Streaming algorithms for 2-coloring uniform hypergraphs

Jaikumar Radhakrishnan, Saswata Shannigrahi

Результат исследований: Публикации в книгах, отчётах, сборниках, трудах конференцийстатья в сборнике материалов конференциинаучнаярецензирование

5 Цитирования (Scopus)

Аннотация

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 оригиналаанглийский
Название основной публикацииAlgorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings
Страницы667-678
Число страниц12
DOI
СостояниеОпубликовано - 1 сен 2011
Событие12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, NY, Соединенные Штаты Америки
Продолжительность: 15 авг 201117 авг 2011

Серия публикаций

НазваниеLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Том6844 LNCS
ISSN (печатное издание)0302-9743
ISSN (электронное издание)1611-3349

Конференция

Конференция12th International Symposium on Algorithms and Data Structures, WADS 2011
СтранаСоединенные Штаты Америки
ГородNew York, NY
Период15/08/1117/08/11

Предметные области Scopus

  • Теоретические компьютерные науки
  • Компьютерные науки (все)

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  • Цитировать

    Radhakrishnan, J., & Shannigrahi, S. (2011). Streaming algorithms for 2-coloring uniform hypergraphs. В Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings (стр. 667-678). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Том 6844 LNCS). https://doi.org/10.1007/978-3-642-22300-6_57