Greg N. Frederickson
Abstract
An algorithm is presented for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with real-valued edge costs but no negative cycles. The algorithm runs in O(pn) time, where n is the number of vertices in G, and p is the minimum cardinality of a subset of the faces that cover all vertices, taken over all planar embeddings of G. The algorithm is based on a decomposition of the graph into O(pn) outerplanar subgraphs satisfying certain separator properties. Linear-time algorithms are presented for various subproblems including that of finding an appropriate embedding of G and a corresponding face-on-vertex covering of cardinality O(p), and of generating all pairs shortest path information in a directed outerplannar graph.
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Index Terms
- Planar graph decomposition and all pairs shortest paths
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Reviews
William Fennell Smyth
Let G denote a directed planar graph of order n with real edge weights and no negative cycles, and let G denote an embedding of G in the plane. A face-on-vertex covering (FOVC) of G is a set of faces of G that cover all vertices of G. Let p denote the minimum cardinality of any FOVC over all embeddings of G. This paper describes a method of computing all pairs shortest path data for G in time O ( pn ). The shortest path data are encoded into routing tables, which are based on a naming of the vertices of G with distinct integers in [1, n ]. For each vertex v , these tables give the label of each arc vw , that is, a list of ranges of names of vertices, each of which has a shortest path from v that begins with arc vw . The method breaks down into two main phases. The first phase takes O ( n ) time to find an embedding G of G and an FOVC of G that is of cardinality q 4 p . This phase uses a generic algorithm for NP-hard problems that, given an embedding G , yields an FOVC of cardinality less than twice that of a minimum-cardinality FOVC of G . The second phase generates routing tables for G based on a given embedding G and corresponding FOVC of cardinality q. This phase consists of five main steps: Name the vertices of G in clockwise order around each face of the FOVC. This step requires O ( n ) time. Decompose G into q hammocks (subgraphs of G consisting of exactly two disjoint simple paths together with additional edges joining vertices in one path to vertices in the other). Each hammock has exactly four attachment vertices (the ends of the two paths) that connect it to the rest of G . This step requires O ( n ) time. Compute the shortest paths between every pair of attachment vertices, using a technique for hammock compression and the monotonicity of a certain distance function over the faces of a planar graph. This step requires O ( q 2 ) time. Using the shortest path data for attachment vertices, compute routing tables for all shortest paths from vertices in one hammock to vertices in another. This step requires O ( qn ) time. Employing an algorithm to compute shortest paths in outer planar graphs (those for which q = 1), and making use of deques with heap order, compute routing tables for all shortest paths between vertices in a single hammock. This step requires O ( n ) time. For anyone interested in planar graphs or shortest path problems, this paper is important. It combines many new insights with a variety of results that have appeared over the last decade to produce a lengthy and intricate algorithm of major theoretical interest. The paper is not easy to read, but the exposition is well organized and, with one or two minor lapses, clear. Examples and figures are used to good effect.
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