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Journal of the ACM (JACM)
The spatial complexity of oblivious k-probe Hash functions
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SIAM Journal on Discrete Mathematics
Journal of the ACM (JACM)
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Approximation Algorithms for Some Parameterized Counting Problems
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Algorithmic construction of sets for k-restrictions
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IEEE Transactions on Information Theory - Part 2
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Proceedings of the forty-first annual ACM symposium on Theory of computing
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SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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The construction of perfect hash functions is a well-studied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to [k] is a δ-balanced (n, k)-family of perfect hash functions if for every S ⊆ [n], |S| = k, the number of functions that are 1-1 on S is between T/δ and δT for some constant T 0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on S, for each S of size k. In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking δ to be close to 1) for every such S. Our main result is that for any constant δ 1, a δ-balanced (n, k)-family of perfect hash functions of size 2O(k log log k) log n can be constructed in time 2O(k log log k)n log n. Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length k and the number of simple cycles of size k for any k ≤ O(log n/log log log n) in a graph with n vertices. The approximation is up to any fixed desirable relative error.