The space complexity of approximating the frequency moments
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Improved bounds and algorithms for hypergraph 2-coloring
Random Structures & Algorithms
A simple algorithm for finding frequent elements in streams and bags
ACM Transactions on Database Systems (TODS)
Equitable Coloring of k-Uniform Hypergraphs
SIAM Journal on Discrete Mathematics
Coloring uniform hypergraphs with few colors
Random Structures & Algorithms
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
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We consider the problem of two-coloring n-uniform hypergraphs. It is known that any such hypergraph with at most 1/10√n/ln n 2n hyperedges can be two-colored [7]. In fact, there is an efficient (requiring polynomial time in the size of the input) randomized algorithm that produces such a coloring. As stated [7], this algorithm requires random access to the hyperedge set of the input hypergraph. In this paper, we show that a variant of this algorithm can be implemented in the streaming model (with just one pass over the input), using space O(|V|B), where V is the vertex set of the hypergraph and each vertex is represented by B bits. (Note that the number of hyperedges in the hypergraph can be superpolynomial in |V|, and it is not feasible to store the entire hypergraph in memory). We also consider the question of the minimum number of hyperedges in non-two-colorable n-uniform hypergraphs. Erdös showed that there exist non-2-colorable n-uniform hypegraphs with O(n22n) hyperedges and Θ(n2) vertices. We show that the choice Θ(n2) for the number of vertices in Erdös's construction is crucial: any hypergraph with at most 2n2/t vertices and 2n exp(t/8) hyperedges is 2-colorable. (We present a simple randomized streaming algorithm to construct the two-coloring.) Thus, for example, if the number of vertices is at most n1.5, then any non-2- colorable hypergraph must have at least 2n exp(√n/8) ≫ n22n hyperedges. We observe that the exponential dependence on t in our result is optimal up to constant factors.