On approximating shortest paths in weighted triangular tessellations

Approximating Weighted Shortest Paths on Polyhedral Surfaces yz

Consider a simple polyhedron P, possibly non-convex, composed of n triangular regions (faces), in which each region has an associated positive weight. The cost of travel through each region is the distance traveled times its weight. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, 0 (s; t), between two points s and t on the surfac...

Approximating Shortest Paths on an Nonconvex Polyhedron

We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + "), where is the length of the shortest path between s and t on @P , and " > 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) tim...

Approximating Geometric Bottleneck Shortest Paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: Given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms f...

Shortest Weighted Paths II

Lemma 1.1. Consider a (directed) weighted graph G = (V, E), w : E → R+ ∪ {0} with no negative edge weights, a source vertex s and an arbitrary distance value d ∈ R+ ∪ {0}. Let X = X ′ ∪ X ′′ , where X ′ = {v ∈ V : δ(s, v)< d} be the set of vertices that are less than d from s, X ′′ ⊆ {v ∈ V : δ(s, v) = d} be the set of vertices that are exactly d from s. Also, let d ′ = min{δ(s, u) : u ∈ V \ X ...

Approximating Shortest Paths in Arrangements of Lines