Abstract
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 loglogk) logn can be constructed in time 2O(k loglogk) n logn. 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 \(k \leq O(\frac{\log n}{\log \log \log n})\) in a graph with n vertices. The approximation is up to any fixed desirable relative error.
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Alon, N., Gutner, S. (2007). Balanced Families of Perfect Hash Functions and Their Applications. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_39
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DOI: https://doi.org/10.1007/978-3-540-73420-8_39
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