Geometric probing

Abstract

We consider problems in geometric probing, the algorithmic study of determining a geometric structure or some aspect of that structure from the results of a mathematical or physical measuring device. A variety of problems from robotics, medical instrumentation, mathematical optimization, integral and computational geometry, graph theory, and other areas fit into this paradigm. Finger probes return the first point of intersection between a directed line l and an object P. Chapter 2 presents results on finger probing convex polygons. We consider related problems in higher dimensions and with different classes of objects. Hyperplane probes return the first hyperplane moving perpendicular to itself which is tangent to P. Chapter 3 discusses the duality relationship between finger and hyperplane probes. We establish the connection between hyperplane probes and certain algorithmic problems and consider the related silhouette and supporting line probe models. X-ray probes return the length of intersection between P and l. Chapter 4 surveys the field of tomography and presents results for x-ray probes, which was inspired by it. We give linear bounds on determination and verification with x-ray probes in two and higher dimensions. Half-space probes return the volume of intersection between a half-space h and P. Chapter 5 presents our linear determination and verification results for two dimensions and discussed the difficulties of determination in higher dimensions. Chapter 6 considers the power of infinite collections of these probes. We discuss Hammer's x-ray problem, presenting new proofs for convex polygons. Also, we discuss the combinatorial geometry problem of k-projections, which arises from aggregate probing. Finally, we consider other aggregate problems such as probing in rounds. Chapter 7 extends probing to an object which is not usually considered geometric. Cut-set probes return the size of a cut-set of a graph. We present surprising results using these to reconstruct and thus represent graphs. Each chapter concludes with relevant open problems.