A Reliable Randomized Algorithm for the Closest-Pair Problem

The following two computational problems are studied:Duplicate grouping:Assume thatnitems are given, each of which is labeled by an integer key from the set {0,?,U?1}. Store the items in an array of sizensuch that items with the same key occupy a contiguous segment of the array.Closest pair:Assume that a multiset ofnpoints in thed-dimensional Euclidean space is given, whered?1 is a fixed integer. Each point is represented as ad-tuple of integers in the range {0,?,U?1} (or of arbitrary real numbers). Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.In 1976, Rabin described a randomized algorithm for the closest-pair problem that takes linear expected time. As a subroutine, he used a hashing procedure whose implementation was left open. Only years later randomized hashing schemes suitable for filling this gap were developed.In this paper, we return to Rabin's classic algorithm to provide a fully detailed description and analysis, thereby also extending and strengthening his result. As a preliminary step, we study randomized algorithms for the duplicate-grouping problem. In the course of solving the duplicate-grouping problem, we describe a new universal class of hash functions of independent interest.It is shown that both of the foregoing problems can be solved by randomized algorithms that useO(n) space and finish inO(n) time with probability tending to 1 asngrows to infinity. The model of computation is a unit-cost RAM capable of generating random numbers and of performing arithmetic operations from the set {+,?,?,div,log2,exp2}, wheredivdenotes integer division andlog2andexp2are the mappings from N to N?{0} withlog2(m)=?log2m? andexp2(m)=2mfor allm?N. If the operationslog2andexp2are not available, the running time of the algorithms increases by an additive term ofO(loglogU). All numbers manipulated by the algorithms consist ofO(logn+logU) bits.The algorithms for both of the problems exceed the time boundO(n) orO(n+loglogU) with probability 2?n?(1). Variants of the algorithms are also given that use onlyO(logn+logU) random bits and have probabilityO(n??) of exceeding the time bounds, where ??1 is a constant that can be chosen arbitrarily.The algorithms for the closest-pair problem also works if the coordinates of the points are arbitrary real numbers, provided that the RAM is able to perform arithmetic operations from {+,?,?,div} on real numbers, whereadivbnow means ?a/b?. In this case, the running time isO(n) withlog2andexp2andO(n+loglog(?max/?max)) without them, where ?maxis the maximum and ?minis the minimum distance between any two distinct input points.