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Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

Authors Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, Minshen Zhu



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Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, IL, USA
Elena Grigorescu
  • Purdue University, West Lafayette, IN, USA
Gabriel Istrate
  • West University of Timişoara, Romania
  • e-Austria Research Institute, Timişoara, Romania
Shubhang Kulkarni
  • University of Illinois, Urbana-Champaign, IL, USA
Young-San Lin
  • Purdue University, West Lafayette, IN, USA
Minshen Zhu
  • Purdue University, West Lafayette, IN, USA

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Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu. Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.IPEC.2020.7

Abstract

A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence and maximum-sized binary tree. We introduce alphabet size as a new parameter in the study of computational problems in permutation DAGs and show that this parameter with respect to a fixed topological ordering admits a complete characterization and a polynomial time algorithm. We believe that this parameter is likely to be useful in the context of optimization problems defined over permutation DAGs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • maximum binary tree
  • heapability
  • permutation directed acyclic graphs

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References

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