James K. Park

Improved algorithms for economic lot size problems

Many problems in inventory control, production planning, and capacity planning can be formulated in terms of a simple economic lot size model proposed independently by A. S. Manne (1958) and by H. M. Wagner and T. M. Whitin (1958). The Manne-Wagner-Whitin model and its variants have been studied widely in the operations research and management science communities, and a large number of algorithms have been proposed for solving various problems expressed in terms of this model, most of which assume concave costs and rely on dynamic programming. In this paper, we show that for many of these concave cost economic lot size problems, the dynamic programming formulation of the problem gives rise to a special kind of array, called a Monge array. We then show how the structure of Monge arrays can be exploited to obtain significantly faster algorithms for these economic lot size problems. We focus on uncapacitated problems, i.e., problems without bounds on production, inventory, or backlogging; capacitated problem...

Parallel searching in generalized Monge arrays with applications

This paper investigates the parallel time and processor complexities of several searching problems involving Monge and Monge-composite arrays. We present array-searching algorithms for concurrent-readconcurrent-write (CRCW) PRAMS, concurrent-readexclusive-write (CREW) PRAMS, hypercubes, cubeconnected-cycles, and shuffle-exchange networks. All these algorithms run in optimal time, and their processor-time products are all within an O(lg n) factor of the worst-case sequential bounds. Several applications of these algorithms are also given. Two applications improve previous results substantially, and the others provide novel parallel algorithms for problems not previously considered.

Selection and sorting in totally monotone arrays

A two-dimensional arrayA={a[i, j]} is calledtotally monotone if, for alli1<i2 andj1<j2,a[i1,j1]<a[i1,j2] impliesa[i2,j1]<a[i2,j2]. Totally monotone arrays were introduced in 1987 by Aggarwal, Klawe, Moran, Shor, and Wilber, who showed that several problems in computational geometry and VLSI river routing could be reduced to the problem of finding a maximum entry in each row of a totally monotone array. In this paper we consider several selection and sorting problems involving totally monotone arrays and give a number of applications of solutions for these problems. In particular, we obtain the following results for anm × n totally monotone arrayA:1.Thek largest (ork smallest) entries in each row ofA can be computed inO(k(m + n)) time. This result allows us to determine thek farthest (ork nearest) neighbors of each vertex of a convexn-gon inO(kn) time.2.Provided the transpose ofA is also totally monotone, thek largest (ork smallest) entries overall inA can be computed inO(m + n + k lg(mn/k)) time. This result allows us to find thek farthest (ork nearest) pairs of vertices from a convexn-gon inO(n + k lg(n2/k)) time.3.The rows ofA can be sorted inO(mn) time whenm ≥n and inO(mn(1 + lg(n/m))) time whenm < n. This result allows us to solve the of Ω(S) on the number of combinations of row permutations possible for a totally monotone array would imply an Ω(lgS) lower bound on the time necessary to sort the arrays rows in a linear decision tree model.)4.In Subsection 4.2 we applied our algorithm for sorting the rows of a totally monotone array to the neighbor-ranking problem for the vertices of a convex polygonP. We then extended this technique to arbitrary point sets. It remains open whether our two selection algorithms for totally monotone arrays, which we also applied to the vertices of a convex polygon, can likewise be applied to arbitrary point sets.

IMPROVED SELECTION IN TOTALLY MONOTONE ARRAYS

This papers main result is an -time algorithm for computing the kth smallest entry in each row of an m×n totally monotone array. (A two-dimensional array A={a[i, j]} is totally monotone if for all i1<i2 and j1<j2, a[i1, j1]<a[i1, j2] implies a[i2, j1]<a[i2, j2].) For large values of k (in particular, for k=⌈n/2⌉), this algorithm is significantly faster than the O(k(m + n))-time algorithm for the same problem due to Kravets and Park. An immediate consequence of this result is an O(n3/2lg1/2 n)-time algorithm for computing the kth nearest neighbor of each vertex of a convex n-gon. In addition to the main result, we also give an O(n lg m)-time algorithm for computing an approximate median in each row of an m×n totally monotone array; this approximate median is an entry whose rank in its row lies between ⌊n/4⌋ and ⌈3n/4⌉ − 1.

Parallel Searching in Generalized Monge Arrays

Abstract. This paper investigates the parallel time and processor complexities of several searching problems involving Monge, staircase-Monge, and Monge-composite arrays. We present array-searching algorithms for concurrent-read-exclusive-write (CREW) PRAMs, hypercubes, and several hypercubic networks. All these algorithms run in near-optimal time, and their processor-time products are all within an

Improved Selection on Totally Monotone Arrays

O (\lg n)

Parallel searching in multidimensional monotone arrays

factor of the worst-case sequential bounds. Several applications of these algorithms are also given. Two applications improve previous results substantially, and the others provide novel parallel algorithms for problems not previously considered.

Sequential Searching in Multidimensional Monotone Arrays

This papers main result is an O((√m lg m)(n lg n)+m lg n)-time algorithm for computing the kth smallest entry in each row of an m×n totally monotone array. (A two-dimensional array A = {a[i,j]} is totally monotone if for all i1<i2 and j1<j2, a[i1,j1]<a[i1,j2] implies a[i2,j1<a[i2,j2.) For large values of k (in particular, for k=[n/2]), this algorithm is significantly faster than the O(k(m+n))-time algorithm for the same problem due to Kravets and Park (1991). An immediate consequence of this result is an O(n3/2 lg2n)-time algorithm for computing the kth nearest neighbor of each vertex of a convex n-gon. In addition to the main result, we also give an O(n lg m)-time algorithm for computing an approximate median in each row of an m×n totally monotone array; this approximate median is an entry whose rank in its row lies between [n/4] and [3n/4].