Statistics Papers

Document Type

Journal Article

Date of this Version

2016

Publication Source

The Annals of Applied Probability

Volume

26

Issue

6

Start Page

3286

Last Page

3318

DOI

10.1214/16-AAP1176

Abstract

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

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aoap/1481792586#abstract

Comments

At the time of publication, author Bhaswar B. Bhattacharya was affiliated with Stanford University. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania

Keywords

Discrete Logarithm, Graph coloring, Limit theorems, Poisson Embedding

Share

COinS
 

Date Posted: 27 November 2017

This document has been peer reviewed.