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Balanced allocation on graphs: A random walk approach
Author(s) -
Pourmiri Ali
Publication year - 2019
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20875
Subject(s) - combinatorics , mathematics , random walk , vertex (graph theory) , ball (mathematics) , discrete mathematics , bin , node (physics) , integer (computer science) , random graph , graph , upper and lower bounds , computer science , algorithm , statistics , mathematical analysis , structural engineering , engineering , programming language
We propose algorithms for allocating n sequential balls into n bins that are interconnected as a d ‐regular n ‐vertex graph G , where d ≥ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let ℓ > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ≤ t ≤ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length ℓ from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of ℓ , with high probability.