A note on finding minimum cuts in directed planar networks by parallel computations

Abstract

In this note we show that the result of Johnson and Venkatesan [4] can be improved using similar ideas as in [3]. The result of concern can be stated as follows: The minimum cut in general planar networks can be found in O(log n 2 ) time by O(n 6 ) processors. We reduce the exponent to 3 in the case when the network is embedded in the plane beforehand and to 4 otherwise. The reader is supposed to be familiar with graphs, planar graphs, multigraphs and so on (see [1]). We use the well-known method in a similar way as in [4] which consists in the fact that the problem of finding minimum cuts in a network can be reduced to the shortest path prob- lem in its dual. While the notion of the dual multigraph of an undirected planar graph is clear and well known, the dual of a directed graph is not so easy to define. For this reason (to avoid techni- cal problems) we work with undirected graphs only. The problem of finding minimum cuts in net- works is a rather popular one since it is strongly connected to the problem of finding maximum flows. In fact, these two problems nearly coincide. The only difference consists in the complexity of these two problems. A fast algorithm for finding minimum cuts does not imply the existence of an other equally fast algorithm for finding maximum flows. This is the case (to our current knowledge) of general (i.e., not s-t-planar) planar networks. Nonetheless, the value of minimum cut in a net- work is of great importance anyway and can be used when finding effective maximum flow finding algorithms as shown in [5], for example, where the authors have found an effective procedure to find a flow of given value if such a flow exists at all. This procedure yields obviously to fast two stage algorithms, finding the minimum cut value first and then the maximum flow. The notions cut and disconnecting set [6] coin- cide, the first becoming more used in the current literature. 1. Definition. Let (V, E) be a connected, planar and undirected graph. For a, b ~ V, a-b will stand for the edge {a, b} and a -~ b respectively b ~ a will denote the two orientations of a-b. 2. Definition. A