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).
This research was supported by the Leonardo Fibonacci Institute in Trento, Italy, and by the National Science Foundation under Grant CCR-9202807. Part of this research was done while the first author was visiting LIPN, Paris, France.
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© 1993 Springer-Verlag Berlin Heidelberg
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Atallah, M.J., Chen, D.Z. (1993). Computing the all-pairs longest chains in the plane. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_229
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DOI: https://doi.org/10.1007/3-540-57155-8_229
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