Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program

Suchenek, Marek A.

doi:10.3233/fi-2012-751

Cited by 3 publications

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References 3 publications

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“…The details are as follows: For each chromosome, convert the chromosome’s peak associations into an array. For each chromosome, convert the chromosome’s array into a binary max-heap keyed on the –Log10( P ) of each genotype/phenotype pair, using the O ( n ) running time heapify algorithm described in 59 . For each chromosome, while the chromosome’s heap is not empty, extract the maximum genetic variant/phenotype pair from the heap to create a new cluster.…”

Section: Methodsmentioning

confidence: 99%

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An expanded set of genome-wide association studies of brain imaging phenotypes in UK Biobank

et al. 2021

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UK Biobank is a major prospective epidemiological study, including multimodal brain imaging, genetics and ongoing health outcomes. Previously, we published genome-wide associations of 3,144 brain imaging-derived phenotypes, with a discovery sample of 8,428 subjects. Here we present a new open resource of GWAS summary statistics, using the 2020 data release, almost tripling the discovery sample size. We now include the X chromosome, and new classes of image derived phenotypes (subcortical volumes and tissue contrast). Previously we had found 148 replicated clusters of associations between genetic variants and imaging phenotypes; here we find 692, including 12 on the X chromosome. We describe some of the newly found associations, focussing on the X chromosome and autosomal associations involving the new classes of imaging-derived phenotypes. Our novel associations implicate e.g. pathways involved in the rare X-linked syndrome STAR (syndactyly, telecanthus and anogenital and renal malformations), Alzheimer’s disease and mitochondrial disorders.

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Section: Methodsmentioning

confidence: 99%

“…For each chromosome, convert the chromosome’s array into a binary max-heap keyed on the –Log10( P ) of each genotype/phenotype pair, using the O ( n ) running time heapify algorithm described in 59 .…”

Section: Methodsmentioning

confidence: 99%

An expanded set of genome-wide association studies of brain imaging phenotypes in UK Biobank

et al. 2021

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Optimizing Binary Heaps

2017

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Worst-Case Analysis of Heapsort, Exactly

Suchenek

2024

The Computer Journal

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This paper completes analysis of the worst-case running time of $ {\tt{Heapsort}}$ measured by the number of comparisons of keys performed while sorting. A derivation of an exact formula for the maximum number of comparisons of keys performed by $ {\tt{Heapsort}} $ on any array of size $ N \geq 2 $ is presented. It is equal to $$ \begin{align}& \nonumber 2(N-1)\lceil \lg N \rceil - 2 ^{\lceil \lg N \rceil +1} - 2 s_2(N) - e_2(N) + \min (\lceil \lg N \rceil, 3) + 5 + c, \end{align} $$ where $s_2(N)$ is the sum of digits of the binary representation of $N$, $e_2(N)$ is the exponent of $2$ in the $N$’s prime factorization and $ c $ is a binary function on the set of positive integers defined by $$ \begin{align}& \nonumber c = \left\{ \begin{array}{@{}ll} 1 \text{ if} \; N \leq 2 ^{\lceil \lg N \rceil} - 4 \\[4pt] 0 \text{ otherwise}. \end{array} \right. \end{align} $$ The above formula allows for deciding, in $ O(N \log N) $ time, if any given $ N $-element array is a worst-case array for $ {\tt{Heapsort}} $. Its proof yields an algorithm for construction, in $ O(N \log N) $ time, of worst-case arrays of arbitrary sizes $ N \geq 2 $ for $ {\tt{Heapsort}} $.

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Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program

Cited by 3 publications

References 3 publications

An expanded set of genome-wide association studies of brain imaging phenotypes in UK Biobank

An expanded set of genome-wide association studies of brain imaging phenotypes in UK Biobank

Optimizing Binary Heaps

Worst-Case Analysis of Heapsort, Exactly

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