Improved Algorithms for Largest Cardinality 2-Interval Pattern Problem

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 $R \subseteq \{<, \sqsubset, \between \}$. 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 $O(n {\rm log} n+\mathcal{L})$ algorithm is proposed for the case $R = \{\between\}$, where $\mathcal{L}$ = 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(n log n + dn) algorithm for the case $R = \{ <, \sqsubset\}$, which improves previous O(n ²) result. Finally, we present another $O(n {\rm log} n + \mathcal{L})$ algorithm for the case $R = \{\sqsubset, \between\}$ with disjoint support(interval ground set), which improves previous $O(n^{2}\sqrt{n})$ upper bound.