Fragile Complexity of Adaptive Algorithms

Prosenjit Bose; Pilar Cano; Rolf Fagerberg; John Iacono; Riko Jacob; Stefan Langerman

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arXiv · 2102.00338

Fragile Complexity of Adaptive Algorithms

Prosenjit Bose·Pilar Cano·Rolf Fagerberg·John Iacono·Riko Jacob·Stefan Langerman

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Abstract

The fragile complexity of a comparison-based algorithm is $f(n)$ if each input element participates in $O(f(n))$ comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size n. We show that searching for the predecessor in a sorted array has fragile complexity ${\Theta}(\log k)$, where $k$ is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the $k$-th smallest element has expected fragile complexity $O(\log \log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity ${\Theta}(\log(Inv))$ and ${\Theta}(\log(Runs))$, where $Inv$ is the number of inversions in a sequence and $Runs$ is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity $O(\log(Runs) + \log \log n)$ and ${\Theta}(\log(Inv))$. Deterministic sorting has fragile complexity ${\Theta}(\log(Inv))$ but it has fragile complexity ${\Theta}(\log n)$ regardless of the number of runs.

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BibTeX RIS

Prosenjit Bose, Pilar Cano, Rolf Fagerberg, John Iacono, Riko Jacob, Stefan Langerman. 2021-01-30. Fragile Complexity of Adaptive Algorithms. https://arxiv.org/abs/2102.00338

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