Incremental algorithms for minimal length paths

We consider the problem of maintaining on-line a solution to the All Pairs Shortest Paths Problem in a directed graph G = (V,E) where edges may be dynamically inserted or have their cost decreased. For the case of integer edge costs in a given range [1…C], we introduce a new data structure which is able to answer queries concerning the length of the shortest path between any two vertices in constant time and to trace out the shortest path between any two vertices in time linear in the number of edges reported. The total time required to maintain the data structure under a sequence of at most O(n2) edge insertions and at most O(Cn2) edge cost decreases is O(Cn3 log(nC)) in the worst case, where n is the total number of vertices in G. For the case of unit edge costs, the total time required to maintain the data structure under a sequence of at most O(n2) insertions of edges becomes O(n3 logn) in the worst case. The same bounds can be achieved for the problem of maintaining on-line longest paths in directed acyclic graphs. All our algorithms improve previously known algorithms and are only a logarithmic factor away from the best possible bounds.