Maximum Flow in Planar Networks

Abstract

Efficient algorithms for finding maximum flow in planar networks are presented. These algorithms take advantage of the planarity and are superior to the most efficient algorithms to date, If the source and the terminal are on the same face, an algorithm of Berge is improved and its time complexity is reduced to O(n log n). In the general case, for a given D > 0 a flow of value D is found if one exists; otherwise, it is indicated thatno such flow exists. This algorithm requires O(n 2 log n) time. If the network is undirected a minimum cut may be found in O(n log n) time. All algorithms require O(n)space.

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

“…The fastest previous algorithm for computing a minimum s-t cut of a planar undirected network [Gomory and Hu, 1961] and [Itai and Shiloach, 1979] has time O(n 2 log(n)) and the best previous time bound for minimum s-t cut of a planar graph (Cheston, Probert, and Saxton, 1977] was O(n2).…”

“…This paper is concerned with a planar undirected network N, which occurs in many practical applications. Ford and Fulkerson [1956] have an elegant minimum s-t cut algorithm for the case N is (s,t)-planar (both s and t are on the same face) which efficiently implemented by Gomory and Hu [1961] and Itai and Shiloach [1979] has time O(n log(n)).…”

“…For the case L contains only integers ~n 0(1) , the algorithm runs in time O(n log(n)loglog(n))° Our algorithm also constructs a minimum s-t cut of a planar graph (i.e., for the case L = {i}) in time 0(n log(n)).The fastest previous algorithm for computing a minimum s-t cut of a planar undirected network [Gomory and Hu, 1961] and [Itai and Shiloach, 1979] has time O(n 2 log(n)) and the best previous time bound for minimum s-t cut of a planar graph (Cheston, Probert, and Saxton, 1977] was O(n2). …”

The Maximum Flows in Planar Dynamic Networks

“…The computation of a maximum flow in a planar static network has been investigated by many researchers starting from the work of Ford and Fulkerson [5] who developed an O(n 2 ) time algorithm for (1, n) networks when the source node 1 and sink node n are on the same face. This algorithm was later improved to O(n log n) time by Itai and Shiloach [8]. By introducing the concept of potentials, Hassin [6] gave an algorithm that run in O(n log 0.5 n) time using Frederickson ′ s shortest path algorithm [4].…”

Counting Minimum (s,t)-Cuts in Weighted Planar Graphs in Polynomial Time

“…Building on the work of Itai and Shiloach [12], Reif [20] developed an O(n log 2 n) divide-andconquer algorithm for undirected graphs. Janiga and Koubek [13] designed an O(n log 2 n log log n) algorithm for directed planar graphs.…”