BFS Structure Merav Parter and David Peleg Weizmann Institute of Science Israel SODA 2014 Breadth First Search BFS Trees ShortestPath Tree BFS rooted at s Sparse solution ID: 1048706
Download Presentation The PPT/PDF document "Fault Tolerant Approximate" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Fault Tolerant Approximate BFS StructureMerav Parter and David PelegWeizmann Institute of Science, IsraelSODA 2014
2. Breadth First Search (BFS) TreesShortest-Path Tree (BFS) rooted at s.Sparse solution: n-1 edges. Problem: Not robust against edge and vertex faults.Unweighted graph G=(V,E), source vertex s V. sv1v2v4v3v5
3. Fault Tolerant BFS TreesObjective:Add a few edges to the BFS tree to make it robust against single edge (or vertex) fault.sv1v2v4v3v5
4. Fault Tolerant BFS TreesObjective:Purchase a collection of edges (BFS + backup edges) that is robust against edge faults.sv1v2v4v3v5
5. (Exact) Fault-Tolerant BFS TreesSubgraph H that contains a BFS tree in G\{e}for every edge failure e in G. sv1v2v4v3v5sv1v2v4v3v5e1e2e3e4e5
6. Fault-Tolerant (FT) BFS TreesSubgraph H that contains a BFS tree in G\{e}for every edge failure e in G. sv1v2v4v3v5sv1v2v4v3v5e1e2e3e4e5
7. Fault Tolerant (FT) BFS TreesSubgraph H that contains a BFS tree in G\{e}for every edge failure e in G. sv1v2v4v3v5sv1v2v4v3v5e1e2e3e4e5
8. Fault Tolerant (FT) BFS TreesSubgraph H that contains a BFS tree in G\{e}for every edge failure e in G. sv1v2v4v3v5sv1v2v4v3v5e1e2e3e4e5
9. FT-BFS Tree - Formal Definition Consider an unweighted graph G=(V,E) and a source vetrex s. A subgraph H is an FT-BFS of G and s if for every v in V and e in E: d(s,v, H\{e}) = d(s,v, G\{e})
10. Exact FT-BFS Trees [P, Peleg, ESA’13]Upper Bound Theorem:For every graph G=(V,E) and every source s ϵ V there exists a (polynomially constructible) FT-BFS tree H with O() edges.
11. Lower Bound Theorem:For every integer n, there exists an n-vertex graph G=(V,E) and a source vertex s ϵ V such that every FT-BFS tree H has edges. Exact FT-BFS Trees Are DenseResorting to approximate distances!
12. FT-ABFS Tree - Formal DefinitionA subgraph H is an (,) FT-ABFS of G and s if for every v in V and e in E: d(s,v, H\{e}) = d(s,v, G\{e})+
13. FT-ABFS Tree - Formal DefinitionA subgraph H is an (, ) FT-ABFS of G and s if for every v in V and e in E: d(s,v,H\{e}) = d(s,v,G\{e})+ Can be viewed as FT single source spanners
14. Related Work Replacement paths.[Gupta et al. 1989; Hershberger and Suri, 2001; Roditty and Zwick, 2005; Weimann and Yuster 2010] Single source distance oracle.[Kahanna and Baswana, 2010; Gradoni, Williams, 2012] Fault tolerant spanners .[Chechik et al., 2009; Dinitz and Krauthgamer, 2011;Braunschvig et al., 2012]
15. Problem definition: Given a source s, destination t, for every e ϵ π(s,t) , compute Ps,t,e the shortest s-t path that avoids e.A related problem: the replacement path problem π(s,t)stePs,t,ePs,t,e: s-t shortest path in G\{e}
16. The structure of a replacement pathP(s,t,e) : s-t shortest path in G\{e} estP(s,t) Detour
17. Better bounds available for replacement paths problem forUndirected graphs: Time complexity: O(m+n log n) [Gupta et al. 1989] [Hershberger and Suri, 2001] Unweighted directed graphs: Time complexity: O(m ) (Randomized MonteCarlo algorithm) [Roditty and Zwick 2005] The replacement paths problem
18. Problem definition: Given a source s, destination t, for every e ϵ π(s,t) , compute an approximate s,t,e in G\{e} such that| s,t,e | dist(s,t, G\{e}) A related problem: Approximate replacement path problem
19. Single source approximate replacement pathData structure setting (Distance oracle):Query : (s,t,e). Answer: -approximate shortest path between s and t when e fails.Complexity measure: query & preproc’ time, DS size.[Kahanna, Baswana STACS’10] FT-ABFS tree revisited:An FT-ABFS tree H contains the collection of allsingle source approximate replacement paths. Complexity measure: size of H (#edges).New!
20. Fault-Tolerant SpannersA subgraph H is an fault tolerant spanner if for every u,v in V and every edge ed(u,v,H\{e}) ·d(u,v,G\{e}).
21. Fault-Tolerant SpannersRobust to f-edge faults:Stretch: 2k-1#edges: [Chechik et al., 2009] d(u,v,H \F) ·d(u,v,G\F) for all u,v in V FT-ABFS tree revisited:An FT-ABFS tree H is a single source FT-spanner.
