We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ > 0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ = 1 the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n → ∞. Mainly expectations, variances and asymptotic distributions of these parameters are derived.
Tries are among the most versatile and widely used data structures on words. They are pertinent to the (internal) structure of (stored) words and several splitting procedures used in diverse contexts ranging from document taxonomy to IP addresses lookup, from data compression (i.e., LempelZiv'77 scheme) to dynamic hashing, from partial-match queries to speech recognition, from leader election algorithms to distributed hashing tables and graph compression. While the performance of tries under a realistic probabilistic model is of significant importance, its analysis, even for simplest memoryless sources, has proved difficult. Rigorous findings about inherently complex parameters were rarely analyzed (with a few notable exceptions) under more realistic models of string generations. In this paper we meet these challenges: By a novel use of the contraction method combined with analytic techniques we prove a central limit theorem for the external path length of a trie under a general Markov source. In particular, our results apply to the Lempel-Ziv'77 code. We envision that the methods described here will have further applications to other trie parameters and data structures.
We consider systems of stochastic fixed point equations that arise in the asymptotic analysis of random recursive structures and algorithms such as Quicksort, large Pólya urn processes, and path lengths of random recursive trees and split trees. The main result states sufficient conditions on the fixed point equations that imply the existence of bounded, smooth, rapidly decreasing Lebesgue densities.
In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex v. In this model, the probability of choosing v depends on its depth in the tree. In particular, we assume that there is a function f such that if v has depth k then its probability of being chosen is proportional to ffalse(kfalse). We consider the expected value of the diameter of this model as determined by f, and for various increasing f we find expectations that range from polylogarithmic to linear.
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