FMC-Properties Main Tools Construction of 2-FTRS Reference Fault-Tolerant Subgraph For Single-Source Reachability: General and Optimal Surender Baswana, Keerti Choudhary, Liam Roditty Presented by: Santhini K A and Sampriti Roy CS18D013 CS18S007 Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 1 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Surender Baswana Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 2 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Surender Baswana Keerti Choudhary Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 2 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Surender Baswana Keerti Choudhary Liam Roditty Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 2 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Reachability? A fundamental problem of graph Is d reachable from s ? a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 3 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove sa a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 4 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove sb a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 5 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove sa , sb . a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 6 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove any 2 edges. Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 7 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove any 2 edges. Can you conclude something?? . Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 7 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Is d reachable? Remove cb , bd a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 8 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Motivation Single source reachability in the presence of failures Any set of k vertex/edge failures Fault-tolerant reachability Previously known result only when k = 1 Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 9 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference k - FTRS Input: G = ( V, E ) , k , and source s ∈ V Output: H ⊆ G Goal: construct a sparse subgraph H such that for every vertex v , for any set F of failure edges with | F | ≤ k , v is reachable from s in G \ F if and only if v is reachable from s in H \ F . H is k - FTRS ( k -fault tolerant reachability subgraph) for G Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 10 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Related Works k - FTTS (fault tolerant shortest path subgraph) 1- FTTS by Demetrescu et al., 1- FTTS by Parter and Peleg 2- FTTS by Parter Extension from 1- FTTS to 2- FTTS is complicated For every constant k , there is a k - FTRS Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 11 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( P )- edges lying on a path P Consider path P a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 12 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( P )- edges lying on a path P Consider path P P a c s t b d E ( P ) = { sa, ac, ct } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 12 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( f )- edges of G carrying nonzero flow for a given flow f Consider the flow f from s to t a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 13 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( f )- edges of G carrying nonzero flow for a given flow f Consider the flow f from s to t f a c s t b d E ( f ) = { sa, ac, ct } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 13 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( A )- edges of G whose end points both lie in set A , where A ⊆ V Consider set A = { s, a, b } a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 14 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology E ( A )- edges of G whose end points both lie in set A , where A ⊆ V Consider set A = { s, a, b } a c s t b d E ( A ) = { sa, sb, ab, ba } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 14 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology G ( A )- subgraph of G induced by the vertices lying in a subset A of V Consider set A = { s, a, b } a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 15 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology G ( A )- subgraph of G induced by the vertices lying in a subset A of V Consider set A = { s, a, b } a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 15 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology P [ s, c ]- subpath of path P lying between s and c Q [ c, t ]- subpath of path Q lying between c and t a c s t b d Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 16 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology P [ s, c ]- subpath of path P lying between s and c Q [ c, t ]- subpath of path Q lying between c and t P a c s t b d E ( P ) = { sa, ac } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 16 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology P [ s, c ]- subpath of path P lying between s and c Q [ c, t ]- subpath of path Q lying between c and t P a c Q s t b d E ( P ) = { sa, ac } E ( Q ) = { ct } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 16 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Terminology P [ s, c ]- subpath of path P lying between s and c Q [ c, t ]- subpath of path Q lying between c and t P :: Q -path formed by concatenating paths P and Q in G P :: Q a c s t b d E ( P ) = { sa, ac } E ( Q ) = { ct } E ( P :: Q ) = { sa, ac, ct } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 17 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Preliminaries Visualize graph G as network with unit edge capacities For any positive integer α , there is a flow from a source set S to a destination vertex t of α if and only if there are α edge disjoint paths from S to t . Farthest min-cut: -unique S - T min-cut for which the set containing source is of largest size Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 18 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference What are the mincuts? a c f g s e t b d h Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 19 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference What are the mincuts? a c f g s e t b d h Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 20 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference What are the mincuts? a c f g s e t b d h Min-cuts { s, a, b, c } { d, e, f, g, h, t } { s, a, b, c, d } { e, f, g, h, t } Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 20 / 66
FMC-Properties Main Tools Construction of 2-FTRS Reference Farthest min-cut a c f g s e t b d h Fault-Tolerant Subgraph For Single-Source Reachability: General Presented by: Santhini K A and Sampriti Roy 21 / 66
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