Abstract. In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε −1 n 2/3 log 2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph.
Dynamic all-pairs shortest path problem in planar graphs was initiated by Klein and Subramanian REF , who showed how to maintain a (1 + ε)-approximation to shortest paths in O(n 2/3 log 2 n log D) amortized update time and O(n 2/3 log 2 n log D) worst-case query time, where D denotes the sum of edge lengths.
A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs
Algorithmica
Algorithmica
Mathematics Computer Science