Dynamic tree algorithms

In this paper, a general tree algorithm processing a random flow of arrivals is analyzed. Capetanakis–Tsybakov–Mikhailov’s protocol in the context of communication networks with random access is an example of such an algorithm. In computer science, this corresponds to a trie structure with a dynamic input. Mathematically, it is related to a stopped branching process with exogeneous arrivals (immigration). Under quite general assumptions on the distribution of the number of arrivals and on the branching procedure, it is shown that there exists a positive constant $\lambda_c$ so that if the arrival rate is smaller than $\lambda_c$, then the algorithm is stable under the flow of requests, that is, that the total size of an associated tree is integrable. At the same time, a gap in the earlier proofs of stability in the literature is fixed. When the arrivals are Poisson, an explicit characterization of $\lambda_c$ is given. Under the stability condition, the asymptotic behavior of the average size of a tree starting with a large number of individuals is analyzed. The results are obtained with the help of a probabilistic rewriting of the functional equations describing the dynamics of the system. The proofs use extensively this stochastic background throughout the paper. In this analysis, two basic limit theorems play a key role: the renewal theorem and the convergence to equilibrium of an auto-regressive process with a moving average.

💡 Research Summary

The paper studies a class of dynamic tree algorithms that process a random flow of arrivals. An initial set of items of size n is recursively split into G sub‑sets whenever its size exceeds a fixed threshold D. After each split, each sub‑set receives an independent random number of new items A_i (i.i.d.) and the procedure repeats on every sub‑set. This construction naturally yields a rooted tree: the root contains the original n items, internal nodes contain at least D items, and leaves contain fewer than D items.

Two equivalent representations are introduced. The first is a distributional recursion for the total number of nodes R_{A,n}: \