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Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach

Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach. Feodor F. Dragan Department of Computer Science Kent State University Ohio, USA. The APSP Problem. APSP (a classical fundamental problem): Given a graph,

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Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach

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  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. 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. Two Ways To Go • 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. • 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. Special Graph Classes • 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.

  5. Distances in Polygons via Distances in Visibility Graphs. (a part of motivation) • visibility graphs of spiral polygons are • interval graphs [Everett/Corneil’90] • [Motwani/Ragunathan/Saran’89] link-distance N N Dent orientations W E W E S S W E S A class 3 polygon ( no N dent )

  6. Distances in Chordal Graphs • [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)

  7. Our Contribution • 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) 2 Chordal Bipartite 3 Trees House-Hole-free 1 Strongly Chordal Chordal Weakly Chordal 0 HHD-free Interval Distance-Hereditary Weakly chordal hierarchy

  8. The Method fori=1 tondo fori=n-1 downto 1 do forj=ndowntoi+1 do if then else return distances • We assume that our graph is given with a vertex ordering • Algorithm APASP: …

  9. Results k-chordal AT-free Weakly Chordal Chordal Bipartite Hole-free House-Hole-free Strongly Chordal Chordal HHD-free Interval Distance-Hereditary bounds are tight k-1 BFS 3 BFS BFS 0, if is given 2 LBFS BFS LBFS 1 LBFS BFS lex 0 lex lex

  10. Proof Technique (LBFS and lex) Let G be an arbitrary graph with a vertex ordering. Lemma1. Assume there exist integers such that Then, • 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 x mn(x) y

  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?

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