This paper gives approximation algorithms of solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an e-short path, that is, a path with length within (1 + e) of shortest. Let n be the total number of faces of the polyhedral obstacles, and e a given value satisfying O e ≤ p. The algorithm for the planar case requires O(n log n)/e time to build a data structure of size O(n/e). Given points s and t, and e-short path from s to t can be found with the use of the data structure in time O(n/e + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ E3 and points s, t e E3, an e-short path between s and t can be found in O(n2l(n) log(n/e)/e4 + n2 lognp log(n logp)) time, where p is the ratio of the length of the longest obstacle edge to the distance between s to t. The function l(n) = a(n)O(a(n)O(1)), where the a(n) is a form of inverse of Ackermann's function. For log(1/e) and log p that are O(log n), this bound is O(log n2(n)l(n)/e4).

Approximation algorithms for shortest path motion planning

Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less than √10 times |ab|, where |ab| is the straight-line Euclidean distance between a and b. Thus, a shortest path in this planar graph is less than about 3 times longer than the corresponding straight-line distance. The triangulation that has this property is the L1 metric Delaunay triangulation for the set 5. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O(n log n) time.

There is a planar graph almost as good as the complete graph

Methods are described to unfold and flatten the curved, convoluted surfaces of the brain in order to study the functional architectures and neural maps embedded in them. In order to do this, it is necessary to solve the general mapmaker's problem for representing curved surfaces by planar models. This algorithm has applications in areas other than computer-aided neuroanatomy, such as robotics motion planning and geophysics. The algorithm maximizes the goodness of fit distances in these surfaces to distances in a planar configuration of points. It is illustrated with a flattening of monkey visual cortex, which is an extremely complex folded surface. Distance errors in the range of several percent are found, with isolated regions of larger error, for the class of cortical surfaces studied so far. >

A numerical solution to the generalized mapmaker's problem: flattening nonconvex polyhedral surfaces

Computing an optimal path in a geometric domain is a fundamental problem in computational geometry, with applications in robotics, geographic information systems (GIS), wire routing, etc.

Shortest paths and networks

A survey of motion planning and related geometric algorithms

We consider the problem of computing the Euclidean shortest path between two points in three-dimensional space which must avoid the interiors of k given disjoint convex polyhedral obstacles, having altogether n faces. Although this problem is hard to solve when k is arbitrarily large, it had been efficiently solved by Mount [Mo84] (cf. also Sharir and Schorr [SS84]) for k = 1, i.e. in the presence of a single convex polyhedral obstacle, in time O(n2log n). In this paper we consider the generalization of this technique to the cases k = 2 and k > 2. In the first part of this presentation we describe an algorithm which calculates shortest paths amidst two convex polyhedral obstacles in time O(n3a(n)O(a(n)7)log n), where a(n) is the functional inverse of Ackermann's function (and is thus extremely slowly growing). This result is achieved by constructing a new kind of Voronoi diagram, called peeper's Voronoi diagram, which is introduced and analyzed here. In the second part we show that shortest paths amidst k > 2 disjoint convex polyhedral obstacles can be calculated in time polynomial in the total number n of faces of these obstacles (but exponential in the number of obstacles). This is a consequence of the following result: Let K be a 3-D convex polyhedron having n vertices. Then the number of shortest-path edge sequences on K is polynomial in n (specifically O(n7)), where a shortest-path edge sequence x is a sequence of edges of K for which there exist two points X, Y on the surface S of K such that x is the sequence of edges crossed by the shortest path from X to Y along S.

A Sharir

Papers

On Shortest Paths Amidst Convex Polyhedra