The Depoissonisation Quintet: Rice-Poisson-Mellin-Newton-Laplace

Author Brigitte Vallée

Thumbnail PDF


  • Filesize: 469 kB
  • 20 pages

Document Identifiers

Author Details

Brigitte Vallée
  • GREYC Laboratory, CNRS and University of Caen, France

Cite AsGet BibTex

Brigitte Vallée. The Depoissonisation Quintet: Rice-Poisson-Mellin-Newton-Laplace. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


This paper is devoted to the Depoissonisation process which is central in various analyses of the AofA domain. We first recall in Section 1 the two possible paths that may be used in this process, namely the Depoissonisation path and the Rice path. The two paths are rarely described for themselves in the literature, and general methodological results are often difficult to isolate amongst particular results that are more directed towards various applications. The main results for the Depoissonisation path are scattered in at least five papers, with a chronological order which does not correspond to the logical order of the method. The Rice path is also almost always presented with a strong focus towards possible applications. It is often very easy to apply, but it needs a tameness condition, which appears a priori to be quite restrictive, and is not deeply studied in the literature. This explains why the Rice path is very often undervalued. Second, the two paths are not precisely compared, and the situation creates various "feelings": some people see the tools that are used in the two paths as quite different, and strongly prefer one of the two paths; some others think the two paths are almost the same, with just a change of vocabulary. It is thus useful to compare the two paths and the tools they use. This will be done in Sections 2 and 3. We also "follow" this comparison on a precise problem, related to the analysis of tries, introduced in Section 1.7. The paper also exhibits in Section 4 a new framework, of practical use, where the tameness condition of Rice path is proven to hold. This approach, perhaps of independent interest, deals with the shifting of sequences and then the inverse Laplace transform, which does not seem of classical use in this context. It performs very simple computations. This adds a new method to the Depoissonisation context and explains the title of the paper. We then conclude that the Rice path is both of easy and practical use: even though (much?) less general than the Depoissonisation path, it is easier to apply.

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Analysis of Algorithms
  • Poisson model
  • Mellin transform
  • Rice integral
  • Laplace transform
  • Newton interpolation
  • Depoissonisation
  • sources
  • trie structure
  • algorithms on words


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Eda Cesaratto and Brigitte Vallée. Gaussian distribution of trie depth for strongly tame sources. Combinatorics, Probability and Computing, 24, issue 01:54-103, 2014. Google Scholar
  2. Julien Clément, Jim Fill, Thu Hien Nguyen Thi, and Brigitte Vallée. Towards a realistic analysis of the Quickselect algorithm. Theory of Computing Systems, 58(4):528-578, 2016. Google Scholar
  3. Julien Clément, Philippe Flajolet, and Brigitte Vallée. Dynamical sources in information theory: a general analysis of trie structures. Algorithmica, 29, 1-2:307-369, 2001. Google Scholar
  4. Julien Clément, Thu Hien Nguyen Thi, and Brigitte Vallée. Towards a realistic analysis of some popular sorting algorithms. Combinatorics, Probability and Computing, 24 issue 01:104-144, 2014. Google Scholar
  5. Philippe Flajolet. Singularity analysis and asymptotics of Bernoulli sums. Theoret. Comput. Sci., 215 1-2:371-381, 1999. Google Scholar
  6. Philippe Flajolet. A journey to the outskirts of Quickbits. Personal Communication, 2010. Google Scholar
  7. Philippe Flajolet, Xavier Gourdon, and Philippe Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144 1-2:3-58, 1995. Google Scholar
  8. Philippe Flajolet, Mireille Régnier, and Wojtek Szpankowski. Some uses of the Mellin integral transform in the analysis of algorithms. In Combinatorial Algorithms on Words, 12:241-254, 1995. Google Scholar
  9. Philippe Flajolet and Robert Sedgewick. Digital search trees revisited. SIAM J. Comput., 15 (3):48-67, 1986. Google Scholar
  10. Philippe Flajolet and Robert Sedgewick. Mellin transforms and asymptotics: finite differences and Rice’s integrals. Theoretical Computer Science, 144 (1-2):101-124, 1995. Google Scholar
  11. Walter Kurt Hayman. A generalisation of Stirling’s formula. J. Reine Angew. Math., 196:67-95, 1956. Google Scholar
  12. Kanal Hun and Brigitte Vallée. Typical depth of a digital search tree built on a general source. In Proceedings of ANALCO, 2014. Google Scholar
  13. Hsien-Kuei Hwang, Michael Fuchs, and Vytas Zacharovas. Asymptotic variance of random symmetric digital search trees. Discrete Mathematics and Theoretical Computer Science, 12(2):103-166, 2010. Google Scholar
  14. Philippe Jacquet. Trie structure for graphs sequences. In Proceedings of AofA 2014, volume 950 of DMTCS Proceedings, pages 181-192, 2014. Google Scholar
  15. Philippe Jacquet and Wojtek Szpankowski. An analytic approach to the asymptotic variance of trie statistics and related structures. Theoretical Computer Science, 201 1-2:1-62, 1998. Google Scholar
  16. Philippe Jacquet and Wojtek Szpankowski. Entropy computations via Analytic depoissonization. IEEE Transactions on Information Theory, 45 (4):1072-1081, 1999. Google Scholar
  17. Niels Erik Nörlund. Vorlesungen über Differenzenrechnung. Chelsea Publishing Company, New-York, 1954. Google Scholar
  18. Nils Erik Nörlund. Leçons sur les équations linéaires aux différences finies. In Collection de monographies sur la théorie des fonctions. Gauthier-Villars, Paris, 1929. Google Scholar
  19. Brigitte Vallée. Dynamical sources in information theory: Fundamental intervals and word prefixes. Algorithmica, 29 (1/2):262-306, 2001. Google Scholar