1. WG’2001 June 14 Boltenhagen near Rostock, Germany 1 Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach Feodor F. Dragan Department of Computer Science Kent State University Ohio, USA

2. WG’2001 June 14 Boltenhagen near Rostock, Germany 2 The APSP Problem APSP (a classical fundamental problem): Given a graph, find shortest paths between all pairs of vertices in the graph. There has been a renewed interest in it recently for general graphs as well as for special graph classes. We consider unweighted, undirected graphs. naïve approach: O(nm) ( for dense graphs) via matrix multiplications: O(M(n) log n) [Seidel’92] [Coppersmith/Winograd’ 87] not practical, large hidden constants best combinatorial: [Basch/Khanna/Motwani’ 95] A better ( ) combinatorial algorithm  similar time bound for Boolean matrix multiplication.

3. WG’2001 June 14 Boltenhagen near Rostock, Germany 3 Two Ways To Go 1. consider the APASP problem stretch t all pairs paths: [Awerbuch/Berger/Cowen/Peleg’93], [Cohen’93] (via t-spanners, for ) distances with an additive one-sided error: [Aingworth/Chekuri/Indyk/Motwani’ 96] 2 [Dor/Halperin/Zwick’ 96] 2 Computing all distances with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. 2.consider special graph classes design simple and efficient (optimal time) algorithms for special graph classes which are interesting from practical point of view) error

4. WG’2001 June 14 Boltenhagen near Rostock, Germany 4 Optimal algorithms are known for interval graphs [Attalah/Chen/Lee’93, Mirchandani’96, Ravi/Marathe/Rangan’96, Sridhar/Joshi/Chandrasekharan’93] circular-arc graphs [ Attalah/Chen/Lee’93, Sridhar/Joshi,Chandrasekharan’93] permutation graphs [Dahlhaus’92] strongly chordal graphs [Balachandhran, Rangan’96, Han/Chandrasekharan/Sridhar’97, Dahlhaus’92] chordal bipartite graphs [Ho/Chang’99] distance hereditary graphs [Dahlhaus’92] dually chordal graphs [Brandstaedt/Chepoi/Dragan’98] Parallel algorithms for some graph classes are also considered. Special Graph Classes

5. WG’2001 June 14 Boltenhagen near Rostock, Germany 5 visibility graphs of spiral polygons are interval graphs [Everett/Corneil’90] [Motwani/Ragunathan/Saran’89] Distances in Polygons via Distances in Visibility Graphs. (a part of motivation) link-distance Polygon classs-visibility graph Star-graph Class 2 permutationchordal Class 3 weakly chordalchordal Class 4 not perfectweakly chordal W S E A class 3 polygon ( no N dent ) S W N E S W N E Dent orientations

6. WG’2001 June 14 Boltenhagen near Rostock, Germany 6 [Han/Chandrasekharan/Sridhar’97] APSP can be solved in for G if is given computing for chordal graph is as hard as for general graphs [Sridhar/Han/Chandrasekharan’95] after linear time (sophisticated) preprocessing step, for any a value such that can be computed in O(1) time  all distances with one-sided error of at most 1 in time From [Brandstaedt/Chepoi/Dragan’99] it also follows for any chordal graph G=(V,E) there is a tree T=(V,U) such that (tree T can be constructed in linear time) Distances in Chordal Graphs

7. WG’2001 June 14 Boltenhagen near Rostock, Germany 7 a very simple and efficient approach for solving APASP problem on weakly chordal graphs and subclasses. the same approach works well also on graphs with small size of largest induced cycle it gives a unified way to solve the APSP and APASP problems on different graph classes (including chordal, AT-free, strongly chordal, chordal bipartite, and distance hereditary graphs) Our Contribution Weakly Chordal House-Hole-free HHD-free Chordal Bipartite Distance-Hereditary Chordal Interval Strongly Chordal Weakly chordal hierarchy 3 2 1 0 Trees

8. WG’2001 June 14 Boltenhagen near Rostock, Germany 8 We assume that our graph is given with a vertex ordering The Method for i=1 to n do for i=n-1 downto 1 do for j=n downto i+1 do if then else return distances Algorithm APASP: …

9. WG’2001 June 14 Boltenhagen near Rostock, Germany 9 Results Weakly ChordalHouse-Hole-freek-chordalChordal BipartiteDistance-HereditaryChordalStrongly Chordal 3 2 1 0 Hole-freeAT-freeHHD-freeInterval k-1 LBFS BFS lex 0, if is given bounds are tight

10. WG’2001 June 14 Boltenhagen near Rostock, Germany 10 Let G be a House-Hole-free (HH-free) graph with a LBFS-ordering. Lemma2.  if is given then error is 0.  we need to examine only for vertices x,y with d(x,y)=2 Let d(v,u)=2. G is HH-free with LBFS ordering , i.e., s=2 G is HHD-free (or Chordal) with LBFS  s=1 G is distance-hereditary with LBFS  s=0 G is strongly chordal (or chordal bipartite) with lex-ordering  s=0 Proof Technique (LBFS and lex) Let G be an arbitrary graph with a vertex ordering. Lemma1. Assume there exist integers such that Then, xymn(x)

11. WG’2001 June 14 Boltenhagen near Rostock, Germany 11 Concluding Remarks and Open Problems we presented a very simple and efficient ( time) approach for solving APASP problem on weakly chordal graphs and subclasses. the same approach works well also on graphs with small size of largest induced cycle it gives a unified way to solve the APSP and APASP problems on different graph classes (including chordal, AT-free, strongly chordal, chordal bipartite, and distance hereditary graphs) with one shoot we obtained many known results on distances in particular graph classes for which other graph classes (with special vertex orderings) can this approach give good results (approximations)? can these ideas be used for designing good routing/ labeling schemes in those graph classes?