Improved Algorithms for Largest Cardinality 2-Interval Pattern Problem

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 ⊆ {<, @, () }. 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 = { () }, where L = O(dn) = O(n) 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 log n) algorithm. Secondly, we use dynamic programming techniques to obtain an O(n log n+dn) algorithm for the case R = {<, @ }, which improves previous O(n) result. Finally, we present another O(n log n+L) algorithm for the case R = {@, () } with disjoint support(interval ground set), which improves previous O(n √ n) upper bound.