Showing papers by "Mark Daniel Ward published in 2005"
TL;DR: In this article, it was shown that under a Markovian model of order one, the average depth of suffix trees of index n is asymptotically similar to the average depths of tries (a.k.a. digital trees) built on n independent strings.
Abstract: In this report, we prove that under a Markovian model of order one, the average depth of suffix trees of index n is asymptotically similar to the average depth of tries (a.k.a. digital trees) built on n independent strings. This leads to an asymptotic behavior of $(\log{n})/h + C$ for the average of the depth of the suffix tree, where $h$ is the entropy of the Markov model and $C$ is constant. Our proof compares the generating functions for the average depth in tries and in suffix trees; the difference between these generating functions is shown to be asymptotically small. We conclude by using the asymptotic behavior of the average depth in a trie under the Markov model found by Jacquet and Szpankowski ([JaSz91]).
29 citations
TL;DR: In this paper, the authors consider a sequence of n geometric random variables and interpret the outcome as an urn model, and derive asymptotic equivalents for all (centered or uncentered) moments in a fairly automatic way.
Abstract: We consider a sequence of n geometric random variables and interpret the outcome as an urn model. For a given parameter m, we treat several parameters like what is the largest urn containing at least (or exactly) m balls, or how many urns contain at least m balls, etc. Many of these questions have their origin in some computer science problems. Identifying the underlying distributions as (variations of) the extreme value distribution, we are able to derive asymptotic equivalents for all (centered or uncentered) moments in a fairly automatic way.
24 citations
TL;DR: This work proves that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations, and compares the distribution of the M MP in suffix trees to its distribution in tries built over independent strings.
Abstract: In a suffix tree, the multiplicity matching parameter (MMP) $M_n$ is the number of leaves in the subtree rooted at the branching point of the $(n+1)$st insertion. Equivalently, the MMP is the number of pointers into the database in the Lempel-Ziv '77 data compression algorithm. We prove that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations. In the proof we compare the distribution of the MMP in suffix trees to its distribution in tries built over independent strings. Our results are derived by both probabilistic and analytic techniques of the analysis of algorithms. In particular, we utilize combinatorics on words, bivariate generating functions, pattern matching, recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis.
8 citations
01 Jan 2005
TL;DR: In a suffix tree, the multiplicity matching parameter (MMP) Mn is the number of leaves in the subtree rooted at the branching point of the (n + 1)st insertion and it is proved that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations.
Abstract: In a suffix tree, the multiplicity matching parameter (MMP) Mn is the number of leaves in the subtree rooted at the branching point of the (n + 1)st insertion Equivalently, the MMP is the number of pointers into the database in the Lempel-Ziv '77 data compression algorithm We prove that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations In the proof we compare the distribution of the MMP in suffix trees to its distribution in tries built over independent strings Our results are derived by both probabilistic and analytic techniques of the analysis of algorithms In particular, we utilize combinatorics on words, bivariate generating functions, pattern matching, recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis
1 citations