Optimal constrained graph exploration
ACM Transactions on Algorithms (TALG)
Survey: Online algorithms for searching and exploration in the plane
Computer Science Review
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We consider the following scenario. A point robot is placed at some start location s in a 2-dimensional scene containing oriented rectangular obstacles. The robot must repeatedly travel back and forth between s and a second location t in the scene. The robot knows the coordinates of s and t but initially knows nothing about the positions or sizes of the obstacles. It can only determine the obstacles' locations by bumping into them. We would like an intelligent strategy for the robot so that its trips between s and t both are relatively fast initially and improve as more trips are taken and more information is gathered.In this paper we describe an algorithm for this problem with the following guarantee: in the first $k \leq n$ trips, the average distance per trip is at most $O(\sqrt{n/k})$ times the length of the shortest s-t path in the scene, where n is the Euclidean distance between s and t. We also show a matching lower bound for deterministic strategies. These results generalize known bounds on the one-trip problem. Our algorithm is based on a novel method for making an optimal trade-off between search effort and the goodness of the path found. We improve this algorithm to a "smooth" variant having the property that for every $i \leq n,$ the robot's ith trip length is $O(\sqrt{n/i})$ times the shortest s-t path length.A key idea of this paper is a method for analyzing obstacle scenes using a tree structure that can be defined based on the positions of the obstacles.