A linear-time algorithm for computing the Voronoi diagram of a convex polygon
Discrete & Computational Geometry
The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract)
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Computing the optimal bridge between two convex polygons
Information Processing Letters
Approximation algorithms for geometric shortest path problems
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On optimal bridges between two convex regions
Information Processing Letters
On computing the optimal bridge between two convex polygons
Information Processing Letters
Efficient algorithms for the minimum diameter bridge problem
Computational Geometry: Theory and Applications - Special issue on Discrete and computational geometry
Improved Optimal Weighted Links Algorithms
ICCS '02 Proceedings of the International Conference on Computational Science-Part III
An epsilon-Approximation for Weighted Shortest Paths on Polyhedral Surfaces
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
Determining approximate shortest paths on weighted polyhedral surfaces
Journal of the ACM (JACM)
k-link shortest paths in weighted subdivisions
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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The computation of shortest paths, distances and feature relationships is a key problem in many applications. In finding shortest distances or paths one often must respect features of the domain. For example, in medical applications such as radiation therapy, the features may include tissue density, risk to radiation exposure, etc. In computing an optimal treatment plan, one can think of these features as weights that effect a cost per unit travel distance function. In this model, the cost of travelling through 2 cm of dense bone might be more than the cost of travelling through 5 cm of very soft tissue. One possible way to model such problems is as shortest path problems in weighted regions.A special case of shortest path problems in weighted regions is that of computing an optimal weighted bridge between two regions. In this version, we are given two disjoint convex polygons P and Q in a weighted subdivision R. A weighted bridge, Bw, is a path from a point p ∈ P to a point q ∈ Q that connects P and Q such that the sum of the weighted length of Bw and the maximum weighted distance from any point in P to p and from any point in Q to q is minimized. The goal is to compute an optimal weighted bridge between P and Q.In this paper, we describe 2-factor and (1 + ∈)-factor approximation schemes for finding optimal 1-link weighted bridges between a pair of convex polygons. We also describe how these techniques can be extended to k-link weighted bridges and weighted bridges where the number of links is not restricted.