Double hashing thresholds via local weak convergence

Leconte, Mathieu

doi:10.1109/allerton.2013.6736515

Cited by 4 publications

(4 citation statements)

References 26 publications

(30 reference statements)

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“…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]. Recently, Mitzenmacher has shown that double hashing asymptotically has no effect on the load distribution in the setting of balanced allocations [13]; we describe this result further in the related work below.…”

Section: Introductionmentioning

confidence: 97%

More Analysis of Double Hashing for Balanced Allocations

Mitzenmacher

2015

2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Abstract. With double hashing, for a key x, one generates two hash values f (x) and g(x), and then uses combinations (f (x) + ig(x)) mod n for i = 0, 1, 2, . . . to generate multiple hash values in the range [0, n − 1] from the initial two. For balanced allocations, keys are hashed into a hash table where each bucket can hold multiple keys, and each key is placed in the least loaded of d choices. It has been shown previously that asymptotically the performance of double hashing and fully random hashing is the same in the balanced allocation paradigm using fluid limit methods. Here we extend a coupling argument used by Lueker and Molodowitch to show that double hashing and ideal uniform hashing are asymptotically equivalent in the setting of open address hash tables to the balanced allocation setting, providing further insight into this phenomenon. We also discuss the potential for and bottlenecks limiting the use this approach for other multiple choice hashing schemes.

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Section: Introductionmentioning

confidence: 97%

More Analysis of Double Hashing for Balanced Allocations

Mitzenmacher

2015

2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…This reduces the amount of entropy in a key's hash values from k log m to 2 log m with no apparent downsides. In particular the thresholds c * k,ℓ remain the same [47,52]. Unaligned Blocks.…”

Section: Creatively Wielding the Powermentioning

confidence: 98%

“…More progress towards proving the conjecture would be exciting. Maybe techniques from statistical physics, which have been a powerful tool for determining thresholds in the static case [49,47,46] can help with the dynamic case as well.…”

mentioning

confidence: 99%

Peeling Close to the Orientability Threshold – Spatial Coupling in Hashing-Based Data Structures

Walzer

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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The celebrated power of two choices paradigm underlies cuckoo hash tables as follows: If you have n balls and m = (2 + ε)n bins and throw each ball into a bin at random, then likely some bin will receive Ω( log n log log n ) balls. If, however, you can choose between two random bins for each ball, you can likely arrange for a private bin for each ball.In the first part of this article, we review some related space-efficient data structures on a high level. We'll find that the additional power afforded by more than 2 choices is often outweighed by the additional costs they bring. In the second part, we present a data structure where choices play a role at coarser than per-ball granularity. In some sense, we rely on the power of 1 + ε choices per ball.This article was written for the algorithms column of the bulletin of the EATCS. It is a "best-of" of my dissertation and related work reviewed from a fresh perspective. I've tried to make it a pleasant read conveying intuition while being unencumbered by technical details. So allow me to be your guide through the garden of my interests, present and past. Our winding path will circle a recent construction called Bumped Ribbon Retrieval [24], with which our tour will conclude.

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“…We still have to exclude the possibility of a gap of size o(n) on the right hand side, imagine for instance M (G n,cn ) = ( − 1 + c)n − √ n to appreciate the difference. In the setting of cuckoo hashing with double hashing (see [16]), it is actually the analogue of this pesky distinction that seems to be in the way of proving precise thresholds for perfect orientability, so we should treat this seriously.…”

Section: :7mentioning

confidence: 99%

Load Thresholds for Cuckoo Hashing with Overlapping Blocks

Walzer

2023

ACM Trans. Algorithms

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We consider a natural variation of cuckoo hashing proposed by Lehman and Panigrahy (2009). Each of cn objects is assigned k = 2 intervals of size l in a linear hash table of size n and both starting points are chosen independently and uniformly at random. Each object must be placed into a table cell within its intervals, but each cell can only hold one object. Experiments suggested that this scheme outperforms the variant with blocks in which intervals are aligned at multiples of l . In particular, the load threshold is higher, i.e. the load c that can be achieved with high probability. For instance, Lehman and Panigrahy (2009) empirically observed the threshold for l = 2 to be around \(96.5\% \) as compared to roughly \(89.7\% \) using blocks. They pinned down the asymptotics of the thresholds for large l , but the precise values resisted rigorous analysis. We establish a method to determine these load thresholds for all l ≥ 2, and, in fact, for general k ≥ 2. For instance, for k = l = 2 we get \(\approx 96.4995\% \) . We employ a theorem due to Leconte, Lelarge, and Massoulié (2013), which adapts methods from statistical physics to the world of hypergraph orientability. In effect, the orientability thresholds for our graph families are determined by belief propagation equations for certain graph limits. As a side note we provide experimental evidence suggesting that placements can be constructed in linear time using an adapted version of an algorithm by Khosla (2013).

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Double hashing thresholds via local weak convergence

Cited by 4 publications

References 26 publications

More Analysis of Double Hashing for Balanced Allocations

More Analysis of Double Hashing for Balanced Allocations

Peeling Close to the Orientability Threshold – Spatial Coupling in Hashing-Based Data Structures

Load Thresholds for Cuckoo Hashing with Overlapping Blocks

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