{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

max-flow-applications-4up

# max-flow-applications-4up - Maximum Flow Applications...

This preview shows pages 1–4. Sign up to view the full content.

Princeton University COS 423 Theory of Algorithms Spring 2001 Kevin Wayne Maximum Flow Applications 2 Maximum Flow Applications Contents Max flow extensions and applications. Disjoint paths and network connectivity. Bipartite matchings. Circulations with upper and lower bounds. Census tabulation (matrix rounding). Airline scheduling. Image segmentation. Project selection (max weight closure). Baseball elimination. 3 Disjoint path network: G = (V, E, s, t). Directed graph (V, E), source s, sink t. Two paths are edge-disjoint if they have no arc in common. Disjoint path problem: find max number of edge-disjoint s-t paths. Application: communication networks. s 2 3 4 Disjoint Paths 5 6 7 t 4 Disjoint path network: G = (V, E, s, t). Directed graph (V, E), source s, sink t. Two paths are edge-disjoint if they have no arc in common. Disjoint path problem: find max number of edge-disjoint s-t paths. s 2 3 4 Disjoint Paths 5 6 7 t

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
5 Max flow formulation: assign unit capacity to every edge. Theorem. There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. Suppose there are k edge-disjoint paths P 1 , . . . , P k . Set f(e) = 1 if e participates in some path P i ; otherwise, set f(e) = 0. Since paths are edge-disjoint, f is a flow of value k. Disjoint Paths s t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 Max flow formulation: assign unit capacity to every edge. Theorem. There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. Suppose max flow value is k. By integrality theorem, there exists {0, 1} flow f of value k. Consider edge (s,v) with f(s,v) = 1. by conservation, there exists an arc (v,w) with f(v,w) = 1 continue until reach t, always choosing a new edge Produces k (not necessarily simple) edge-disjoint paths. Disjoint Paths s t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 Network connectivity network: G = (V, E, s, t) . Directed graph (V, E), source s, sink t. A set of edges F E disconnects t from s if all s-t paths uses at least on edge in F. Network connectivity: find min number of edges whose removal disconnects t from s. Network Connectivity s 2 3 4 5 6 7 t 8 Network connectivity network: G = (V, E, s, t) . Directed graph (V, E), source s, sink t. A set of edges F E disconnects t from s if all s-t paths uses at least on edge in F. Network connectivity: find min number of edges whose removal disconnects t from s. Network Connectivity s 2 3 4 5 6 7 t
9 Disjoint Paths and Network Connectivity Menger’s Theorem (1927). The max number of edge-disjoint s-t paths is equal to the min number of arcs whose removal disconnects t from s. s 2 3 4 5 6 7 t s 2 3 4 5 6 7 t 10 Disjoint Paths and Network Connectivity Menger’s Theorem (1927). The max number of edge-disjoint s-t paths is equal to the min number of arcs whose removal disconnects t from s.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

max-flow-applications-4up - Maximum Flow Applications...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online