Approximation algorithms for computing minimum exposure paths in a sensor field

The exposure of a path p in a sensor field is a measure of the likelihood that an object traveling along p is detected by at least one sensor from a network of sensors, and is formally defined as an integral over all points x of p of the sensibility (the strength of the signal coming from x) times the element of path length. The minimum exposure path (MEP) problem is, given a pair of points x and y inside a sensor field, to find a path between x and y of minimum exposure. In this article we introduce the first rigorous treatment of the problem, designing an approximation algorithm for the MEP problem with guaranteed performance characteristics. Given a convex polygon P of size n with O(n) sensors inside it and any real number &epsis;0, our algorithm finds a path in P whose exposure is within an 1+&epsis; factor of the exposure of the MEP, in time O(n/&epsis;2 ψlog n), where ψ is a geometric characteristic of the field. We also describe a framework for a faster implementation of our algorithm, which reduces the time by a factor of approximately θ(1/&epsis;), while keeping the same approximation ratio.