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Improved Randomized Algorithms for Path Problems in Graphs

This research explores efficient randomized algorithms for various path problems in graphs, including all-pairs shortest paths and transitive closure. It presents complex query paradigms that involve online answering for dynamic graph settings, as well as preprocessing strategies for static scenarios to achieve optimal query times. Addressing algorithms with both additive and multiplicative errors, the study reviews existing methodologies and proposes improvements in data structures that enable fast responses to path queries. The implications of these algorithms extend to significant advancements in algorithmic graph theory.

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Improved Randomized Algorithms for Path Problems in Graphs

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  1. Improved Randomized Algorithms for Path Problems in Graphs Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science

  2. Path Problems in Graph Given a graph, report a path with certain characteristics

  3. Path Problems in Graph Given a graph, report a path with certain characteristics Transitive closure

  4. Path Problems in Graph Given a graph, report a path with certain characteristics Transitive closure All-pairs shortest paths

  5. Query paradigm of a path problem Given a graph G=(V,E)

  6. Query paradigm of a path problem Given a graph G=(V,E) • A sequence of path queries

  7. Query paradigm of a path problem Given a graph G=(V,E) • A sequence of path queries • Answer each path query online

  8. Path Problems in static settings • Preprocess a given graph G=(V,E) to form a data-structure that can answer a path query in optimal time.

  9. Path Problems in dynamic settings A graph G=(V,E) ,

  10. Path Problems in dynamic settings A graph G=(V,E) , q, q,u ,u , q ,u,u,u,u , q , q ,u, …

  11. Path Problems in dynamic settings AIM : Maintain a data-structure

  12. Path Problems in dynamic settings AIM : Maintain a data-structure • Answer each query in optimal time

  13. Path Problems in dynamic settings AIM : Maintain a data-structure • Answer each query in optimal time • Requires less update time

  14. Organization of the talk • Static algorithms • All-pairs approximate shortest paths • Graph Spanners • Dynamic algorithms • Transitive closure • All-pairs shortest paths • Open Problems

  15. Organization of the talk • Static algorithms • All-pairs approximate shortest paths • Graph Spanners • Dynamic algorithms • Transitive closure • All-pairs shortest paths • Open Problems

  16. All-pairs approximate shortest paths

  17. All-Pairs Shortest Paths • Given a graph G=(V,E), preprocess it to compute shortest-path for every pair of vertices u,vЄV. |V|=n, |E|=m δ(u,v) : distance from u to v One of the most fundamental algorithmic graph problem

  18. Existing algorithms for APSP • Floyd and Warshal O(n3) • Pettie [ICALP 2002, TCS 2004] O(mn+n2log logn) • Zwick [ISAAC 2004] O(n3 (log log n)1/2/log n)

  19. Sub-cubic algorithms for APSP Distance matrix G=(V,E) Fast Matrix Multiplication Subroutine

  20. Are there simple and efficient algorithms for APSP with • sub-cubic preprocessing time ? • sub-quadratic space data-structure ?

  21. Are there simple and efficient algorithms for APSP with • sub-cubic preprocessing time ? • sub-quadratic space data-structure ? At the cost of approximation

  22. All-pairs approximate shortest Paths • Some error ε in the distance δ* • Additive errorε= k δ(u,v) ≤ δ*(u,v) ≤ δ(u,v) + k • Multiplicative errorε= t δ(u,v) ≤ δ*(u,v) ≤ t δ(u,v)

  23. Existing algorithms for APASP in undirected graph • Algorithms with additive error • Aingworth et al. [SODA 1996, SICOMP 1999] • Dor et al. [FOCS 1996, SICOMP 1999] • Algorithms with multiplicative error • Cohen [SICOMP 1998] • Cohen & Zwick [SODA 1997, J. Algo. 2001] • Thorup & Zwick [STOC 2001] • Algorithms with additive and multiplicative error • Elkin [PODC 2001]

  24. Approximate Distance Oracle[Thorup & Zwick, STOC 2001] Given a positive integer k • Preprocessing timeO(kmn1/k) • Data-structure of sizeO(kn1+1/k) • Answer any approximate distance query inO(k)time δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v)

