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Unformatted text preview: Chapter 12 Network Flow By Sariel Har-Peled , October 9, 2007 Version: 0.25 12.1 Network Flow We would like to transfer as much merchandise as possible from one point to another. For example, we have a wireless network, and one would like to transfer a large file from s to t . The network have limited capacity, and one would like to compute the maximum amount of information one can transfer. Specifically, there is a network and capacities associated with each connection in the network. The question is how much flow can you transfer from a source s into a sink t . Note, that here we think about the flow as being splitable, so that it can travel from the source to the sink along several parallel paths simultaneously. So, think about our network as being a network of pipe moving water from the source the sink (the capacities are how much water can a pipe transfer in a given unit of time). On the other hand, in the internet tra ffi c is packet based and splitting is less easy to do. s 13 4 10 14 t 7 4 12 20 9 16 u v w x Definition 12.1.1 Let G = ( V , E ) be a directed graph. For every edge ( u v ) E ( G ) we have an associated edge ca- pacity c ( u , v ), which is a non-negative number. If the edge ( u v ) < G then c ( u , v ) = 0. In addition, there is a source vertex s and a target sink vertex t . The entities G , s , t and c ( ) together form a flow network or just a network . An example of such a flow network is depicted on the right. This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. 1 s 1 1 / 1 6 t 10 8 / 1 3 1 / 4 4 / 9 7 / 7 1 5 / 2 12 / 12 4 / 4 11 / 14 u v w x We would like to transfer as much flow from the source s to the sink t . Specifically, all the flow starts from the source vertex, and ends up in the sink. The flow on an edge is a non- negative quantity that can not exceed the capacity constraint for this edge. One possible flow is depicted on the left figure, where the numbers a / b on an edge denote a flow of a units on an edge with capacity at most b . We next formalize our notation of a flow. Definition 12.1.2 (flow) A flow in a network is a function f ( , ) on the edges of G such that: (A) Bounded by capacity : For any edge ( u v ) E , we have f ( u , v ) c ( u , v ). Specifically, the amount of flow between u and v on the edge ( u v ) never exceeds its capac- ity c ( u , v ). (B) Anti symmetry : For any u , v we have f ( u , v ) =- f ( v , u ). (C) There are two special vertices: (i) the source vertex s (all flow starts from the source), and the sink vertex t (all the flow ends in the sink)....
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