An O(n3loglogn/logn) time algorithm for the all-pairs shortest path problem

In this paper we consider the all-pairs shortest path (APSP) problem, which computes shortest paths between all pairs of vertices of a directed graph with non-negative real numbers as edge costs. We present an algorithm that computes shortest distances between all pairs of vertices, since shortest paths can be computed as by-products in our algorithm. It is well known that the time complexity of (n, n)-distance matrix multiplication (DMM) is asymptotically equal to that of the APSP problem for a graph with n vertices. Thus we concentrate on DMM in this paper. The computational model in this paper is the conventional RAM, where only arithmetic operations, branching operations, and random accessibility with O(log n) bits are allowed. Fredman [4] was the first to break the cubic complexity of O(n3) under RAM, giving O(n3(log log n/ log n)1/3). This complexity was improved to O(n3(log log n/ log n)1/2) by Takaoka [6] with RAM, and to O(n3/(log n)1/2) by Dobosievicz [3] with extended RAM instructions. Recently there have been some more progresses such as O(n3(log log n)/ log n)5/7) [5] and O(n3(log log n)2/ log n) [9]. In this paper we improve the complexity further to O(n3 log log n/ log n). If edge costs are small integers, the complexity becomes more subcubic, i.e., O(n3− ) for some > 0, as shown in [7], [1], and [11]. We follow the same framework as those in [4], [6], and [9]. That is, we take a two level divide-and-conquer approach. To multiply the small matrices resulting from dividing the original matrices, we sort distance data, and use the ranks of those data in the sorted lists. As the ranks are small integers, the multiplication can be done efficiently by looking at some precomputed tables.