1 / 30

Approximate Distance Oracles for Graphs with Dense Clusters

Approximate Distance Oracles for Graphs with Dense Clusters. Mattias Andersson Joachim Gudmundsson Christos Levcopoulos. Motivation. Distance oracles – Dense Clusters. p. q. Distance oracles – Dense Clusters. Distance oracles – Dense Clusters. r(q). p. q. r(p).

amie
Download Presentation

Approximate Distance Oracles for Graphs with Dense Clusters

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Approximate Distance Oracles for Graphs with Dense Clusters Mattias Andersson Joachim Gudmundsson Christos Levcopoulos

  2. Motivation

  3. Distance oracles – Dense Clusters p q

  4. Distance oracles – Dense Clusters

  5. Distance oracles – Dense Clusters r(q) p q r(p)

  6. Distance oracles – Dense Clusters

  7. Distance oracles – Dense Clusters : Airport M : Set of edges connecting airports

  8. Problem definition • Input • Graph G = (V, E) • Islands of t-spanners • Edges connecting Islands • Constant e • Output • Representative points • Distance matrix

  9. Queries • Input • vertices p and q • Output • Path between p and q • Length at most (1+e) * OPT

  10. Distance oracles • [Thorup and Zwick, 2001] • Uses data structure • O(kmn1/k) time • O(kn1+1/k) space • Computes (2k-1) approximate • solutions in O(k) time • distance oracle

  11. Distance oracles – Geometric version • [Gudmundsson et al., 2002] • distance oracle, geometric query version. • Input: t-spanner graph • Data structure • O(n log n) space • O(m + n log n) time • Answers approx. queries in constant time

  12. Theorem: A distance oracle for graphs consisting • of islands of t-spanners • answers (1+ε)-approx. queries in O(1) time • datastructure: • O(M2 + n log n) space. • O(m + (M2 + n) log n) time. Results

  13. Representative points r(p) p

  14. O(# airports) repr. points O (n log n) time Representative points

  15. Distance matrix • Using Dijkstras on input graph G Too time consuming

  16. Approximate graph • For each island: • Construct distance oracle A • Consider Airports and Repr. Points • Compute WSPD • Add egde e between each pair u,v • |e| = A(u, v)

  17. Approximate graph • Add M (edges connecting airports) • Graph G’: • O(|M|) edges and vertices

  18. Distance oracles – Dense Clusters • Dijkstras • O(|M|2 log (|M|) time • O(|M|2) space

  19. q p Query: same island • Use [Gudmundsson et al., 2002] • Each island is a t-spanner

  20. Query: different islands p r(p) r(q) q |p, r(p)|ε + |r(p), r(q)|ε + |r(q), q|ε [Gudmundsson et al., 2002] [Gudmundsson et al., 2002]

  21. Summary Total: O(m+(|M2|+n) log n) time O(|M2| + n log n) space

  22. Proof r(p) r(q) p q

  23. O(# airports) repr. points O (n log n ) time Representative points V’ V

  24. [Krznaric, Levcopoulos, 1995] : Hierarchy of clusters can be computed in O(n log n) time and space. Hierarchy of clusters

  25. Repr. points - proof

  26. Repr. points - proof

  27. Doughnuts O(#airports) clusters => O(#airports) doughnuts

  28. Inner cells Each cluster ’pays for’ a constant number of inner cells

  29. Theorem: A distance oracle for graphs consisting • of islands of t-spanners • answers (1+ε)-approx. queries in O(1) time • datastructure: • O(M2 + n log n) space. • O(m + (M2 + n) log n) time. Results

  30. Thank you!

More Related