We present an algorithm to find an Euclidean Shortest Path from a source vertex s to a sink vertex t in the presence of obstacles in R. Our algorithm takes O(T +m(lgm)(lg n)) time and O(n) space. Here, O(T ) is the time to triangulate the polygonal region, m is the number of obstacles, and n is the number of vertices. This bound is close to the known lower bound of O(n+m lgm) time and O(n) spac...

Chazelle’s triangulation [1] forms today the common basis for linear-time Euclidean shortest path (ESP) calculations (where start and end point are given within a simple polygon). This paper provides an alternative method for subdividing a simple polygon into “basic shapes”, using trapezoids instead of triangles. The authors consider the presented method as being substantially simpler than the ...

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k-1) trees (each of which contains a previous best path). An O( ( log ) m K n K   ) time and O(K+m) ...

Given a network modeled as a graph G with each link associated with a cost and k weights, the Constrained Shortest Path (CSP(k)) problem asks for computing a minimum cost path from a source node s to a target node t satisfying pre-specified bounds on path weights. This problem is NP-hard. In this paper we propose a new approximation algorithm called GEN-LARAC for CSP(k) problem based on Lagrang...

Given an edge-weighted undirected graph and two vertices s and t, the next-to-shortest path problem is to find an st-path whose length is minimum among all st-paths of lengths strictly larger than the shortest path length. The problem is shown to be polynomially solvable if all edge weights are positive, while the complexity status for the nonnegative weight case was open. In this paper we show...

Abstract This paper studies a generalization of the shortest path tour problem with time windows ( GSPTPTW ). The aim is to find single-origin single-destination path, which has pass through an ordered sequence not necessarily disjoint node-subsets. Each node window for each node-subset it belongs. We investigate theoretical properties and propose dynamic programming approach solve it. Numerica...

Given an edge-weighted undirected graph and two vertices s and t, the next-to-shortest path problem is to find an st-path whose length is minimum among all st-paths of lengths strictly larger than the shortest path length. The problem is shown to be polynomially solvable if all edge weights are positive, while the complexity status for the nonnegative weight case was open. In this paper we show...

Suppose one has a line arrangement in which many lines are parallel, so that the number of diierent line orientations is k, and one wants to nd a shortest path from one point on a line in the arrangement to another such point. Using known techniques one can nd the shortest path in time and space O(n 2). We present an algorithm that can nd the shortest path in time and space O(n + k 2).