@TECHREPORT{Nakama09analysisof, author = {Takéhiko Nakama}, title = {Analysis of Execution Costs for QuickSelect}, institution = {}, year = {2009} }

Share

OpenURL

Abstract

QuickSelect is a search algorithm widely used for finding a key of target rank in a file of keys. When the algorithm compares keys during its execution, 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 dissertation, we conduct analyses of running costs for the algorithm that take into account not only the number of key comparisons but also the cost of each key comparison. First we suppose that keys are independent and uniformly distributed in the unit interval (0, 1) and are represented as their binary expansions, and we derive exact and asymptotic expressions for the expected number of bit comparisons required by QuickSelect. We also establish a closed formula for the expectation that only involves finite summation and use it to compute the expected cost for various values of the target rank and the number of keys. Then we investigate execution costs for the algorithm applied to keys that are represented by more general sequences of symbols, and we identify limiting distributions associated with the costs. Further, we derive integral and series expressions for the expectations of the limiting distributions and use them to recapture previously obtained results on the number of key comparisons required by QuickSelect.