Geodesic ham-sandwich cuts

Abstract

Let P be a simple polygon with m vertices, k of which are reflex, and which contains r red points and b blue points in its interior. Let n=m+r+b. A ham-sandwich geodesic is a shortest path in P between any two points on the boundary of P that simultaneously bisects the red points and the blue points. We present an O (n log k)-time algorithm for finding a ham-sandwich geodesic. We also show that this algorithm is optimal in thealgebraic computation tree model when parameterizing the running time with respect to n and k.