A Delaunay triangulation-based shortest path algorithm with O(n log n) time in the Euclidean plane (original) (raw)

An $\bm{O(n\log n)}$ Shortest Path Algorithm Based on Delaunay Triangulation

IEEE/ASME Transactions on Mechatronics, 2000

In Euclidean and/or λ-geometry planes with obstacles, the shortest path problem involves determining the shortest path between a source and a destination. There are three different approaches to solve this problem in the Euclidean plane: roadmaps, cell decomposition, and potential field. In the roadmaps approach, a visibility graph is considered to be one of the most widely used methods. In this paper, we present a novel method based on the concepts of Delaunay triangulation, an improved Dijkstra algorithm and the Fermat points to construct a reduced visibility graph that can obtain the near-shortest path in a very short amount of computational time. The length of path obtained using our algorithm is the shortest in comparison to the other fastest algorithms with O(n log n) time complexity. The proposed fast algorithm is especially suitable for those applications which require determining the shortest connectivity between points in the Euclidean plane, such as the robot arm path planning and motion planning for a vehicle.

Shortest Paths Among Obstacles in the Plane

International Journal of Computational Geometry & Applications, 1996

We give a subquadratic (O(n3/2+∊) time and O(n) space) algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles; previous time bounds were at least quadratic in n, in the worst case. The method avoids use of visibility graphs, relying instead on the continuous Dijkstra paradigm. The output is a shortest path map (of size O(n)) with respect to a given source point, which allows shortest path length queries to be answered in time O( log n). The algorithm extends to the case of multiple source points, yielding a method to compute a Voronoi diagram with respect to the shortest path metric.

Improving Shortest Paths in the Delaunay Triangulation

International Journal of Computational Geometry & Applications, 2012

We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set S, we look for a new point p ∉ S that can be added, such that the shortest path from s to t, in the Delaunay triangulation of S∪{p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed.

An Optimal Algorithm for Euclidean Shortest Paths in the Plane

SIAM Journal on Computing, 1999

We propose an optimal-time algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worst-case time O(n logn) and requires O(n logn) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an e cient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a xed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a signi cant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving non-point and multiple sources.

Two-step and Apex-angle Routing Algorithms for Delaunay Triangulations

2010

ABSTRACT Memoryless online routing (MOR) algorithms are suitable for the applications only using local information to discover paths, and Delaunay triangulations are the class of geometric graphs widely proposed as wireless network topologies. Motivated by these two facts, this paper reports two new MOR algorithms for the Delaunay triangulations: the Two-step Routing algorithm and the Apex-angle Routing algorithm, thus greatly enriching the family of such algorithms.

An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane

Discrete and Computational …, 1997

1Department of Computer Science and Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, India {skapoor, snm}@cse.iitd.ernet.in 2Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794–3600, USA ...

Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

SIAM Journal on Computing, 2015

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-θ 6-graph 1 (the half-θ 6-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most 5/ √ 3 ≈ 2.887 times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-θ 6-graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-θ 6-graph using O(log n) bits per vertex coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of 15/ √ 3 ≈ 8.660 on two bounded degree subgraphs of the half-θ 6-graph.

Computing Shortest Paths in the Plane with Removable Obstacles

2018

We consider the problem of computing a Euclidean shortest path in the presence of removable obstacles in the plane. In particular, we have a collection of pairwise-disjoint polygonal obstacles, each of which may be removed at some cost c_i > 0. Given a cost budget C > 0, and a pair of points s, t, which obstacles should be removed to minimize the path length from s to t in the remaining workspace? We show that this problem is NP-hard even if the obstacles are vertical line segments. Our main result is a fully-polynomial time approximation scheme (FPTAS) for the case of convex polygons. Specifically, we compute an (1 + epsilon)-approximate shortest path in time O({nh}/{epsilon^2} log n log n/epsilon) with removal cost at most (1+epsilon)C, where h is the number of obstacles, n is the total number of obstacle vertices, and epsilon in (0, 1) is a user-specified parameter. Our approximation scheme also solves a shortest path problem for a stochastic model of obstacles, where each ...

Ellipse and Arc Routing Algorithms for Delaunay Tringulations

2009

Abstract—Oblivious online routing (OOR) algorithms are suitable when applications only have local information available to make routing decisions. This paper presents two new OOR algorithms for Delaunay triangulations: the Ellipse Routing algorithm and the Arc Routing algorithm. Both of their names come from the shapes of their searching areas for the next-hop neighbor. This paper also evaluates and compares the presented algorithms with three existing OOR algorithms in terms of link distances and Euclidean distances.

Minimum-link paths among obstacles in the plane

Algorithmica, 1992

Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in time O(E (n) log 2 n) (and space O(E)), where n is the total number of edges of the obstacles, E is the size of the visibility graph, and (n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted at s) of minimum-link paths from s to all obstacle vertices. This leads to a method of solving the query version of our problem (for query points t).