Subcubic Cost Algorithms for the All Pairs Shortest Path Problem

Abstract

In this paper we give three sub-cubic cost algorithms for the all pairs shortest distance APSD and path APSP problems. The rst is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O log 2 n time with O n p log n processors where = 2:688 on an EREW-PRAM. The second parallel algorithm solves the APSP, and consequently APSD, problem for a directed graph with non-negative general costs real numbers in O log 2 n time with o n 3 subcubic cost. Previously this cost was greater than O n 3 . Finally we improve with respect to M the complexity O M n of a sequential algorithm for a graph with edge costs up to M into O M 1=3 n 6+! =3 log n 2=3 log log n 1=3 in the APSD problem, where ! = 2 :376.

All pairs shortest paths using bridging sets and rectangular matrix multiplication

“…Alon, Galil and Margalit [AGM97] (see also Takaoka [Tak98]) describe a way of using fast matrix multiplication, and fast integer multiplication, to compute distance products of matrices whose elements are taken from the set {−M, . .…”

“…, M }, i.e., integer weights of absolute value at most M . The running time of their algorithm is thenÕ( [Tak98] obtained an algorithm whose running time isÕ(M 1/3 n (6+ω)/3 ). The bound of Takaoka is better than the bound of Alon, Galil and Margalit for larger values of M .…”

“…To convert distance products of matrices into normal algebraic products we use a technique suggested in [AGM97] (see also Takaoka [Tak98]), based on a previous idea of Yuval [Yuv76]. Suppose that A = (a ij ) and B = (b ij ) are two n × n matrices with elements taken from the set {−M, .…”

Efficient Algorithms for the 2-Center Problems

“…If edge costs are small non-negative integers, the complexity for APSP becomes deeply sub-cubic, i.e., O(n 3−ǫ ) for some ǫ > 0, as shown in [14], [2], [17] and [19]. It is interesting to investigate whether we can use those sub-cubic algorithms for the APSP problem for the 2-center problems.…”

Soundness of Resource-Constrained Workflow Nets

“…We now check our condition: Observe that the computation proposed strongly resembles the All-Pairs Shortest Paths problem (Floyd-Warshal algorithm, see [7]; for a more efficient algorithm see [15]; also see [16] for a survey). Hence, our computation can benefit from efficient matrix multiplication algorithms.…”