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Balls into bins via local search: Cover time and maximum load
Author(s) -
Bringmann Karl,
Sauerwald Thomas,
Stauffer Alexandre,
Sun He
Publication year - 2016
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.20602
Subject(s) - ball (mathematics) , combinatorics , bounded function , upper and lower bounds , bin , mathematics , homogeneous , vertex (graph theory) , transitive relation , vertex cover , discrete mathematics , graph , algorithm , geometry , mathematical analysis
– We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G . Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m = n , we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m ≥ n . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 681–702, 2016