Fault-Tolerant Subgraph For Single-Source Reachability: General and - - PowerPoint PPT Presentation

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

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 1 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Surender Baswana

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 2 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Surender Baswana Keerti Choudhary

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 2 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Surender Baswana Keerti Choudhary Liam Roditty

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 2 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Reachability?

A fundamental problem of graph Is d reachable from s?

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 3 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove sa

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 4 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove sb

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 5 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove sa, sb.

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 6 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove any 2 edges.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 7 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove any 2 edges. Can you conclude something?? .

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 7 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Is d reachable?

Remove cb , bd

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 11 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Terminology

E(P)- edges lying on a path P Consider path P

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 12 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Terminology

E(P)- edges lying on a path P Consider path P

s b d a c t P

E(P) = {sa, ac, ct}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t f

E(f) = {sa, ac, ct}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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}

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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}

s b d a c t

E(A) = {sa, sb, ab, ba}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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}

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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}

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t P

E(P) = {sa, ac}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t P Q

E(P) = {sa, ac} E(Q) = {ct}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

s b d a c t P :: Q

E(P) = {sa, ac} E(Q) = {ct} E(P :: Q) = {sa, ac, ct}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 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

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 18 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

What are the mincuts?

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 19 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

What are the mincuts?

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 20 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

What are the mincuts?

s b d a c e f h g t

Min-cuts {s, a, b, c} {d, e, f, g, h, t} {s, a, b, c, d} {e, f, g, h, t}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 20 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Farthest min-cut

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 21 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Farthest min-cut

s b d a c e f h g t

Farthest min-cut {s, a, b, c, d} {e, f, g, h, t}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 21 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Compute maxflow

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 22 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Compute maxflow

s b d a c e f h g t

maxflow value=2

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 22 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Add e to the source. What is new maxflow??

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 23 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Add e to the source. What is new maxflow??

s b d a c e f h g t

maxflow value=3

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 23 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Add e to the source. What is new maxflow??

s b d a c e f h g t

maxflow value=3

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 23 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 1: Farthest (S, t)-min-cut is independent of the choice of initial max-flow.

s b d a c e f h g t

S = {s, e}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 24 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 1: Farthest (S, t)-min-cut is independent of the choice of initial max-flow.

s b d a c e f h g t

S = {s, e}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 24 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 2: If you add an edge(not present in original graph) from a vertx in S to a vertex in B(A-B is the partition induced by FMC(G, S, t)), then maxflow value of new graph increases by 1. Compute maxflow

s b d a c e f h g t

S = {s}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 25 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 2: If you add an edge(not present in original graph) from a vertx in S to a vertex in B(A-B is the partition induced by FMC(G, S, t)), then maxflow value of new graph increases by 1. Compute maxflow

s b d a c e f h g t

maxflow value=2 S = {s}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 25 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Add edge from s to e

s b d a c e f h g t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 26 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Add edge from s to e

s b d a c e f h g t

maxflow value=3

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 26 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 3: If fs is a maxflow from S to t, then there is a maxflow f from s to t such that E(f) ⊆ E(A) ∪ E(fs)

s b d a c e f h g t

value of maxflow fS=3 S = {s, e} E(fs) = {ef, eg, eh, ft, gt, ht} =red edges E(A) = E\{ft, gt, ht}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 27 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 3: If fs is a maxflow from S to t, then there is a maxflow f from s to t such that E(f) ⊆ E(A) ∪ E(fs)

s b d a c e f h g t

value of maxflow fS=3 S = {s, e} E(fs) = {ef, eg, eh, ft, gt, ht} =red edges E(A) = E\{ft, gt, ht}

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 27 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

s b d a c e f h g t

value of maxflow f=2 E(f) ⊆ E(A) ∪ E(fs)

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 28 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

FMC-Properties

Property 4: Let ( ˆ A, ˆ B) be the partition induced by FMC(s, t) and (A, B) be the partition of V induced by FMC(S, t). Then B ⊆ ˆ B. ˆ A = {s, a, b, c, d}, ˆ B = {e, f, g, h, t} A = {s, a, b, c, d, e, f, g, h}, B = {t} Clearly, B ⊆ ˆ B.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 29 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why FMC?

