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

@article{Basdevant2017AlmostsureAF, title={Almost-sure asymptotic for the number of heaps inside a random sequence}, author={Anne-Laure Basdevant and Arvind Singh}, journal={arXiv: Probability}, year={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 number of heaps grows logarithmically. In this note, we improve on the previous result by establishing the almost-sure and L 1 convergence.

## 4 Citations

### 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.

### Computing a minimal partition of partial orders into heapable subsets

- Mathematics, Computer ScienceArXiv
- 2017

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.

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

- Mathematics, Computer ScienceIPEC
- 2020

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,…

### 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.

## References

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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.

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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 $\frac{1+\sqrt{5}}{2}\cdot \ln(n)$.

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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.