A Space Saving Trick for Directed Dynamic Transitive Closure and Shortest Path Algorithms

We present a simple space saving trick that applies to many previous algorithms for transitive closure and shortest paths in dynamic directed graphs. In these problems, an update can change all edges incident to a node. The basic queries on reachability and distances should be answered in constant time, but also paths should be produced in time proportional to their length. For: Transitive closure of Demetrescu and Italiano (FOCS 2000) Space reduction from O(n3) to O(n2), preserving an amortized update time of O(n2). Exact all-pairs shortest dipaths of King (FOCS 1999) Space reduction from O(n3) to O(n2√nb), preserving an amortized update time of O(n2√nb), where b is the maximal edge weight. Approximate all-pairs shortest dipaths of King (FOCS 1999) Space reduction from O(n3) to On2, preserving an amortized update time of On2. Several authors (Demetrescu and Italiano, FOCS 2000, and Brown and King, Oberwolfach 2000) had discovered techniques to give a corresponding space reduction, but these techniques could be used to show only the existence of a desired dipath, and could not be used to produce the actual path.