Two familiar transitive closure algorithms which admit no polynomial time, sublinear space implementations

Abstract

Any Boolean straight-line program which computes the transitive closure of an nxn Boolean matrix by successive squaring requires time exceeding any polynomial in n if the space used is o(n). This is the first demonstration of a "natural" algorithm which (1) has a polynomial time implementation and (2) has a small (e.g., O(log2n)) space implementation, but (3) has no implementation running in polynomial time and small space simultaneously. It is also shown that any implementation of Warshall's transitive closure algorithm requires Ω(n) space.