, Volume 13, Issue 3, pp 263-275

Improved algorithms for largest cardinality 2-interval pattern problem

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Abstract

The 2-INTERVAL PATTERN problem is to find the largest constrained pattern in a set of 2-intervals. The constrained pattern is a subset of the given 2-intervals such that any pair of them are R-comparable, where model \(R \subseteq \{ <, \sqsubset, \mathtt{(\hspace{-3.5pt})} \}\) . The problem stems from the study of general representation of RNA secondary structures. In this paper, we give three improved algorithms for different models. Firstly, an O(n{log} n +L) algorithm is proposed for the case \(R= \{ \mathtt{(\hspace{-3.5pt})} \} \) , where \({\cal L}=O(dn)=O(n^2)\) is the total length of all 2-intervals (density d is the maximum number of 2-intervals over any point). This improves previous O(n 2log n) algorithm. Secondly, we use dynamic programming techniques to obtain an O(nlog n + dn) algorithm for the case R = { <, ⊏ }, which improves previous O(n 2) result. Finally, we present another \(O(n {\rm log} n +{\cal L})\) algorithm for the case \(R = \{\sqsubset, \mathtt{(\hspace{-3.5pt})} \}\) with disjoint support(interval ground set), which improves previous O(n 2n) upper bound.

A preliminary version of this article appears in Proceedings of the 16th Annual International Symposium on Algorithms and Computation, Springer LNCS, Vol. 3827, pp. 412–421, Hainan, China, December 19–21, 2005.