### Distributional convergence for the number of symbol comparisons used by QuickSelect

#### Abstract

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

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

Dates
First available in Project Euclid: 10 June 2013

https://projecteuclid.org/euclid.aap/1370870125

Digital Object Identifier
doi:10.1239/aap/1370870125

Mathematical Reviews number (MathSciNet)
MR3102458

Zentralblatt MATH identifier
1278.68354

Subjects
Primary: 60F25: $L^p$-limit theorems

#### Citation

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. https://projecteuclid.org/euclid.aap/1370870125

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