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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/1n 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. Erdos showed that there exist non-2-colorable n-uniform hypegraphs with O(n2 2n) 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 n 1.5, then any non-2-colorable hypergraph must have at least 2 n exp(√n/8) ≫ n22n hyperedges. We observe that the exponential dependence on t in our result is optimal up to constant factors.
Язык оригинала | английский |
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Название основной публикации | Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings |
Страницы | 667-678 |
Число страниц | 12 |
DOI | |
Состояние | Опубликовано - 1 сен 2011 |
Событие | 12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, NY, Соединенные Штаты Америки Продолжительность: 15 авг 2011 → 17 авг 2011 |
Название | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Том | 6844 LNCS |
ISSN (печатное издание) | 0302-9743 |
ISSN (электронное издание) | 1611-3349 |
конференция | 12th International Symposium on Algorithms and Data Structures, WADS 2011 |
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Страна/Tерритория | Соединенные Штаты Америки |
Город | New York, NY |
Период | 15/08/11 → 17/08/11 |
ID: 49849561