# Computing the all-pairs longest chains in the plane

- First Online:

DOI: 10.1007/3-540-57155-8_229

- Cite this paper as:
- 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

## 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*^{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}).

## Preview

Unable to display preview. Download preview PDF.