A New Binomial Recurrence Arising in a Graphical Compression Algorithm - Archive ouverte HAL Access content directly
Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2012

A New Binomial Recurrence Arising in a Graphical Compression Algorithm

Abstract

In a recently proposed graphical compression algorithm by Choi and Szpankowski (2009), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability $p$) or the right subtree (with probability 1-$p$). A new node is created as long as there is at least one ball in that node. Furthermore, a nonnegative integer $d$ is given, and at level $d$ or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed (i.e., after $n+d$ steps). Observe that when $d=∞$ the above tree can be modeled as a $\textit{trie}$ that stores $n$ independent sequences generated by a memoryless source with parameter $p$. Therefore, we coin the name $(n,d)$-tries for the tree just described, and to which we often refer simply as $d$-tries. Parameters of such a tree (e.g., path length, depth, size) are described by an interesting two-dimensional recurrence (in terms of $n$ and $d$) that – to the best of our knowledge – was not analyzed before. We study it, and show how much parameters of such a $(n,d)$-trie differ from the corresponding parameters of regular tries. We use methods of analytic algorithmics, from Mellin transforms to analytic poissonization.
Fichier principal
Vignette du fichier
dmAQ0101.pdf (109.33 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive
Loading...

Dates and versions

hal-01197247 , version 1 (11-09-2015)

Identifiers

Cite

Yongwook Choi, Charles Knessl, Wojciech Szpankowski. A New Binomial Recurrence Arising in a Graphical Compression Algorithm. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), 2012, Montreal, Canada. pp.1-12, ⟨10.46298/dmtcs.2979⟩. ⟨hal-01197247⟩

Collections

TDS-MACS
198 View
648 Download

Altmetric

Share

Gmail Facebook X LinkedIn More