Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time

weight. We also present an even faster (1 + ɛ) approximate algorithm for the simpler problem of approximating the k shortest simple s − t paths in a directed graph with positive edge weights. That is, our algorithm outputs k different simple s−t paths, where the kth path we output is a (1 + ɛ

We describe a new algorithm to enumerate the k shortest simple (loopless) paths in a directed graph and report on its implementation. Our algorithm is based on a replacement paths algorithm proposed recently by Hershberger and Suri [7], and can yield a factor #(n) improvement for this problem

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(|V |) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so

motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [21, 13] repeatedly computes the replacement paths from s to t. Its running time is O(kn(m + n log n)). Second, the computation of Vickrey pricing of edges

Given a simple polygon P and a point p 2 P , we show how to maintain the visibility polygon from p, the shortest path tree from p, and the corresponding shortest path partition as p is translated inside P . Given a direction of motion of p, we can determine how far p can move until the first

problem on an n-node m-edge undirected graph (respectively, directed acyclic graph) to the single-source shortest-paths problem on an O(n)-node O(m)-edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacement-paths problem is no harder than the single

shortest paths of a given FST as fast as possible. Before we actually look at the algorithms that find top k paths on a given graph, we should be aware of that an FST is a directed acyclic graph (DAG) with extra input and output labels attached to each edge, and it also has multiple edges between 2 nodes

shortest path from u to v in G. Add the edges of this path to H. We show that, with a very simple initial graph H, this naïve method gives additive 6- and 2-spanners of sizes matching the best known upper bounds. For additive 2-spanners we start with H “ H and end with Opn3{2q edges in the spanner

the results reported in the following sections utilize the binary matrix, we note that all our results are robust to using network centrality measures calculated from valued matrices. 6 Unlike the undirected matrix, the directed matrix does not record a tie between VCs j and k who were members of the same