Key ideas: F-set of failing edges P- path from s to t in G\F P passes through edge (ai, bi) of FMC Adding bi to the source set increases maxflow ( value k to k + 1) k + 1 edge disjoint paths from the set {s, bi} to t One of these path must be intact even after k failures We can construct k-FTRS(t)

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 30 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Assumption: out-degree of all vertices in G is at most 2 .

s b d a c

deg+(s) = 3 deg+(a) = deg+(b) = 2 deg+(c) = deg+(d) = 1 Otherwise we can transform G into a new graph H = (V ′, E′)

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 31 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

H = (V ′, E′) An edge in G is mapped to a path in H (u, vi) → (u, ur) :: (path from ur to ul

i in Bu) :: (ul i, vi)

.

s ur ul

a

ul

c

ul

b

b d a c

Computing k-FTRS(t) in G is equivalent to computing k-FTRS(t) in H

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 32 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Locality Lemma

A is an algorithm, ck is an integer Given G = (V, E) and a vertex v ∈ V , A computes a subgraph H such that

i H is a k-FTRS(v) ii deg−(v) in H is bounded by ck

Then we can compute a k-FTRS for G with atmost ckn edges.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 33 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Proof idea: {v1, v2, . . . vn} is an arbitrary sequence of n vertices in G k-FTRS is computed in n rounds Start with G0 = G. Compute k-FTRS(i) for Gi−1 , say H using A Set Gi be the graph obtained from Gi−1 by restricting incoming edges of vi to only those edges present in H. Use mathematical induction to show G0, G1, . . . Gn are k-FTRS for G.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 34 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Construction of 2-FTRS with 4n edges

Size of the failure edge set = 2 Recall: Out-degree of all the vertices is at most 2 ⇒ max flow value from s to t is either 2 or 1. Two cases to consider:

Case I: Max-Flow(G,s,t)=2 Case II: Max-Flow(G,s,t)=1

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 35 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 36 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Construction contd...

Let C = {(a, b), (a′, b′)} be the farthest (s, t) min-cut. Let a set of edges |F| = 2 got failed. If there is a path from s to t in G \ F, it should pass through C. There are two possibilities:

Passes through (a, b) OR Passes through (a′, b′)

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 37 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

If it passes through (a, b) construct H by adding (s, b). Else add (s, b′) to construct H.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 38 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 39 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

Max-Flow(H,s,t)=3 WHY? Lemma: C be FMC(G, S, t), and (A, B) be the partition of V corresponding to cut C:

Let (s, w) ∈ (S × B) be any arbitrary edge, and G′ = G + (s, w) be a new graph Max-Flow(G′, S, t) = 1 + Max-Flow(G, S, t)

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 40 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

There will be 3 edge disjoint paths from s to t in H. Recall: For any positive integer α, there is a flow from source set S to destination vertex t of value α if and only if there are α edge disjoint paths from S to t.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 41 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

There will be 3 edge disjoint paths from s to t in H. Recall: For any positive integer α, there is a flow from source set S to destination vertex t of value α if and only if there are α edge disjoint paths from S to t.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 41 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 42 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 42 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 42 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 42 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

P0, P1, P2 are edge disjoint paths if H = G + (s, b). P ′

0, P ′ 1.P ′ 2 are edge disjoint paths if H = G + (s, b′).

Use these paths to compute 2-FTRS

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 43 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

The set Et

Define the set Et : Add the last edges on paths P0, P1, P2 in case b = t, and add the edge (a, b) in case b = t. Add the last edges on paths P ′

0, P ′ 1, P ′ 2 in case b′ = t, and add the

edge (a′, b′) in case b′ = t. Then the graph G∗ formed by restricting the incoming edges of t in G to Et is a 2-FTRS.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 44 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Proof

R be the path from s to t and it passes through edge (a, b) of the cut C. H = G + (s, b) If b = t, R is in G∗ \ F.

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 45 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

Consider the case b = t As P0, P1, P2 are edge disjoint, at least one of them will be in the graph H \ F Let the path be P0 Two cases to consider:

P0 is in G \ F, OR First edge of P0 must be (s, b)

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

P0 is in G \ F Path in G \ F ⇔ Path in H \ F

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

First edge of P0 is (s, b) Replace the edge (s, b) by R[s, b] The path R[s, b] :: P0[b, t] is in G \ F Path in G \ F ⇔ Path in H \ F

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 48 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Consequence

In-degree of t in G∗ can be 6, Why? Worst case: Include last edges of all six paths P0, P1, P2, P ′

0, P ′ 1, P ′ 2

Aim: Achieve a degree bound of 4

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 49 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Consequence

In-degree of t in G∗ can be 6, Why? Worst case: Include last edges of all six paths P0, P1, P2, P ′

0, P ′ 1, P ′ 2

Aim: Achieve a degree bound of 4

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 49 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 50 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Achieve the degree bound

Construct a source set S = A ∪ ({b, b′} \ {t} Compute a max-flow fs from S to t Define the set ˜ E by adding the incoming edges to t carrying non-zero flow

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 51 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why ˜ E?

