Network flow and generalized path compression

Abstract

An 0(EViog2V) algorithm for finding the maximal flow in networks is described. It is asymptotically better than the other known algorithms if E = O(V 2-~) for some e>0. The analysis of the running time exploits the discovery of a phenomenon similar to (but more general thanl path com~ pression, although the union find algorithm is not used. The time bound is shown to be tight in terms of V and E by exhibiting a family of networks that require ~(_EViog2V) ++ time.

Quantum Algorithms for Matching and Network Flows

“…Each of them can be processed in time O(n 2 ). For sparse real networks, the fastest known algorithm runs in time O(nm(log n) 2 ) [GN79]. Let us restrict the setting to integer capacities bounded by U .…”

“…For networks with real capacities, the fastest algorithms run in time O(n 3 ) [Kar74,MKM78]. If the network is sparse, one can achieve a faster time O(nm(log n) 2 ) [GN79]. If all capacities are integers bounded by U , the maximal flow can be found in time O(min(n 2/3 m, m 3/2 ) log(n 2 /m) log U ) [GR98].…”

Minimum s-t cut of a planar undirected network in o(n log2(n)) time

“…Our algorithm runs in an order of magnitude less time than previous algorithms for this problem. An additional attractive feature of this algorithm is its simplicity, as compared to other algorithms for computing minimum s-t cuts for sparse networks (Galil, Naamad, 1979] and [Shiloach, 1978]. …”

“…The best known algorithms [Galil, Naamad, 1979;Shiloach, 1978] for computing the max flow or minimum s-t cut of a sparse directed or undirected network (with n vertices and O(n) edges) has time O(n 2 log2(n)). This paper is concerned with a planar undirected network N, which occurs in many practical applications.…”

Testing Deadlock-Freedom of Computer Systems

“…Recently a number of new algorithms for finding a maximal network flow has been published (see [10,11] for references). Among these we fred Dinic's algorithm [7] most useful for our purpose.…”