Cores of random r-partite hypergraphs

Abstract

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.

Information and data management at PUC-rio and UFMG

“…The design of new hashing methods for static sets of keys is strongly related to the generation of indexes for Information Retrieval systems, since a significant portion of the time is spent in hash operations. We proposed a construction for minimal perfect hash functions that combines theoretical analysis, practical performance, expected linear constructing time and near optimal space consumption for the data structure [23,24,25,26,27,28]. The space consumption for the data structure considering n keys and m = n (m the size of the hash table) ranges from 2.62n to 3.3n bits, and for m = 1.23n it ranges from 1.95n to 2.7n bits.…”

Simple multi-party set reconciliation

“…5 For simplicity, we assume random hash functions here, and for technical reasons we assume that the hashes of each key yield distinct locations; hence the k hashes yield a uniform subset of k distinct cells from the m k possibilities. Alternatively, this could easily be accomplished, for instance, by splitting the table into k subtables and having the ith hash function choose a location independently and uniformly at random in the ith subtable, which would not asymptotically change the thresholds [4].…”

“…While the theory set reconciliation among two parties has been widely studied, there appears to be no substantial prior work (that we are aware of) specifically examining the theory of multi-party reconciliation schemes, although the question of how to implement them was raised in [29]. 4 Special cases, such as rumor spreading (see, e.g., [10,12,14,18,22,34]), where one (or more) parties have a single key to share with everyone, have been studied, however.…”

Practical perfect hashing in nearly optimal space

“…For this reason the proof was not included in this paper. Instead it has been published in a more appropriated forum [9]. For the sake of completeness we present a simplified version of Theorem 3.4 in [9].…”