22. Fault-Tolerant Spannersd(u,v,H \F) ·d(u,v,G\F) for all u,v in V Robust to f-vertex faults:Stretch: 2k-1#edges: [Chechik et al., 2009] [Dinitz and Krauthgamer, 2011]
23. Our Results Multiplicative stretch: linear upper bound. Additive stretch: superlinear lower bound.
24. Multiplicative stretch: (,0) FT-ABFS structuresUpper Bound:For every graph G=(V,E) and every source s ϵ V there exists a (polynomially constructible) (3,0) FT-BFS structure H with at most 4n edges.FT multiplicative single source spanners are linear!* O(log n) improvement from [Kahanna, Baswana STACS’10]
25. Algorithm for constructing (3,0) FT-ABFSInput: unweighted graph G=(V,E), source vertex s.Output: (3,0) FT-ABFS structure H ⊆ G.Add BFS(s, G) to Hsuv
26. Algorithm for constructing (3,0) FT-ABFSAdd BFS(s, G) to HFor every t, and every edge e in choose a specific s-t replacementpath Ps,t,e in G\{e} π(s,t)stePs,t,eNew edge not in T0
27. Algorithm for constructing (3,0) FT-ABFSAdd BFS(s, G) to HFor every t, and every edge e in carefully choose a specific s-t replacement path Ps,t,e in G\{e} π(s,t)stePs,t,eNew edge not in BFS(s,G)
28. Algorithm for constructing (3,0) FT-ABFSInput: unweighted graph G=(V,E), source vertex s.Output: (3,0) FT-BFS tree H ⊆ G.Add BFS(s, G) to HFor every u, and every edge e on choose a specific s-u replacement path Ps,t,e in G\{e} Add to H, the first new edge on Ps,t,e that is missing on T0=BFS(s,G).
29. Picking the first edge is criticalTaking the last new edge Exact FT-BFS with edges. π(s,t)stePs,t,eTaking the first new edge (3,0) FT-BFS with edges. ESA’13SODA’14
30. CorrectnessLemma: Here:Show that s and t are connectedin (BFS ) \ {e} Ps,t,eestue'
31. CorrectnessClaim:The failing edge e appears on Ps,t,esutfFirst edge not in BFSee'
32. CorrectnessClaim:The failing edge e appears on Ps,t,esutfFirst edge not in BFS(s,G)ee'
33. CorrectnessClaim:The failing edge e appears on Ps,t,esutfHence u and t are in BFS\{e}First edge not in BFS(s,G)ee'
34. CorrectnessClaim:The failing edge e appears on Ps,t,esutfFirst edge not in T0ee'
35. Size AnalysisClaim:Every vertex u is adjacent to at most3 first new edges. sue1e2e3e4abcdPs,t,e1Prove thatdist(s,a)>dist(s,b)>dist(s,c)>dist(s,d)Ps,t,e2Ps,t,e3Ps,t,e4By contradiction.
36. Size AnalysisClaim:Every vertex u is adjacent to at most3 first new edges. sue1e2e3e4abcdPs,t,e1Prove thatdist(s,a)>dist(s,b)>dist(s,c)>dist(s,d)
37. Additive stretch: (1,) FT-ABFS structuresLower Bound:For every integer n, andadditive stretch 1 o(log n) n-vertex graph G=(V,E) and a source s ϵ V s.tevery (1, ) FT-ABFS H has ) edges for )>0. FT additive single source spanners are superlinear!
38. The Lower Bound Construction () Let B(d,d) := Dense bipartite graph with girth* 6 and dges. copies of B(d,d) B1(d,d)B2(d,d)Bd(d,d)* Girth is the minimal cycle length in the graph.
39. The Lower Bound Construction () B1(d,d)B2(d,d)Bd(d,d)vX|X|=d2=|Z|=d= Z+1
40. The Lower Bound Construction () B1(d,d)B2(d,d)Bd(d,d)vXZsCollection ofpaths which areVertex disjointMonotone increasing. 20) ) Girth=6+1 21+
41. The Lower Bound Construction () B1(d,d)B2(d,d)Bd(d,d)vXZsCollection ofpaths which areVertex disjointMonotone increasing. 2021+ ) ) Girth=6+1
42. The ConstructionTotal number of vertices: n ) copies of graphs.) vertex disjoint pathsof increasing length contain) vertices. B1B2BdZs) +1
43. The ConstructionB1B2BdZs) Total number of edges: In each Bi () edges.Overall, ()= ()edges.
44. Cl. : Every (1,3) FT-ABFS tree H must contain ALL the edges of the Bi graphsThe Construction By contradiction:Assume there exists anedge ei,j that is not in H.Consider the case where fi fails. B1BdZs) ei,jfi+1
45. The Constructiond(s,xj, H \{fi}) >d(s,xj, G\{fi})+3Contradiction since H is an (1,3) FT-ABFS tree.B1BdZsfiei,jxj+1
46. Additive stretch: (1,) FT-ABFS structuresUpper Bound:For every graph G=(V,E) and every source s ϵ V there exists a (polynomially constructible) (1,4) FT-BFS tree H with at most O() edges. Main technical contribution:Adapting the Path-Buying scheme to the FT-setting.
47. Dichotomy between additive and multiplicative stretch Multiplicative stretchAdditive stretch 13134O(log n)4n , Size of (,) FT-ABFS structure O(nlogn)2
48. Thanks!