The Monge array-an abstraction and its applications

This thesis develops a body of versatile algorithmic techniques. We demonstrate the power and generality of these techniques by applying them to a wide variety of problems. These problems are drawn from such diverse areas of study as computational geometry, VLSI theory, operations research, and molecular biology. The algorithmic techniques described in this thesis are centered around a family of highlystructured arrays known as Monge arrays. An m x n array A = {a[i,j]} is called Monge if a[i,j] + a[k, < a[i, + a[k,j] for all i, j, k, and I such that 1 < i < k < m and 1 < j < e < n. We will show that Monge arrays capture the essential structure of many practical problems, in the sense that algorithms for searching in the abstract world of Monge arrays can be used to obtain efficient algorithms for these practical problems. The first part of this thesis describes the basic Monge-array abstraction. We begin by defining several different types of Monge and Monge-like arrays. These definitions include a generalization of the notion of two-dimensional Monge arrays to higher-dimensional arrays. We also present several important properties of Monge and Monge-like arrays and introduce a computational framework for manipulating such arrays. We then develop a variety of algorithms for searching in Monge arrays. In particular, we give efficient sequential and parallel (PRAM) algorithms for computing minimal entries in Monge arrays and efficient sequential algorithms for selection and sorting in Monge arrays. Highlights include an O(dn lgd2 n)-time sequential algorithm for computing the minimum entry in an n x n x . -x n d-dimensional Monge array, an O(n3/2lg 2 n)-time sequential algorithm for computing the median entry in each row of an n x n two-dimensional Monge array, and an optimal O(lg n)-time, (n2 / lg n)-processor CREWPRAM algorithm for computing the minimum entry in each 1 x n x 1 subarray of an n x n x n three-dimensional Monge array. The second part of this thesis investigates the diverse applications of the Monge-array abstraction. We first consider a number of geometric problem relating to convex polygons in the plane. Specifically, we use Monge-array techniques to develop efficient algorithms for several proximity problems involving the vertices of a convex polygon, as well as the maximumperimeter-inscribed-k-gon problem and the minimum-area-circumscribing-k-gon problem. We