Dynamic algorithms for shortest paths in planar graphs

Abstract

We propose data structures for maintaining shortest path in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3). Our data structure can be also applied to the problem of maintaining the maximum flow problem in an s - t planar network. As far as the all pairs shortest path problem is concerned, we are interested in computing the shortest distances between q pairs of nodes after the weight of an edge has been modified. We obtain different bounds depending on q. We also obtain an algorithm that compares favourably with the best off-line algorithm for computing the all pairs shortest path if we are interested in computing only a subset of the O(n2) possible pairs. Namely, we show how to obtain an o(n2) algorithm for computing the shortest path between q pairs of nodes whenever q = o(n2).