Collision Times in Multicolor Urn Models and Sequential Graph Coloring With Applications to Discrete Logarithms

Consider an urn model where at each step one of $q$ colors is sampled according to some probability distribution and a ball of that color is placed in an urn. The distribution of assigning balls to urns may depend on the color of the ball. Collisions occur when a ball is placed in an urn which already contains a ball of different color. Equivalently, this can be viewed as sequentially coloring a complete $q$-partite graph wherein a collision corresponds to the appearance of a monochromatic edge. Using a Poisson embedding technique, the limiting distribution of the first collision time is determined and the possible limits are explicitly described. Joint distribution of successive collision times and multi-fold collision times are also derived. The results can be used to obtain the limiting distributions of running times in various birthday problem based algorithms for solving the discrete logarithm problem, generalizing previous results which only consider expected running times. Asymptotic distributions of the time of appearance of a monochromatic edge are also obtained for other graphs.Comment: Minor revision. 35 pages, 2 figures. To appear in Annals of Applied Probabilit

A fourth moment phenomenon for asymptotic normality of monochromatic subgraphs

Given a graph sequence $\{G_n\}_{n\ge1}$ and a simple connected subgraph $H$, we denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. In this article, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $H$ that we call good joins. Counts of good joins are closely related to the fourth moment of a normalized version of $T(H,G_{n})$, and that connection allows us to show a fourth moment phenomenon for the central limit theorem. Precisely, for $c\ge 30$, we show that $T(H,G_n)$ (appropriately centered and rescaled) converges in distribution to $\mathcal{N}(0,1)$ whenever its fourth moment converges to 3 (the fourth moment of the standard normal distribution). We show the convergence of the fourth moment is necessary to obtain a normal limit when $c\ge 2$. The combination of these results implies that the fourth moment condition characterizes the limiting normal distribution of $T(H,G_n)$ for all subgraphs $H$, whenever $c\ge 30$.Comment: 25 pages, 2 figures; comments welcome

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum