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引用次数: 1

摘要

Chazelle [JACM00]介绍了软堆作为高效最小生成树算法的构建块,最近Kaplan等人[SOSA2019]展示了如何应用软堆来实现各种选择问题的更简单算法。软堆通过允许在总共$N$次插入之后损坏堆中的$\epsilon N$项来权衡准确性和效率,其中损坏的项是人为增加键的项,$0 < \epsilon \leq 1/2$是固定的错误参数。Chazelle的软堆是基于二叉树的,支持在平摊$O(\lg(1/\epsilon))$时间内插入和在平摊$O(1)$时间内提取最小值操作。本文对软堆的设计空间进行了探讨。本文的主要贡献是一种基于合并排序序列的软堆实现,其时间界限与Chazelle软堆的时间界限相匹配。我们还讨论了Kaplan等人[SICOMP2013]的软堆变体,其中我们避免了惰性插入。它基于三叉树而不是二叉树,并且匹配Kaplan等人的时间限制,即平摊$O(1)$插入和平摊$O(\lg(1/\epsilon))$ extract-min。我们的两种数据结构只在提取最小操作之后引入损坏,提取最小操作返回被操作损坏的项集。
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Soft Sequence Heaps
Chazelle [JACM00] introduced the soft heap as a building block for efficient minimum spanning tree algorithms, and recently Kaplan et al. [SOSA2019] showed how soft heaps can be applied to achieve simpler algorithms for various selection problems. A soft heap trades-off accuracy for efficiency, by allowing $\epsilon N$ of the items in a heap to be corrupted after a total of $N$ insertions, where a corrupted item is an item with artificially increased key and $0 < \epsilon \leq 1/2$ is a fixed error parameter. Chazelle's soft heaps are based on binomial trees and support insertions in amortized $O(\lg(1/\epsilon))$ time and extract-min operations in amortized $O(1)$ time. In this paper we explore the design space of soft heaps. The main contribution of this paper is an alternative soft heap implementation based on merging sorted sequences, with time bounds matching those of Chazelle's soft heaps. We also discuss a variation of the soft heap by Kaplan et al. [SICOMP2013], where we avoid performing insertions lazily. It is based on ternary trees instead of binary trees and matches the time bounds of Kaplan et al., i.e. amortized $O(1)$ insertions and amortized $O(\lg(1/\epsilon))$ extract-min. Both our data structures only introduce corruptions after extract-min operations which return the set of items corrupted by the operation.
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