Finding nearest neighbours

In designing an algorithm to find pairs of points that are within Euclidean distance d it is effective to use a screening procedure to reject most pairs of points that are far apart. A procedure based on multiple, overlapping lattices can efficiently identify close points and exclude distant ones.

In this paper, we pursue the use of probalistic (randomized) algorithms in VLSI architectures, in order to reduce the amount of computation, and, correspondingly, the time of computation as well as chip area.

As a case example, we propose two VLSI solutions to the well-known problem of the nearest pair of points in computational geometry. These implementations are based on Rabin's and Weide's probabilistic algorithms. The chip area and time of computation which result are each O(n). This is a marked improvement over both the straightforward deterministic approach (leading to an O(n²) computational time) and the deterministic algorithms known as being the best (leading to O(n log n)computational time).

In suggesting a VLSI solution to the nearest-pair problem, we introduce two new systolic structures, a systolic grouper and a systolic minimum-distance processor. We also make use of a new class of systolic arrays introduced earlier, probabilistic systolic arrays.

Given an N × N array of 0s and 1s, the closest pair problem is to determine the minimum distance between any pair of ones. Let D be this minimum distance (or D = 2N if there are fewer than two 1s). Two solutions to this problem are given, one requiring O(log(N) + D) time and the other O(log(N)). These solutions are for two types of parallel computers arranged in a pyramid fashion with the base of the pyramid containing the matrix. The results improve upon an algorithm of Dyer that requires O(N) time on a more powerful computer.

We analyze the average-case behavior of a general class of algorithms to which the method of buckets can be applied. In particular, we give sufficient conditions under which an algorithm $\text{A}$ that solves a problem $\text{P}$ of size n in O(n^k), k > 1, worst-case time, can be made to run in linear expected time.

This chapter focuses on the computational algorithms for the single-link clustering method that is one of the oldest methods of cluster analysis. This clustering method is also known by many other names because of the fact that it has been reinvented in different application areas and that there exist many very different computational algorithms corresponding to the single-link clustering model. Often this identity has gone unnoticed, as the new clustering methods are not always compared with the existing ones. Different clustering methods imply different definitions of what constitutes a “cluster” and should, thus, be expected to give different results for many data sets. A variety of algorithms to serve as a convenient source of algorithms for the single-link method are presented in the chapter. While the algorithms differ considerably in terms of their computational efficiency, even the least efficient algorithm may sometimes be useful for small data sets.