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.
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