Algorithms and Data Structures
Volume 709 of the series Lecture Notes in Computer Science pp 113
Computing the allpairs longest chains in the plane
 Mikhail J. AtallahAffiliated withDept. of Computer Science, Purdue University
 , Danny Z. ChenAffiliated withDepartment of Computer Science and Engineering, University of Notre Dame
Abstract
Many problems on sequences and on circulararc graphs involve the computation of longest chains between points in the plane. Given a set S of n points in the plane, we consider the problem of computing the matrix of longest chain lengths between all pairs of points in S, and the matrix of “parent” pointers that describes the n longest chain trees. We present a simple sequential algorithm for computing these matrices. Our algorithm runs in O(n ^{2}) time, and hence is optimal. We also present a rather involved parallel algorithm that computes these matrices in O(log ^{2}n) time using O(n ^{2}/log n) processors in the CREW PRAM model. These matrices enables us to report, in O(1) time, the length of a longest chain between any two points in S by using one processor, and the actual chain by using k processors, where k is the number of points of S on that chain. The space complexity of the algorithms is O(n ^{2}).
 Title
 Computing the allpairs longest chains in the plane
 Book Title
 Algorithms and Data Structures
 Book Subtitle
 Third Workshop, WADS '93 Montréal, Canada, August 11–13, 1993 Proceedings
 Pages
 pp 113
 Copyright
 1993
 DOI
 10.1007/3540571558_229
 Print ISBN
 9783540571551
 Online ISBN
 9783540479185
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 709
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Mikhail J. Atallah ^{(1)}
 Danny Z. Chen ^{(2)}
 Author Affiliations

 1. Dept. of Computer Science, Purdue University, 47907, West Lafayette, IN
 2. Department of Computer Science and Engineering, University of Notre Dame, 46556, Notre Dame, IN
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