An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

Abstract

We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain D, consisting of n tetrahedra with positive weights, and a real number ε ∈ (0, 1), our algorithm constructs paths in D from a fixed source vertex to all vertices of D, whose costs are at most 1 + ε times the costs of (weighted) shortest paths, in O(C(D) n ε 2.5 log n ε log 3 1 ε ) time, where C(D) is a geometric parameter related to the aspect ratios of tetrahedra.The efficiency of the proposed algorithm is based on an in-depth study of the local behavior of geodesic paths and additive Voronoi diagrams in weighted threedimensional domains, which are of independent interest. The paper extends the results of Aleksandrov et al.[4] to three dimensions.

Computing highly occluded paths on a terrain

“…We ran our experiments on a computer with an Intel Core i7-3770 CPU running at 3.40 GHz and 32GB of 2 The fact that observer at height h above point p 1 ∈ Σ can see point p 2 ∈ Σ does not imply that an observer at height h above p 2 can see internal memory. The machine has a NVIDIA GeForce GTX 660 graphics card which we interfaced with using OpenGL.…”

“…There is also some work on the so-called weighted-region problem, where a weighted planar subdivision is given and the goal is to find a path of the minimum weighted length. The exact algorithms are quite expensive in 2D, but faster approximation algorithms are known; see [2] and references therein. One can assign the weights of a region based on visibility of that region from a given set of observers and can formulate the problem of computing a highly occluded path as a weighted-region problem.…”

Navigating Weighted Regions with Scattered Skinny Tetrahedra

“…11,27 There are also successful discretization schemes whose running time is linear in the input size and dependent on some geometric parameter of the polygonal domain. 5,33 In contrast, only one algorithm for the weighted region problem in 3D has been proposed (Aleksandrov et al 6 ). The authors present a (1 + ε)-approximation algorithm whose running time is O(Knε −2.5 log n ε log 3 1 ε ), where K is asymptotically at least the cubic power of the maximum aspect ratio of the tetrahedra in the worst case.…”

“…It runs in O(2 time is linear in n. In comparison, the running time of the algorithm by Aleksandrov et al 6 has the advantage of being independent from N and W , but K can be arbitrarily large even if there are only O(1) skinny tetrahedra. Putting their result in our model, K is a function of N and n in the worst case, and K can be Ω( 1 n N 3 + 1).…”

“…Our algorithm discretizes T and builds an edge-weighted graph G so that the shortest path in G is a 1+O(ε) approximation. This approach is also taken by Aleksandrov et al 6 in 3D. Unlike their approach, in order to allow for skinny tetrahedra, we discretize the fat tetrahedra only, and the edges in G represent approximate shortest paths that may not lie within a single tetrahedron.…”