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

Basdevant, Anne-Laure; Singh, Arvind

doi:10.1214/18-ecp120

Cited by 3 publications

(7 citation statements)

References 3 publications

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“…Heapability was further investigated in [16] (and, independently, in [23]). In particular, a subgroup of the authors of the present papers showed that for permutations one can compute in polynomial time a minimal decomposition into heapable subsequences, and investigated the analog of the Ulam-Hammersley problem for heapable sequences (see also subsequent work in [17,2,3], that extends/confirms some of the conjectures of [16]). As shown in [17], one can meaningfully study the analogs of heapability and the Ulam-Hammesley problem in the context of partial orders.…”

Section: Introductionmentioning

confidence: 80%

On the heapability of finite partial orders

Balogh,

Bonchiş,

Diniş

et al. 2017

Preprint

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We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove that this quantity has a characterization related to (the proof of) Dilworth's theorem. As a byproduct we derive a flow-based algorithm for computing such a minimal decomposition. On the other hand, in the particular cases of sets and sequences of intervals and for trapezoid partial orders we prove that such minimal decompositions can be computed via simple greedy-type algorithms.Second, while the complexity of computing a maximal heapable subsequence of a permutation is still open, we show that this problem has a polynomial time algorithm for sequences of intervals.

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Section: Introductionmentioning

confidence: 80%

On the heapability of finite partial orders

Balogh,

Bonchiş,

Diniş

et al. 2017

Preprint

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“…Thus particles are arranged in the increasing order of values, from the smallest to the largest Example 2. Consider the state s X of the system HAD 2 after all particles with values X = [5,1,4,2,3] have arrived (in this order). Then in the state s X particle 5 has 0 lives, particle 1 has two lives, particle 4 has 0 lives left, particle 2 has two lives, particle 3 has two lives left.…”

Section: Main Definitions and Resultsmentioning

confidence: 99%

“…1 ar 0 br ∈ Σ * 1 (where 0 0 = , the null word). The following combinatorial concept was introduced (for k = 2) in [7] and further studied in [9,15,10,4,5,3]: Definition 17. A sequence X = X 0 , .…”

Section: Motivation and Notationsmentioning

confidence: 99%

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The Language (and Series) of Hammersley-Type Processes

Bonchiş

Istrate

Rochian

2018

Lecture Notes in Computer Science

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We study languages and formal power series associated to (variants of) the Hammersley process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the Hammersley interval process we show that there are two relevant variants of formal languages. One of them leads to the same language as the ordinary Hammersley tree process. The other one yields non-context-free languages. The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the Hammersley process. We employ this algorithm to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.

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“…Heapability of integer sequences was introduced in [BHMZ11] and has been investigated further in [IB15,Por15,IB16,BGGS16,BS18,BIR18,BBD + 20]. Heapability of integer sequences can be decided by a simple greedy algorithm [BHMZ11] (see also [IB16] for an alternate approach based on integer programming, and [BBD + 20] for connections with Dilworth's theorem and an algorithm based on network flows).…”

Section: Related Workmentioning

confidence: 99%

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

Chandrasekaran,

Grigorescu,

Istrate

et al. 2021

Preprint

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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 as well as maximum-sized binary tree. We show the following results:1. The longest heapable subsequence problem can be solved in k O(log k) n time, where k is the number of distinct values in the input sequence.

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

Cited by 3 publications

References 3 publications

On the heapability of finite partial orders

On the heapability of finite partial orders

The Language (and Series) of Hammersley-Type Processes

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

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