Peeling arguments and double hashing

Abstract

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.

Arithmetic Progression Hypergraphs: Examining the Second Moment Method

“…It has been shown that, even with this more limited randomness, not only does the maximum load remain log log n/ log d + O(1) with high probability, but that the asymptotic fraction of bins of each constant load is the same as when the d choices are all perfectly random [21,22]. Similar results have been found for other data structures that use multiple hash functions, including Bloom filters [17] and cuckoo hashing [18,23].…”

More Analysis of Double Hashing for Balanced Allocations

“…We have shown that the coupling argument of Lueker and Molodowitch can, with some modification of the standard random hashing process, yield results for double hashing with the balanced allocations framework. It is worth considering if this approach could be generalized further to handle other processes, most notably cuckoo hashing and peeling processes, where double hashing similarly seems to have the same performance as random hashing [15]. The challenge here for cuckoo hashing appears to be that the state change on entry of a new key is not limited to a single location; while only one cell in the hash table obtains a key, other cells become potential future recipients of the key if it should move, effectively changing the state of those cell.…”

“…Bachrach and Porat use double hashing in a variant of min-wise independent sketches [2]. Mitzenmacher and Thaler show suggestive preliminary results for double hashing for peeling algorithms and cuckoo hashing [15]. Leconte consideres double hashing in the context of the load threshold for cuckoo hashing, and shows that the thresholds are the same if one allows double hashing to fail to place o(n) keys [10].…”

Double hashing thresholds via local weak convergence

“…However, if before throwing each ball we check d bins at random and throw the ball in the least loaded bin, then the most loaded bin will only hold log log n [20], which represents an exponential improvement. In the same setup, where bins have no limit capacity, double hashing achieves the same performances [21], [7] as fully random choices. However, in order to guaranty a worst-case constant look-up time, one cannot allow a key to refer to an unbounded number of items, which motivates the hashing setup presented in Section I.…”