Finding optimal weighted bridges with applications

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. B_w, 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 B_w 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 + ε)-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.