Storing a Sparse Table with 0(1) Worst Case Access Time
Journal of the ACM (JACM)
The spatial complexity of oblivious k-probe Hash functions
SIAM Journal on Computing
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Constructing Small Sample Spaces Satisfying Given Constants
SIAM Journal on Discrete Mathematics
Journal of the ACM (JACM)
Finding Even Cycles Even Faster
SIAM Journal on Discrete Mathematics
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Approximation Algorithms for Some Parameterized Counting Problems
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Splitters and near-optimal derandomization
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Detecting short directed cycles using rectangular matrix multiplication and dynamic programming
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
The Parameterized Complexity of Counting Problems
SIAM Journal on Computing
Algorithmic construction of sets for k-restrictions
ACM Transactions on Algorithms (TALG)
Balanced Hashing, Color Coding and Approximate Counting
Parameterized and Exact Computation
Faster algorithms for finding and counting subgraphs
Journal of Computer and System Sciences
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The construction of perfect hash functions is a well-studied topic. In this article, 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 δ T for some constant T0. 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.