, Volume 75, Issue 4, pp 724–781 | Cite as

A Unified Approach to Linear Probing Hashing with Buckets

  • Svante Janson
  • Alfredo ViolaEmail author


We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on q-calculus) to directly derive the generating functions to analyze.


Hashing Linear probing Buckets Generating functions Analytic combinatorics 

Mathematics Subject Classification

60W40 68P10 68P20 



Philippe Flajolet has had a strong influence in our scientific careers. The core of the use of the symbolic method in hashing problems has been taken from [13]. Thank you Philippe for all the work you have left to inspire our research. We also thank Alois Panholzer for interesting discussions, Mark Wilson for comments on multivariate singularity analysis, and Hsien-Kuei Hwang for suggesting us the derivation that leads to Theorem 50. Last but not least, we thank an anonymous referee for useful comments oriented to improve the presentation of the paper.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Universidad de la RepúblicaMontevideoUruguay

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