Analysis of the Expected Number of Bit Comparisons Required by Quickselect |
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Authors: | James Allen Fill Takéhiko Nakama |
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Affiliation: | 1. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, USA
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Abstract: | When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance
of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number
of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons
but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by
Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest
key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for
the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons
that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and
use it to compute the expectation for each target rank. |
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