In this thesis we consider the problem faced by an agent who is tasked with inspecting certain objects within the plane. In particular, given some starting position not interior to a convex polygon the agent must traverse through the plane to fully see all edges of the polygon. However traversal is expensive for the agent and so the agent would like to determine the shortest path it can take such that all edges of the polygon will be seen from the path. For all polygons and feasible starting positions we present an efficient algorithm for determining what the shortest path is by characterizing properties that any such path must exhibit. Next we consider how the agent should accomplish this task when it is only aware that the polygon it is to inspect is convex, but not its shape (i.e. location of edges.) In such a scenario the agent must search for the edges by traversing through the plane. Yet if such searching is not done in an efficient manner, the distance traversed may be unacceptably large with respect to the length of the optimal path. We discuss several strategies that the agent can employ that will ensure that the distance it traverses to see all edges is at most a constant times the length of the optimal path.

On competitive strategies for external exploration of a convex polygon

A robot is free to move in a non-convex polygonal region, starting against an edge on the boundary. Ahead of the robot at one unit of distance is a corner, i.e. a reflex vertex (see Figure 1). The angle θ is unknown to the robot. The robot’s task is to look at the region around the corner, which it initially cannot see. We let ϕ = π − θ .I fϕ ≥ π/2, the robot is best off moving straight to the vertex. However, if ϕ is close to 0, much shorter paths exist, making this solution suboptimal. Therefore we ask: What is the best path for the robot to follow? In this paper, we look into the problem of finding an optimal average-case strategy under a homogeneous probability distribution for ϕ. The average-case performance of a strategy is measured by its average cost, defined as the expected value of the strategy’s competitive function. Given a value for ϕ, the competitive function of a strategy gives the ratio of the distance the robot travels to look around the corner (as prescribed by the strategy) to the shortest distance it must travel to do so. We give strong evidence that an optimal average-case strategy exists and achieves an average cost of ∼ 1.189.

/pdf/optimal-average-case-strategy-for-looking-around-a-corner-2sdd9ej61a.pdf

Selim Tawfik

Papers

Optimal Average Case Strategy for Looking around a Corner