{"id":161191,"date":"2011-01-01T00:00:00","date_gmt":"2011-01-01T00:00:00","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/msr-research-item\/balls-and-bins-smaller-hash-families-and-faster-evaluation\/"},"modified":"2018-10-16T21:26:19","modified_gmt":"2018-10-17T04:26:19","slug":"balls-and-bins-smaller-hash-families-and-faster-evaluation","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/balls-and-bins-smaller-hash-families-and-faster-evaluation\/","title":{"rendered":"Balls and Bins: Smaller Hash Families and Faster Evaluation"},"content":{"rendered":"<div class=\"asset-content\">\n<p>A fundamental fact in the analysis of randomized algorithm is that when <i>n<\/i> balls are hashed into <i>n<\/i> bins independently and uniformly at random, with high probability each bin contains at most <i>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<p>In this paper we study the size of families (or, equivalently, the description length of their functions) that guarantee a maximal load of <i>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 <i>\u03a9( log n)<\/i> bits, the best upper bound was formerly <i>O(log<sup>2<\/sup> n \/ log log n)<\/i> bits, which is attained by <i>O(log n \/ log log n)<\/i>-wise independent functions. Traditional constructions of the latter offer an evaluation time of <i>O(log n \/ log log n)<\/i>, which according to Siegel&#8217;s lower bound [FOCS &#8217;89] can be reduced only at the cost of significantly increasing the description length.<\/p>\n<p>We construct two families that guarantee a maximal load of <i>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 <i>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 <i>O(log n log log n)<\/i> bits and evaluated in time <i>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 <i>O(log<sup>3\/2<\/sup> n)<\/i> bits and evaluated in time <i>O(\u221alog n)<\/i>. The latter can be compared to Siegel&#8217;s lower bound stating that <i>O(log n \/ log log n)<\/i>-wise independent functions that are evaluated in time <i>O(\u221alog n)<\/i> must be described using <i>\u03a9(2<sup>\u221alog n<\/sup>)<\/i> bits.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 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