A fundamental fact in the analysis of randomized algorithm is that when n<\/i> balls are hashed into n<\/i> bins independently and uniformly at random, with high probability each bin contains at most O(log n \/ log log n)<\/i> balls. In various applications, however, the assumption that a truly random hash function is available is not always valid, and explicit functions are required.<\/p>\n\n\n\n
In this paper we study the size of families (or, equivalently, the description length of their functions) that guarantee a maximal load of O(log n \/ log log n)<\/i> with high probability, as well as the evaluation time of their functions. Whereas such functions must be described using \u03a9( log n)<\/i> bits, the best upper bound was formerly O(log2<\/sup> n \/ log log n)<\/i> bits, which is attained by O(log n \/ log log n)<\/i>-wise independent functions. Traditional constructions of the latter offer an evaluation time of O(log n \/ log log n)<\/i>, which according to Siegel’s lower bound [FOCS ’89] can be reduced only at the cost of significantly increasing the description length.<\/p>\n\n\n\n We construct two families that guarantee a maximal load of O(log n \/ log log n)<\/i> with high probability. Our constructions are based on two different approaches, and exhibit different trade-offs between the description length and the evaluation time. The first construction shows that O(log n \/ log log n)<\/i>-wise independence can in fact be replaced by \u201cgradually increasing independence\u201d, resulting in functions that are described using O(log n log log n)<\/i> bits and evaluated in time O(log n log log n)<\/i>. The second construction is based on derandomization techniques for space-bounded computations combined with a tailored construction of a pseudorandom generator, resulting in functions that are described using O(log3\/2<\/sup> n)<\/i> bits and evaluated in time O(\u221alog n)<\/i>. The latter can be compared to Siegel’s lower bound stating that O(log n \/ log log n)<\/i>-wise independent functions that are evaluated in time O(\u221alog n)<\/i> must be described using \u03a9(2\u221alog n<\/sup>)<\/i> bits.<\/p>\n","protected":false},"excerpt":{"rendered":" A fundamental fact in the analysis of randomized algorithm is that when n balls are hashed into n bins independently and uniformly at random, with high probability each bin contains at most O(log n \/ log log n) balls. In various applications, however, the assumption that a truly random hash function is available is not […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Elisa 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