Point Set pattern matching ind-dimensions

Abstract

In this dissertation, we apply computational geometry techniques to obtain efficient algorithms for several instances of a point set pattern matching problem. The problem is studied in the context of high dimensional Euclidean space for which no polynomial time algorithms were previously known. The problem can be expressed as follows: given a set $S$ of $n$ points and a set $P$ of $k$ points in the $d$-dimensional Euclidean space, determine whether $P$ matches any $k$-subset of $S$. The notion of match is defined in a rather general setting. Namely, the set $P$ is allowed to undergo translation, rotation, reflection, global scaling and local perturbation. To overcome the lack of a total order for points in high dimensional spaces, a crucial mechanism is developed to enable us to traverse the sets in an orderly fashion. We devise what we call the circular sorting of a set of points as a generalization of sorting in the real line. An optimal algorithm (in time and space) is described for circular sorting in arbitrary dimensions. Efficient algorithms of time complexity exponential in the dimension of the space are presented for solving the point set pattern matching problem.