Advances in Applied Probability

Distributional convergence for the number of symbol comparisons used by QuickSelect

James Allen Fill and Takehiko Nakama

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When the search algorithm QuickSelect compares keys during its execution in order to find a key of target rank, it must operate on the keys' representations or internal structures, which were ignored by the previous studies that quantified the execution cost for the algorithm in terms of the number of required key comparisons. In this paper we analyze running costs for the algorithm that take into account not only the number of key comparisons, but also the cost of each key comparison. We suppose that keys are represented as sequences of symbols generated by various probabilistic sources and that QuickSelect operates on individual symbols in order to find the target key. We identify limiting distributions for the costs, and derive integral and series expressions for the expectations of the limiting distributions. These expressions are used to recapture previously obtained results on the number of key comparisons required by the algorithm.

Article information

Adv. in Appl. Probab., Volume 45, Number 2 (2013), 425-450.

First available in Project Euclid: 10 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F25: $L^p$-limit theorems
Secondary: 68W40: Analysis of algorithms [See also 68Q25]

QuickSelect QuickQuant QuickVal limit distribution almost-sure convergence L^p-convergence symbol comparison probabilistic source


Fill, James Allen; Nakama, Takehiko. Distributional convergence for the number of symbol comparisons used by QuickSelect. Adv. in Appl. Probab. 45 (2013), no. 2, 425--450. doi:10.1239/aap/1370870125.

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