Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs

This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b. For approximate shortest paths with an error factor of (2+ε), for any positive constant ε, the amortized update time is O(n² log² n/log log n); for an error factor of (1+ε) the amortized update time is O(n² log ³ (bn)/ε²). For exact shortest paths the amortized update time is O(n^2.5 √(b log n)). Query time for exact and approximate shortest distances is O(1); exact time and approximate paths can be generated in time proportional to their lengths. Also presented is a fully dynamic transitive closure algorithm with update time O(n² log n) and query time O(1). The previously known fully dynamic transitive closure algorithm with fast query time has one-sided error and update time O(n^2.28). The algorithms use simple data structures, and are deterministic