  25. Approximate Distance Oracle[Thorup & Zwick, STOC 2001] Given a positive integer k • Preprocessing timeO(kmn1/k) • Data-structure of sizeO(kn1+1/k) • Answer any approximate distance query inO(k)time δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v) (2k-1)-approx. distance oracle

  26. Approximate Distance Oracle[Thorup & Zwick, STOC 2001] Given a positive integer k • Preprocessing timeO(kmn1/k) • Data-structure of sizeO(kn1+1/k): Optimal • Answer any approximate distance query inO(k)time δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v) (2k-1)-approx. distance oracle

  27. Approximate distance oracles for un-weighted graphs

  28. All-pairs t-approximate shortest paths, for t<3

  29. All-pairs 2-approx. shortest path • A : algorithm for (2-ε)-approximate shortest paths • B,C : two nΧn boolean matrices

  30. All-pairs 2-approx. shortest path • A : algorithm for (2-ε)-approximate shortest paths • B,C : two nΧn boolean matrices BΧC GB,C B,C Linear time A

  31. All-pairs 2-approx. shortest path • A : algorithm for (2-ε)-approximate shortest paths • B,C : two nΧn boolean matrices BΧC GB,C B,C Linear time A All-pairs (2- ε)-approx. paths is as difficult as Boolean matrix multiplication

  32. All-pairs 2-approx. shortest path [Cohen & Zwick, SICOMP 2000] G=(V,E) : Undirected weighted graph • Preprocessing time : O(m1/2n3/2) • Space : O(n2)

  33. All-pairs 2-approx. shortest paths 2.5 Upper bound ω 2.376 2.0 Lower bound

  34. All-pairs 2-approx. shortest paths 2.5 Upper bound ω 2.376 Algorithm 1 Õ(m2/3n + n2) δ(u,v) ≤δ*(u,v) ≤ 2δ(u,v) + 1 2.0 Lower bound

  35. All-pairs 2-approx. shortest paths 2.5 Upper bound ω 2.376 Algorithm 1 Õ(m2/3n + n2) δ(u,v) ≤δ*(u,v) ≤ 2δ(u,v) + 1 2.0 Algorithm 2 Õ(n2) δ(u,v) ≤δ*(u,v) ≤ 2δ(u,v) + 3 Lower bound

  36. Graph Spanners

  37. Graph Spanners Definition : Given a graph G=(V,E), a sub-graph G=(V,Es) that is sparse and yet preserves approximate distances pair-wise.

  38. Graph Spanners Definition : Given a graph G=(V,E), a sub-graph G=(V,Es) that is sparse and yet preserves approximate distances pair-wise. Multiplicative t-spanner : δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)

  39. Graph Spanners Definition : Given a graph G=(V,E), a sub-graph G=(V,Es) that is sparse and yet preserves approximate distances pair-wise. Multiplicative t-spanner : δ(u,v) ≤ δs(u,v) ≤ t δ(u,v) Additive t-spanner : δ(u,v) ≤ δs(u,v) ≤ δ(u,v) + t

  40. Applications of Graph Spanner • Distributed Computing • Design of Synchronizers • Compact routing tables • Computational Biology • Reconstruction of Phylogenetic trees • All-pairs Approximate Shortest Paths

  41. Multiplicative Spanners

  42. Multiplicative Spannersize versus approximation v u

  43. Multiplicative Spannersize versus approximation t v u

  44. Multiplicative Spannersize versus approximation v u girth = t stretch ≥ t-1

  45. Multiplicative Spannersize versus approximation v u girth = t stretch ≥ t-1 Erdös[1960], Bondy & Simonovits [1974], Bollobas [1978] : For all k≥1, there are graphs with Ω(n1+1/k) edges and girth >2k

  46. Multiplicative Spannersize versus approximation For k ≥1, a (2k-1)-spanner may have Ω(n1+1/k) edges

  47. Multiplicative Spanner : Results

  48. Multiplicative Spanner : Results Can we Compute (2k-1)-spanners in linear time ?

  49. Multiplicative Spanner : Results

  50. Local approach u v Edge in Spanner Edge not in Spanner

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