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We consider and analyze a new algorithm for balancing indivisible loads on a distributed network with n processors. The aim is minimizing the discrepancy between the maximum and minimum load. In every time-step paired processors balance their load as evenly as possible. The direction of the excess token is chosen according to a randomized rounding of the participating loads. We prove that in comparison to the corresponding model of Rabani, Sinclair, and Wanka (1998) with arbitrary roundings, the randomization yields an improvement of roughly a square root of the achieved discrepancy in the same number of time-steps on all graphs. For the important case of expanders we can even achieve a constant discrepancy in O(log n (log log n)3) rounds. This is optimal up to loglog-factors while the best previous algorithms in this setting either require ©(log2 n) time or can only achieve a logarithmic discrepancy. Our new result also demonstrates that with randomized rounding the difference between discrete and continuous load balancing vanishes almost completely.