We discuss data mining based on association rules for two numeric attributes and one Boolean attribute. For example, in a database of bank customers, “Age ” and “Balance” are two numeric attributes, and “CardLoan ” is a Boolean attribute. Taking the pair (Age, Balance) as a point in two-dimensional space, we consider an association rule of the form ((Age, Balance) c P) * (CardLoan = Yes), which implies that bank customers whose ages and balances fall in a planar region P tend to use card loan with a high probability. We consider two classes of regions, rectangles and adrmssible (i.e. connected and z-monotone) regions. For each class, we propose efficient algorithms for computing the regions that give optimal association rules for gain, support, and confidence, respectively. We have implemented the algorithms for admissible regions, and constructed a system for visualizing the rules. 1

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

The classical algorithm for computing the similarity between two sequences [36, 39] uses a dynamic programming matrix, and compares two strings of size n in O(n 2 ) time. We address the challenge of computing the similarity of two strings in sub-quadratic time, for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both local and global alignment computations. The speed-up is achieved by dividing the dynamic programming matrix into variable sized blocks, as induced by Lempel-Ziv parsing of both strings, and utilizing the inherent periodic nature of both strings. This leads to an O(n 2 = log n) algorithm for an input of constant alphabet size. For most texts, the time complexity is actually O(hn 2 = log n) where h 1 is the entropy of the text. Institut Gaspard-Monge, Universite de Marne-la-Vallee, Cite Descartes, Champs-surMarne, 77454 Marne-la-Vallee Cedex 2, France, email: mac@univ-mlv.fr. y Department of Computer Science, Haifa University, Haifa 31905, Israel, phone: (972-4) 824-0103, FAX: (972-4) 824-9331; Department of Computer and Information Science, Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201-3840; email: landau@poly.edu; partially supported by NSF grant CCR-0104307, by NATO Science Programme grant PST.CLG.977017, by the Israel Science Foundation (grants 173/98 and 282/01), by the FIRST Foundation of the Israel Academy of Science and Humanities, and by IBM Faculty Partnership Award. z Department of Computer Science, Haifa University, Haifa 31905, Israel; On Education Leave from the IBM T.J.W. Research Center; email: michal@cs.haifa.il; partially supported by by the Israel Science Foundation (grants 173/98 and 282/01), and by the FIRST Foundation of the Israel Academy of Science ...

Improving on a recent breakthrough of Sharir, we find two minimum-radius circular disks covering a planar point set, in randomized expected time O(n log 2 n). 1 Introduction The k-center problem for a point set S is to find k points (called centers, usually not required to be a subset of S) such that the maximum distance from any point in S to the nearest center is minimized. A case of particular interest is the planar two-center problem [4], which can be viewed less abstractly as one of covering a set of points in the plane by two congruent circular disks, in such a way as to minimize the radius r # of the disks. For a long time the best algorithms for this problem had time bounds of the form O(n 2 log c n) [1, 5, 12, 11]. In a recent breakthrough, Sharir [16] greatly improved all of these algorithms, giving a two-center algorithm with running time O(n log c n). The basic idea is to search for different types of partition depending on the relative positions of the two disk...

An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.

Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex k-gon, (2) ~ is an empty convex k-gon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms for solving each of these three problems in time O(kn3). The space complexity is O(n) for k = 4 and O(kn 2) for k> 5. The algorithms are based on a dynamic ptogramming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

this paper we consider the problem of computing two similar recurrences: the one-dimensional case

The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings and a target string. is a common substring of all strings, that is. The goal is to compute the similarity of all strings with, without computing the part of again and again. Using the classical dynamic programming tables, each appearance of in a source string would require the computation of all the values in a dynamic programming table of size where is the size of. Here we describe an algorithm which is composed of an encoding stage and an alignment stage. During the first stage, a data structure is constructed which encodes the comparison of with. Then, during the alignment stage, for each comparison of a source with, the pre-compiled data structure is used to speed up the part of. We show how to reduce the alignment work, for each appearance of the common substring in a source string, to- at the cost of encoding work, which is executed only once.

We give an O(n log² n)-time, linear-space algorithm that, given a directed planar graph with positive and negative arc-lengths, and given a node s, finds the distances from s to all nodes.

Samuel R. Buss # Peter N. Yianilos + Abstract Let G be a complete, weighted, undirected, bipartite graph with n red nodes, n # blue nodes, and symmetric cost function c(x, y) . A maximum matching for G consists of min{n, n # edges from distinct red nodes to distinct blue nodes. Our objective is to find a minimum-cost maximum matching, i.e. one for which the sum of the edge costs has minimal value. This is the weighted bipartite matching problem; or as it is sometimes called, the assignment problem.