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

Abstract

SUMMARYLet N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge's cost) from a set L. This paper presents an algorithm for computing a minimum (cost) s-t cut of N. For general L, this algorithm runs in time O(n log2(n)) time on a (uniform cost criteria) RAM. 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).

Maximum weight independent set in trees

“…If c is marked free in step [10] then at least one of vl, vz ..... Vk is not free and the expression (total + w(c)) is negative or zero. Clearly, by lemma 3.2, there can be no augmenting subtree in the tree T. Now consider the case in which the centroid c gets added to the independent set.…”

“…Let grl be its gain.) [9] if vi is not free then MAX-GAIN-PAT(v, T/, grl) total := total +gr~ end [10] if (totat+w(c)) < 0, then mark c as free else [11] Invert the maximum gain proper alternating trees (possibly none) found in and add c to the maximum weight independent set in T. [12] Reorient the tree to make r the root of T. end {INDSET} step [9], Algorithm 3.2.…”

“…In the next two steps (i.e. [10] and [11]), gain of vertex w is computed after gain of each child of w is determined. Now if gain of w is negative or zero, and w is free then Tw, a subtree consisting of w and its descendants, is deleted from T~ (step [12] of GAIN).…”

“…LEMMA 3.3: If step [10] or step [11] is executed by Algorithm 3.1 then there will be no augmenting subtree of T w.r.t, the resulting independent set.…”

“…Note that, although a tree is a planar graph, after addition of the source (s), the terminal (t), and the dummy edges to the given tree, the resulting graph may' not be planar. This prevents us from using an efficient algorithm for a minimum cut in an undirected planar network [10]. Recently, Coppersmith and Vishkin [1] have developed an algorithm for NPhard problems in "almost trees", but they do not consider the weighted case.…”