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July, 1987 On Probabilistic Analysis of a Coalesced Hashing Algorithm
B. Pittel
Ann. Probab. 15(3): 1180-1202 (July, 1987). DOI: 10.1214/aop/1176992090


An allocation model [$n$ balls, $m (\geq n)$ cells, at most one ball in a cell] related to a hashing algorithm is studied. A ball $x$ goes into the cell $h(x)$, where $h(\cdot): \{1,\cdots, n\} \rightarrow \{1, \cdots, m\}$ is random. In case the cell $h(x)$ is already occupied, the ball $x$ is rejected and moved into the leftmost empty cell. This empty cell is found via the sequential search from left to right starting with the cell occupied by the last (before $x$) rejected ball. Denote $T_2(x)$ the number of the necessary probes. In the end, due to a resulting system of references, the $n$ occupied cells form a disjoint union of ordered chains, and to locate a ball $x$ it suffices to search only the cells of a subchain originating at the cell $h(x)$. Denote $T_1(x)$ the length of this subchain. The main result of the paper is: in probability, $\max T_1(x) = \log_bn - 2\log_b\log n + O(1),$ $\max T_2(x) = \log_bn - \log_b\log n + O(1),$ as $n \rightarrow \infty$, if $n/m$ is bounded away from $0, b = (1 - e^{-n/m})^{-1}$.


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B. Pittel. "On Probabilistic Analysis of a Coalesced Hashing Algorithm." Ann. Probab. 15 (3) 1180 - 1202, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0626.60012
MathSciNet: MR893923
Digital Object Identifier: 10.1214/aop/1176992090

Primary: 60C05
Secondary: 05C80 , 60F99 , 68P10 , 68P20 , 68R05

Keywords: hashing , largest search time , limiting distributions , probabilistic analysis , Search algorithm

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 3 • July, 1987
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