MaxFlow-handouts-2

# MaxFlow-handouts-2 - Maximum Flow 4/3/2006 2:57 PM 1...

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Unformatted text preview: Maximum Flow 4/3/2006 2:57 PM 1 4/3/2006 2:57 PM Maximum Flow 1 Maximum Flow w s v u t z 3 / 3 1 / 9 1 / 1 3 / 3 4 / 7 4 / 6 3 / 5 1 / 1 3 / 5 2 / 2 χ 4/3/2006 2:57 PM Maximum Flow 2 Outline and Reading Flow networks ¡ Flow (§8.1.1) ¡ Cut (§8.1.2) Maximum flow ¡ Augmenting path (§8.2.1) ¡ Maximum flow and minimum cut (§8.2.1) ¡ Ford-Fulkerson’s algorithm (§8.2.2-8.2.3) Maximum Flow 4/3/2006 2:57 PM 2 4/3/2006 2:57 PM Maximum Flow 3 Flow Network A flow network (or just network) N consists of ¡ A weighted digraph G with nonnegative integer edge weights, where the weight of an edge e is called the capacity c ( e ) of e ¡ Two distinguished vertices, s and t of G , called the source and sink, respectively, such that s has no incoming edges and t has no outgoing edges. Example: w s v u t z 3 9 1 3 7 6 5 1 5 2 4/3/2006 2:57 PM Maximum Flow 4 Flow A flow f for a network N is is an assignment of an integer value f ( e ) to each edge e that satisfies the following properties: Capacity Rule: For each edge e , 0 ≤ f ( e ) ≤ c ( e ) Conservation Rule: For each vertex v ≠ s,t where E − ( v ) and E + ( v ) are the incoming and outgoing edges of v , resp. The value of a flow f , denoted | f | , is the total flow from the source, which is the same as the total flow into the sink Example: ∑ ∑ + − ∈ ∈ = ) ( ) ( ) ( ) ( v E e v E e e f e f w s v u t z 3 / 3 2 / 9 1 / 1 1 / 3 3 / 7 2 / 6 4 / 5 1 / 1 3 / 5 2 / 2 Maximum Flow 4/3/2006 2:57 PM 3 4/3/2006 2:57 PM Maximum Flow 5 Maximum Flow A flow for a network N is said to be maximum if its value is the largest of all flows for N The maximum flow problem consists of...
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## This note was uploaded on 09/10/2009 for the course CS 401 taught by Professor Cs401 during the Spring '08 term at Illinois Tech.

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MaxFlow-handouts-2 - Maximum Flow 4/3/2006 2:57 PM 1...

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