Boundary sketching with asymptotically optimal distance and rotation

Abstract

We address the problem of designing a distributed algorithm for two robots that sketches the boundary of an unknown shape. Critically, we assume a certain amount of delay in how quickly our robots can react to external feedback. In particular, when a robot moves, it commits to move along path of length at least λ, or turn an amount of radians at least λ for some positive λ≤1/26, that is normalized based on a unit diameter shape. Then, our algorithm outputs a polygon that is an ϵ-sketch, for ϵ=8λ, in the sense that every point on the shape boundary is within distance ϵ of the output polygon. Moreover, our costs are asymptotically optimal in two key criteria for the robots: total distance traveled and total amount of rotation.Additionally, we implement our algorithm, and illustrate its output on some specific shapes.