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A New Approach to the Maximum Flow Problem

Report ID:
June 1986
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All previously known efficient maximum flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest length augmenting paths at once (using the layered network approach of Dinic). We introduce an alternative method based on
the preflow concept of Karzanov. A preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths.
The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense graphs, achieving an O(n3) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running
in O(nm log(n2/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efficient distributed and parallel implementations. We obtain a parallel implementation running in O(n2 log n) time and
using only O(m) space. This time bound matches that of the Shiloach-Vishkin algorithm, which requires O(n2) space.

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