自适应算法的脆弱复杂性
Fragile Complexity of Adaptive Algorithms
摘要：
如果每个输入元素都参与$O(f(n))$比较，则基于比较的算法的脆弱复杂度为$f(n)$。在本文中，我们探讨了适应各种输入限制的算法的脆弱复杂性，即，具有脆弱复杂度的算法，该脆弱复杂度由输入大小n以外的量参数化。我们发现，在一个有序的数组中搜索的前任有脆弱的复杂性$Θ(\log k)$，其中$k$是查询元素的排名，无论是在一个随机和确定性的设置。对于前任搜索，我们还展示了如何以最佳方式减少阵列中元素的摊销脆弱复杂性。我们还证明了以下结果：选择$k$ -th最小的元素有预期的脆弱复杂性$O(\log \log k)$的元素选择。确定性地找到最小元素具有脆弱的复杂性$Θ(\log(Inv))$和$Θ(\log(Runs))$，其中$Inv$是序列中的反转的数目，而$Runs$是序列中的递增游程的数目。确定性地找到中值具有脆弱的复杂性$O(\log(Runs) + \log \log n)$和$Θ(\log(Inv))$。确定性排序具有脆弱的复杂性$Θ(\log(Inv))$，但无论运行次数如何，它都具有脆弱的复杂性$Θ(\log n)$。
英文摘要：
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 $Θ(\log k)$, 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 $Θ(\log(Inv))$ and $Θ(\log(Runs))$, 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 $Θ(\log(Inv))$. Deterministic sorting has fragile complexity $Θ(\log(Inv))$ but it has fragile complexity $Θ(\log n)$ regardless of the number of runs.
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