# New Results for the 2-Interval Pattern Problem

Abstract : We present new results concerning the problem of finding a constrained pattern in a set of 2-intervals. Given a set of n 2-intervals D and a model R describing if two disjoint 2-intervals can be in precedence order ($<$), be allowed to nest ($\sqsubset$) and/or be allowed to cross $(\between$), we consider the problem of finding a maximum cardinality subset $\D' \subseteq \D$ of disjoint $2$-intervals such that any two $2$-intervals in $\D'$ agree with $R$. We improve the time complexity of the best known algorithm for $R = \{\sqsubset\}$ by giving an optimal O(n log n) time algorithm. Also, we give a graph-like relaxation for $R = \{\sqsubset,\between\}$ that is solvable in $O(n^2 \sqrt{n})$ time. Finally, we prove that the problem is NP-complete for $R = \{<,\between\}$ and in addition to that, we give a fixed-parameter tractability result based on the crossing structure of D.
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Cited literature [9 references]

https://hal-upec-upem.archives-ouvertes.fr/hal-00620366
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Submitted on : Friday, September 30, 2011 - 3:32:06 PM
Last modification on : Thursday, January 20, 2022 - 5:28:05 PM
Long-term archiving on: : Saturday, December 31, 2011 - 2:21:34 AM

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• HAL Id : hal-00620366, version 1

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Guillaume Blin, Guillaume Fertin, Stéphane Vialette. New Results for the 2-Interval Pattern Problem. 15th Symposium on Combinatorial Pattern Matching (CPM'04), Jul 2004, Istanbul, Turkey, Turkey. pp.311-322. ⟨hal-00620366⟩

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