Balanced allocations

• Published in: SIAM journal on computing Vol. 29; no. 1; pp. 180 - 200
• Main Authors: AZAR, Y, BRODER, A. Z, KARLIN, A. R, UPFAL, E
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2000
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Computer science; control theory; systems; Computer systems performance. Reliability; Data processing. List processing. Character string processing; Exact sciences and technology; Mathematics; Memory organisation. Data processing; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Software; Stochastic processes; Theoretical computing; Finite process; Separability; Urn model; Approximation; Planar graph; Resource allocation; Occupancy problem; Graph theory; Balanced conditions; Dynamic allocation; Greedy algorithm; Hashing; Infinite process; Load balancing; Transfer function
• ISSN: 0097-5397; 1095-7111

Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1+o(1)) ln n /ln ln n balls in it. Suppose instead that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only ln ln n/ln 2+O(1) balls - exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and on-line load balancing.