The Annals of Probability

On Probabilistic Analysis of a Coalesced Hashing Algorithm

B. Pittel

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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 $0, b = (1 - e^{-n/m})^{-1}$.

Article information

Ann. Probab., Volume 15, Number 3 (1987), 1180-1202.

First available in Project Euclid: 19 April 2007

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Primary: 60C05: Combinatorial probability
Secondary: 60F99: None of the above, but in this section 68P10: Searching and sorting 68P20: Information storage and retrieval 68R05: Combinatorics 05C80: Random graphs [See also 60B20]

Search algorithm hashing probabilistic analysis limiting distributions largest search time


Pittel, B. On Probabilistic Analysis of a Coalesced Hashing Algorithm. Ann. Probab. 15 (1987), no. 3, 1180--1202. doi:10.1214/aop/1176992090.

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