Invited Presentations

Algorithms and Data Structures

Volume 709 of the series Lecture Notes in Computer Science pp 1-13

Date:

Computing the all-pairs 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

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Abstract

Many problems on sequences and on circular-arc 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 2n) 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).