Computing Minimum-Link Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane

Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct a polygonal path between these points which has the minimum number of segments (links) and the minimum ‘total turn’, that is the sum of absolute values of angles of turns of the consecutive polygon links. We describe an algorithm that solves the problem spending polynomial time to construct one segment of the minimum-link and minimum-turn polygon if to use a modification of real RAMs which permits to handle the solutions of algebraic equations. It is known that the number of segments in such a minimum-link polygon can be exponential as function of the length of the input data or even of the degree of polynomials representing the semi-algebraic set. In fact, we describe how to construct a minimum-link-turn path for a given class of homotopy(whose shortest path has no self-intersections), and provide a rigorous and rather a universal way of reasoning about homotopy classes in contexts related to algorithms. It was previously shown by Heintz-Krick-Slissenko-Solerno that a shortest path in the situation under consideration is semi-algebraic, and an extended real RAM that is able to compute integrals of algebraic functions can find it in polytime.

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