1. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 1 A self-stabilizing algorithm for the maximum flow problem Presented By Niranjan Sukumar Ghosh, Arobinda Gupta and Sriram V. Pemmaraju 6 th May 2004

2. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 2 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

3. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 3 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(n 2 /e)) Parallel Algorithms –Shiloach and Vishkin : O(n 2 logn) 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

4. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 4 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

5. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 5 Preliminaries Skew Symmetry Capacity Constraint Residual Graph Residual Capacity Feasible flow Max-flow, Min-cut theorem These terms are used in the same context

6. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 6 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])

7. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 7 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 G f D(i): [0…n]

8. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 8 Key Idea… For any node i, –Demand(i) = O f (i) – I f (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 G f Use BFS to keep track of shortest paths from s to all nodes in G f

9. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 9 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 b c a y x Residual Graph d = 5 d = 3 d = 4 d = 5 IN(i) = {a, b, c} D(i) = {4, 5, 6}

10. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 10 Algorithm S1: Each node i, computes its d-value by examining the values d(j) for all (j,i)=E f. 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)=E f and d(j)=d(i)-1 S4: if demand(i) and d(i)=n, then there is no path from s to i in G f and it reduces the outflow.

11. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 11 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

12. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 12 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

13. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 13 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

14. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 14 Results

15. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 15 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(n 2 ) in average case settings

16. Outline Max Flow Algorithm Model of Computation Proposed Algorithm Self Stabilization Contribution 16 Thank You ….