An efficient algorithm for the three-guard problem Original Research Article

Given a simple polygon image with two vertices image and image, the three-guard problem asks whether three guards can move from image to image such that the first and third guards are separately on two boundary chains of image from image to image and the second guard is always kept to be visible from two other guards inside image. It is a generalization of the well-known two-guard problem, in which two guards move on the boundary chains from image to image and are always kept to be mutually visible. In this paper, we introduce the concept of link-2-ray shots, which can be considered as ray shots under the notion of link-2-visibility. Then, we show a one-to-one correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards, and generalize the solution for the two-guard problem to that for the three-guard problem. We can decide whether there exists a solution for the three-guard problem in image time, and if so generate a walk in image time, where image denotes the number of vertices of image and image the size of the optimal walk. This improves upon the previous time bounds image and image, respectively. Moreover, our results can be used to solve other more sophisticated geometric problems.