A New Binomial Recurrence Arising in a Graphical Compression Algorithm

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

The expected profile of digital search trees

AbstractA digital search tree (DST) is a fundamental data structure on words that finds various applications from the popular Lempel–Zivʼ78 data compression scheme to distributed hash tables. The profile of a DST measures the number of nodes at the same distance from the root; it depends on the number of stored strings and the distance from the root. Most parameters of DST (e.g., depth, height, fillup) can be expressed in terms of the profile. We study here asymptotics of the average profile in a DST built from sequences generated independently by a memoryless source. After representing the average profile by a recurrence, we solve it using a wide range of analytic tools. This analysis is surprisingly demanding but once it is carried out it reveals an unusually intriguing and interesting behavior. The average profile undergoes phase transitions when moving from the root to the longest path: at first it resembles a full tree until it abruptly starts growing polynomially and oscillating in this range. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, poissonization and depoissonization, the saddle point method, singularity analysis and uniform asymptotic analysis

Asymptotic variance of random symmetric digital search trees

Dedicated to the 60th birthday of Philippe Flajole