Point Set Pattern Matchin in d-Dimensions

In this paper, we apply computational geometry techniques to obtain an efficient algorithm for the following point set pattern matching problem. 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, where a match can be any similarity, i.e., the set P is allowed to undergo translation, rotation, reflection and global scaling. Motivated by the need to traverse the sets in an orderly fashion to shun exponential complexity, we circumvent the lack of a total order for points in high dimensional spaces by presenting an extension of one-dimensional sorting to higher dimensions (called circular sorting). This mechanism, which is of interest in its own right, enables us to achieve the orderly traversal we sought. An optimal algorithm (in time and space) is described for performing circular sorting in arbitrary dimensions. The time complexity of the resulting algorithm for point set pattern matching is O(n log n ..