# Parallel Searching in Generalized Monge Arrays

## Summary (1 min read)

### 1. Introduction

- Larmore and Przytycka in [30] reduce Huffman coding to theConcave Least Weight Subsequence (CLWS) problem(defined in Section 4.2) and then show how to solve CLWS, and thereby Huffman coding, inO( √ n lg n) time usingn processors on a CREW PRAM.
- For this problem, the authors obtain (in Section 4.4) a CREW-PRAM algorithm that takesO(lg n) time and usesn processors.

### 2. CREW-PRAM Algorithms to Compute Row Minima in Staircase-Monge

- In this section the authors give CREW-PRAM algorithms for computing row minima in staircase-Monge arrays.
- From [3], the minima ofAt induce a partitioning ofA such that certain regions can be omitted from further searching for row minima because of the Monge condition.
- Thus, bothµ1 andµ3 are bracketed byµ0, adding regionsF3 andF2, respectively.
- Thus, Atallah and Kosaraju [14] show how to find the row minima for anm×n Monge array inO(lg mn) time usingm+n processors.
- The authors first determine the minima in all the feasible Monge arrays.

### PRAM.

- In this section the authors give anO(lg m lg n)-timemn-processor hypercube algorithm for the string editing problem.
- Respectively, determine, for each vertexx of P, 1. the vertex ofQ nearest tox that is not visible tox, 2. the vertex ofQ farthest fromx that is not visible tox, 3. the vertex ofQ nearest tox that is visible tox, and 4.the authors.

### 4. Applications

- 1. The All Pairs Shortest Path Problem.
- The following theorem is due to Aggarwalet al. [2].
- Given a directed acyclic graph whose edge weights satisfy the Monge condition(or the inverse-Monge condition), the APSP problem can be solved in O(lg2 n) time using n2 processors on a CREW PRAM.
- This decomposition ofD reduces the computation ofDn to lgn array multiplications and additions.
- In this section the authors present the first NC algorithm for the Huffman coding problem that doeso(n2) work.

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### "Parallel Searching in Generalized M..." refers background in this paper

...In [3], Aggarwal and Klawe gave an O((m+ n) lg lg(m+ n))-time sequential algorithm for finding the row minima of an m× n staircase-Monge array....

[...]

864 citations

506 citations

### "Parallel Searching in Generalized M..." refers background or methods in this paper

...Aggarwal et al. [ 4 ] showed that the all-farthest-neighbors problem for the vertices of a convex n-gon can be solved in linear time using Monge arrays....

[...]

...(If a tube has several minima, then we take the one with the minimum second coordinate.) For sequential computation, the result of [ 4 ] can be trivially used to solve the tube-minima problem in O.. pC r/q/ time....

[...]

...However, if we wanted to solve the row-maxima problem (instead of the rowminima problem) for an m£ n staircase-Monge array, then we could, in fact, employ the sequential algorithm given in [ 4 ] and solve the row-maxima problem in O.mC n/ time....

[...]

...Aggarwal et al. [ 4 ] showed that the row-minima problem for an m£ n Monge array can be solved in O.mC n/ time, which is optimal....

[...]

349 citations