We study various computational aspects of the problem of determining whether a given order contains a given sub-order. Formally, given a permutation π on k elements, and a permutation σ on n > k elements, the goal is to determine whether there exists a strictly increasing function f from [1.k] to [1.n] which is order preserving, i.e., f satisfies σ(f (i)) > σ(f(j)) whenever π(i) > π(j). We call this decision problem the Sub-Permutation Problem. The study falls into two parts. In the first part we develop and analyze an algorithm (or, rather, an algorithmic paradigm) for this problem. We show that the complexity of this algorithm is at most O(n1+C(π)), where C(π) is a naturally defined function of the permutation π. In the second part we study C(π). In particular, we show that C(π) ≤0.35k + o(k), implying that the complexity of the Sub-Permutation problem is O(ck + n0.35k+o(k)). On the other hand, we prove that for most π’s, C(π) = Ω(k), establishing a lower bound for our algorithm. In addition, we develop a fast polylogarithmic approximation algorithm for computing C(π), and bound the value of this parameter for some interesting families of permutations.
|Title of host publication||Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings|
|Editors||Magnús M. Halldórsson|
|Number of pages||14|
|ISBN (Print)||3540676902, 9783540676904|
|State||Published - 2000|
|Event||7th Scandinavian Workshop on Algorithm Theory, SWAT 2000 - Bergen, Norway|
Duration: 5 Jul 2000 → 7 Jul 2000
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||7th Scandinavian Workshop on Algorithm Theory, SWAT 2000|
|Period||5/07/00 → 7/07/00|
Bibliographical notePublisher Copyright:
© Springer-Verlag Berlin Heidelberg 2000.
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science (all)