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 \({\cal D}\) 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 \((\between\)), the problem asks to find a maximum cardinality subset \({\cal D}' \subseteq {\cal D}\) such that any two 2-intervals in \({\cal D}'\) 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 \({\cal D}\).
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Blin, G., Fertin, G., Vialette, S. (2004). New Results for the 2-Interval Pattern Problem. In: Sahinalp, S.C., Muthukrishnan, S., Dogrusoz, U. (eds) Combinatorial Pattern Matching. CPM 2004. Lecture Notes in Computer Science, vol 3109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27801-6_23
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DOI: https://doi.org/10.1007/978-3-540-27801-6_23
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