1

Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

A self-stabilizing algorithm for the maximum flow

problem

Presented ByNiranjan

Sukumar Ghosh, Arobinda Gupta and Sriram V. Pemmaraju

6th May 2004

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Paper Outline

• Introduction to maximum flow problem

• Model of computation• Maximum flow algorithm for acyclic

graphs• Proof of correctness• Experimental Evaluation• Conclusion – Failure Model,

Algorithm for arbitrary graph

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Algorithms for Max Flow

• Sequential Algorithms– Ford and Fulkerson : Picks augmented paths

arbitrarily– Edmonds and Karp : Use BFS to construct augmented

paths– Goldmerg and Tarjan : Works in localized manner -

O(ne log(n2/e))

• Parallel Algorithms– Shiloach and Vishkin : O(n2logn) using O(n) processors

• Distributed Algorithms– Based mainly on the Glodmerg and Tarjan

• Self-Stabilized Algorithms– Distributed Algorithms

• Fault Tolerant• Adjust to dynamic changes in the network topology

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Self-Stabilization

• Introduced by Dijkstra [1974]• S is self-stabilizing with respect to

predicate P if it satisfies the following two properties:– Closure: P is closed under the execution of S.– Convergence: Starting from an arbitrary

global state, S is guaranteed to reach a global state satisfying P within a finite number of state transitions

• Failure Model : Transient Failure– An event that may the change the state of

the system, but not its behavior

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Preliminaries

• Skew Symmetry• Capacity Constraint• Residual Graph• Residual Capacity• Feasible flow• Max-flow, Min-cut

theorem

These terms are used in the same context

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Model of Computation

• Each node i in G corresponds to a process called process i that executes a program asynchronously

• Each edge (i, j) corresponds to a bidirectional link b/w process i and process j

• Process i local variables can be read by its neighbors but written only by process i

• Program Model (expressed using guarded commands [Dijkstra 1975])

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Maximum Flow Algorithm for Acyclic Graphs

• G is a acyclic digraph• For each edge (i,j), f(i,j): current flow

from node i to node j

• Both process i and process j can read from and write into the variable f(i,j)

• Each node i contains a single variable d(i): length of the shortest path form s to i in the residual graph Gf D(i): [0…n]

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Key Idea…

• For any node i ,– Demand(i) = Of(i) – If(i)

• Demand(t) = Infinity

• Each node i tries to restore the flow conservation constraint demand(i)=0 by– Reducing its inflow if demand(i) < 0– Increasing its inflow and or reducing its

outflow if demand(i) > 0

• Each node with positive demand attempts to pull flow via a shortest path from s to itself in Gf

• Use BFS to keep track of shortest paths from s to all nodes in Gf

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Notation

• Distinguished Nodes Node s remains idle with d(s) = 0 Node t executes the same program as other processes with demand(t) = INFINITY

i

bc

ay

x

Residual Graph

d = 5

d = 3d = 4

d = 5

IN(i) = {a, b, c}

D(i) = {4, 5, 6}

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Algorithm

• S1: Each node i, computes its d-value by examining the values d(j) for all (j,i)=Ef. d(i) = min{D[i], n}

• S2: If demand(i)<0, then total flow along incoming edges in G is reduced irrespective of d-values

• S3: if demand(i)>0 and d(i)<n, then i pulls flow along an incoming edges (j,i)=Ef and d(j)=d(i)-1

• S4: if demand(i) and d(i)=n, then there is no path from s to i in Gf and it reduces the outflow.

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Algorithm

• Assumption: f(i,j) never exceeds C(i,j) - New action (A5) that appropriately reduces the flow on an incident edge that has flow in excess of capacity

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Example

Node: a Guard: S2

Node: b Guard: S1

Node: t Guard: S3

Node: b Guard: S3

C(s,a) = C(b,t) = 2

C(a,b) = 1

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Example …

Node: a Guard: S3

Node: b Guard: S1

Node: b Guard: S4

Node: t Guard: S1

C(s,a) = C(b,t) = 2

C(a,b) = 1

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Results

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Contribution of this paper

• First distributed self-stabilizing algorithm for max-flow

• Inherently tolerant to transient faults• Automatically adjust to topology

changes– Arbitrary addition or deletion of edges– Addition and deletion of nodes provided that

#nodes in the network is bounded– Arbitrary changes in the capacities of the

edges

• Requires O(n2) in average case settings

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Outline

Max Flow Algorithm

Model of Computation

Proposed Algorithm

Self Stabilization

Contribution

Thank You ….