Contains the incoming edges to t carrying non-zero flow. Does it satisfy the conditions of Et? Yes! When b = t; (a, b) is a directed edge from S to t It is in fS; (a, b) is present in ˜ E

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 52 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Why ˜ E?

Contains the incoming edges to t carrying non-zero flow. Does it satisfy the conditions of Et? Yes! When b = t; (a, b) is a directed edge from S to t It is in fS; (a, b) is present in ˜ E

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 52 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

When b = t To show: ˜ E contains the last edges of the paths P0, P1, P2 Lemma: If fS is max-flow from S to t and A, B are partition of the vertices

We can get a max flow f from s to t where E(f) ⊆ E(A) ∪ E(fS)

Incoming edges to t are from the set of incoming edges to t in fS Recall: |f| = 3; edges in ˜ E are coming from three edge disjoint paths.

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Degree bound

Aim: In-degree of t in G∗ is bounded by 4 Task: Show that the (S, t) max-flow in G is at most 4

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Contd...

b, b′ = t, outgoing edges of b and b′ will form an (S, t) cut. In-degree of t is 4 b = t, (a, b) along with two outgoing edges of b′ will form an (S, t) min-cut. In-degree of t is 3 b′ = t, (a′, b′) along with two outgoing edges of b will form an (S, t) min-cut. In-degree of t is 3

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FMC-Properties Main Tools Construction of 2-FTRS Reference

s a a′ b b′ t

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Recall

Size of the failure edge set = 2 Recall: Out-degree of all the vertices is at most 2 ⇒ max flow value from s to t is either 2 or 1. Two cases to consider:

Case I: Max-Flow(G,s,t)=2 Case II: Max-Flow(G,s,t)=1

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Analysing case II

s x y z1 z2 t

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Analysing case II

C = {(x, y)} be the farthest min-cut Every path from s to t must pass through C Construct H by adding (s, x) If y = t, delete all incoming edges of t except (x, t) When y = t, Analysis same as case I

Presented by: Santhini K A and Sampriti Roy Fault-Tolerant Subgraph For Single-Source Reachability: General 59 / 66

FMC-Properties Main Tools Construction of 2-FTRS Reference

Analysing case II

C = {(x, y)} be the farthest min-cut Every path from s to t must pass through C Construct H by adding (s, x) If y = t, delete all incoming edges of t except (x, t) When y = t, Analysis same as case I

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Generalization for k-FTRS

Start with the source vertex s Iterate for i = 1 to k times: Compute FMC, Ci in each iteration, Ai and Bi be the partition in ith iteration Define the set Si+1 = Ai ∪ out(Ai) \ {t} Finally compute max-flow from Sk+1 to t

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Generalization for k-FTRS

Define the set ˜ E as the incoming edges to t present in the max-flow Return G∗ = (G \ in-edges(t)) + ˜ E

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Time Complexity

Transforming G into a graph with O(m) vertices and edges with

• ut-degree bound 2 ⇒ O(m)

Computing k - farthest min-cuts ⇒ O(k

i=1 m × |Ci|) = O(2km)

Computing k-FTRS for any vertex, needs n rounds ⇒ O(2kmn)

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Future Work

What if we use multiple source set, and an integer k, can we come up up with k-FTRS? Given set of pair of vertices P ⊂ V × V and k, is it possible to compute k-FTRS for each pair (u, v) ∈ P?

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Reference

Surender Baswana, Keerti Choudhary, and Liam Roditty. ”Fault-Tolerant Subgraph For Single-Source Reachability: General and Optimal.” 2018 SIAM J. COMPUTING

• L. R. Ford, Jr., and D. R. Fulkerson, Flows in Networks, Princeton

University Press, 1962; reprinted in 2011 by Princeton University Press, http://press.princeton.edu/titles/9233. html.

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Follow-up works

Surender Baswana , Keerti Choudhary , Liam Roditty. ”An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model”, Algorithmica, March 2019 Amir Abboud , Greg Bodwin. ”Reachability preservers: new extremal bounds and approximation algorithms”, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms,January 2018 Greg Bodwin , Michael Dinitz , Merav Parter , Virginia Vassilevska Williams. ”Optimal vertex fault tolerant spanners (for fixed stretch)”, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, January 2018 Ran Duan , Seth Pettie. ”Connectivity oracles for graphs subject to vertex failures”, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, January 2017

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FMC-Properties Main Tools Construction of 2-FTRS Reference

Thank You!

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