On-line and Dynamic Shortest Paths through Graph Decompositions (Preliminary Version)

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. We give both sequential and parallel algorithms that work on a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. The parallel algorithms presented here are the first known ones for solving this problem. Our results can be extended to hold for digraphs of genus $o(n)$