Cores in random hypergraphs and Boolean formulas

Molloy, Michael

doi:10.1002/rsa.20061

Cited by 154 publications

(247 citation statements)

References 17 publications

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“…The proof of our theorem will be reminiscent of studies of the k-core, the pure literal rule, and other similar problems [14,11,6,7,9]. There, one repeatedly removes vertices (literals, etc.)…”

Section: Introductionmentioning

confidence: 99%

Sets that are connected in two random graphs

Molloy

2013

Random Struct Algorithms

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Section: Introductionmentioning

confidence: 99%

Sets that are connected in two random graphs

Molloy

2013

Random Struct Algorithms

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“…This specific process finds the k-core of a hypergraph. The k-core itself appears in the analysis of several algorithms, and variations on this underlying process have arisen in the analysis of many problems that on their surface appear quite different, such as low-density paritycheck codes [16], cuckoo hashing [23], [5], [7], [9], and the satisfiability of random formulae [2], [18], [22].…”

Section: Introductionmentioning

confidence: 99%

Peeling arguments and double hashing

Mitzenmacher

Thaler

2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-The analysis of several algorithms and data structures can be reduced to the analysis of the following greedy "peeling" process: start with a random hypergraph; find a vertex of degree at most k, and remove it and all of its adjacent hyperedges from the graph; repeat until there is no suitable vertex. This specific process finds the k-core of a hypergraph, and variations on this theme have proven useful in analyzing for example decoding from low-density parity-check codes, several hash-based data structures such as cuckoo hashing, and algorithms for satisfiability of random formulae. This approach can be analyzed several ways, with two common approaches being via a corresponding branching process or a fluid limit family of differential equations.In this paper, we make note of an interesting aspect of these types of processes: the results are generally the same when the randomness is structured in the manner of double hashing. This phenomenon allows us to use less randomness and simplify the implementation for several hash-based data structures and algorithms. We explore this approach from both an empirical and theoretical perspective, examining theoretical justifications as well as simulation results for specific problems.

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“…Analogous results were proved for ordinary (not multipartite) hypergraphs in [4,5,15]. We will assume the reader is familiar with [4], and point out the simple modifications of its proof so as to cover the present setting.…”

Section: Proof Of Theoremmentioning

confidence: 63%

“…The argument of [4] can now be followed from just below its Equation (12). In place of its Equations (15) and (16) we now have…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Cores of random r-partite hypergraphs

Botelho

Wormald

Ziviani

2012

Information Processing Letters

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We show that the threshold c r,k for appearance of a k-core in a random r-partite r-uniform hypergraph G r,n,m is the same as for a random r-uniform hypergraph with cn/r edges without the r-partite restriction, where r, k ≥ 2. In both cases, the average degree is c. This is an important problem in the analysis of the algorithm presented in [2]. The algorithm constructs a family of minimal perfect hash functions based on random r-partite r-uniform hypergraphs with an empty k-core subgraph, for k ≥ 2. The above claim was not proved but was provided with strong experimental evidence. For an input key set S with m keys, the algorithm was the first one capable of constructing a simple and efficient family of minimal perfect hash functions that can be stored in O(m) bits, where the hidden constant is within a factor of two from the information theoretical lower bound. The case r, k = 2 was analyzed in [3] but the general case r ≥ 3, k ≥ 2 was still open.

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Cores in random hypergraphs and Boolean formulas

Cited by 154 publications

References 17 publications

Sets that are connected in two random graphs

Sets that are connected in two random graphs

Peeling arguments and double hashing

Cores of random r-partite hypergraphs

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