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

Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, Minshen Zhu
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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.
最长可堆子序列和最大二叉树的固定参数算法
可堆序列是可以排列在最小堆数据结构中的数字序列。Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011)提出了寻找给定序列的最长可堆积子序列,作为已经得到充分研究的最长递增子序列问题的推广,其复杂性仍然是开放的。最长可堆子序列问题的等效公式是在给定的置换有向无环图(置换DAG)中寻找最大尺寸的二叉树。在这项工作中,我们研究了最长可堆子序列和最大尺寸二叉树的参数化算法。我们将字母表大小作为一个新的参数引入到置换DAGs的计算问题的研究中,并证明了该参数对于固定拓扑排序具有完备的表征和多项式时间算法。我们相信这个参数在排列dag上定义的优化问题的上下文中可能是有用的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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