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

@article{Chandrasekaran2021FixedParameterAF, title={Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree}, author={Karthekeyan Chandrasekaran and Elena Grigorescu and Gabriel Istrate and Shubhang Kulkarni and Young-San Lin and Minshen Zhu}, journal={ArXiv}, year={2021}, volume={abs/2110.00495} }

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…

## References

SHOWING 1-10 OF 36 REFERENCES

### Heapable Sequences and Subsequences

- Mathematics, Computer ScienceANALCO
- 2011

An efficient algorithm is obtained for determining the heapability of a sequence, and it is proved that the question of whether a sequence can be arranged in a complete binary heap is NP-hard.

### Partition into Heapable Sequences, Heap Tableaux and a Multiset Extension of Hammersley's Process

- MathematicsCPM
- 2015

It is shown that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS), and experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2}\cdot \ln(n)$.

### On the heapability of finite partial orders

- Mathematics, Computer ScienceDiscret. Math. Theor. Comput. Sci.
- 2020

A characterization result reminiscent of the proof of Dilworth's theorem is proved, which yields as a byproduct a flow-based algorithm for computing such a minimal decomposition of sets and sequences of intervals.

### From Hammersley’s lines to Hammersley’s trees

- Mathematics, Computer Science
- 2016

It is shown that the number of heaps grows logarithmically with the size of the permutation in Hammersley’s tree process, which is related to the problem of the longest increasing subsequence and has a combinatorial interpretation.

### The Maximum Binary Tree Problem

- Computer Science, MathematicsESA
- 2020

A randomized algorithm is designed to verify if a given directed graph on n vertices contains a binary tree of size k in 2kpoly(n) and constant-factor inapproximability results are shown assuming P≠NP are presented.

### A linear time algorithm for finding tree-decompositions of small treewidth

- MathematicsSTOC '93
- 1993

Every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm that determines whether the treewidth of G is at most k, and if so, finds a treedecomposition of G withtreewidth at mostK.

### Almost-sure asymptotic for the number of heaps inside a random sequence

- Mathematics
- 2017

We study the minimum number of heaps required to sort a random sequence using a generalization of Istrate and Bonchis's algorithm (2015). In a previous paper, the authors proved that the expected…

### Parameterized Algorithms

- Computer ScienceSpringer International Publishing
- 2015

This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.

### The language (and series) of Hammersley-type processes

- Computer Science, MathematicsMCU
- 2018

An algorithm for computing formal power series associated to the variants of the Hammersley's process, that have the formal languages studies in this paper as their support, are employed to settle the nature of the scaling constant.