is ( , γ, r)-decomposable if there exists a vertex-separator S in G such that Balanced separator: each conn. comp. of G-S has ≤  n vertices; Bounded r-dominating set: S has an r-dominating set D in G with |D|≤ γ; Hereditary family: any induced subgraph of G is ( , γ, r)-decomposable. 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 4-chordal graph

17 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Main Results of the Paper Theorem: Any ( , γ, r)-decomposable graph admits a system of at most γ log 1/  n collective additive tree 2r-spanners. Graph classdecomposition planar with genus g W/o an h -vertex minor tw(G) ≤ k-1 cw(G) ≤ k c- chordal + Polynomial time constructions

18 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph Find a good balanced separator S of G. 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515

19 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 Use S as the root of the rooted balanced decomposition tree. Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph 1, 2, 3, 4

20 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 1, 2, 3, 4 For each connected component of G-S, find its good balanced separator. Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph

21 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 Use the separators as nodes of the rooted balanced decomposition tree and let S be their father. Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph

22 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph γ. Recursively repeat previous procedure until each connected component has an r-dominating set of size at most γ.

23 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 Get the rooted balanced decomposition tree. 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 141415157 9 21 222026 24 Constructing a Rooted Balanced Decomposition Tree for an ( , γ, r)- Decomposable Graph

24 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 141415157 9 21 222026 24 Rooted Balanced Decomposition Tree

25 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 for each layer of the decomposition tree, construct local spanning trees (shortest path trees in the subgraph). we use second layer for illustration. 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 141415157 9 21 222026 24 Constructing Local Spanning Trees

26 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 each time, pick a different vertex from the r-dominating set to grow a shortest path tree in the subgraph. 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 141415157 9 21 222026 24 Constructing Local Spanning Trees

27 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 191941 9 21 22 10 11 12 20 161656 1818 3 2171787 25 23 26 24 1313 1414 1515 1, 2, 3, 4 5, 6, 810, 11, 12,1316, 17, 19, 23, 25 141415157 9 21 222026 24 each time, pick a different vertex from the r-dominating set to grow a shortest path tree in the subgraph. Constructing Local Spanning Trees

28 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 Local Spanning Trees 10 11 121313 1414 1515 10 11 121313 1414 1515 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 9 56 87

29 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 Connect local spanning trees to form spanning trees for the original graph. 10 11 121313 1414 1515 10 11 121313 1414 1515 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 9 56 87 Spanning Trees Construction

30 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 10 11 121313 1414 1515 10 11 121313 1414 1515 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 9 56 87 Connect local spanning trees to form spanning trees for the original graph. Spanning Trees Construction

31 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 10 11 121313 1414 1515 10 11 121313 1414 1515 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 1919 21 22 20 1616 1818 1717 25 23 26 24 9 56 87 Connect local spanning trees to form spanning trees for the original graph. Spanning Trees Construction

32 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Three spanning trees for the original graph w.r.t. layer 2 of the decomposition tree. 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 1919 41 9 21 22 10 11 12 20 161656 1818 3 2 1717 87 25 23 26 24 1313 1414 1515 Spanning Trees Construction 3

33 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Analysis Length is at most r+l 2 Length is at most r+l 1 l1l1 l2l2 r Theorem: Any ( , γ, r)-decomposable graph admits a system of at most γ log 1/  n collective additive tree 2r-spanners.

34 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Further Results  Any ( , γ, r)-decomposable graph G admits an additive 2r-spanner with at most γ n log 1/  n edges which can be constructed in polynomial time.  Any ( , γ, r)-decomposable graph G admits a routing scheme of deviation 2r and with labels of size O( γ log 1/  n log 2 n/log log n) bits per vertex. Once computed by the sender in γ log 1/  n time, headers never change, and the routing decision is made in constant time per vertex.

35 ISAAC 2005, SanyaFeodor 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…

36 ISAAC 2005, SanyaFeodor F. Dragan, Kent State University Thank You

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