Alan Frieze, Tony Johansson
Cuckoo Hashing is a hashing scheme invented by Pagh and Rodler. It uses $d\geq 2$ distinct hash functions to insert items into the hash table. It has been an open question for some time as to the expected time for Random Walk Insertion to add items. We show that if the number of hash functions $d=O(1)$ is sufficiently large, then the expected insertion time is $O(1)$ per item.
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We study wear-leveling techniques for cuckoo hashing, showing that it is possible to achieve a memory wear bound of $\log\log n+O(1)$ after the insertion of $n$ items into a table of size $Cn$ for a suitable constant $C$ using cuckoo hashing. Moreover, we study our cuckoo hashing method empirically, showing that it significantly improves on the memory wear performance for classic cuckoo hashing and linear probing in practice.
We show a new proof for the load of obtained by a Cuckoo Hashing data structure. Our proof is arguably simpler than previous proofs and allows for new generalizations. The proof first appeared in Pinkas et. al. \cite{PSWW19} in the context of a protocol for private set intersection. We present it here separately to improve its readability.
O(1) Expected Insertion for Random Walk d-ary Cuckoo Hashing up to the Load Threshold
本文提出了一种在负载阈值内实现随机游走 d-ary 鸽巢 hashing 的 O(1) 插入算法，通过理论分析证明了插入时间的期望为常数。
Although cuckoo hashing has significant applications in both theoretical and practical settings, a relevant downside is that it requires lookups to multiple locations. In many settings, where lookups are expensive, cuckoo hashing becomes a less compelling alternative. One such standard setting is when memory is arranged in large pages, and a major cost is the number of page accesses. We propose the study of cuckoo hashing with pages, advocating approaches where each key has several possible locations, or cells, on a single page, and additional choices on a second backup page. We show experimen…
Cuckoo hashing is a highly practical dynamic dictionary: it provides amortized constant insertion time, worst case constant deletion time and lookup time, and good memory utilization. However, with a noticeable probability during the insertion of n elements some insertion requires Ω(log n) time. Whereas such an amortized guarantee may be suitable for some applications, in other applications (such as high-performance routing) this is highly undesirable. Recently, Kirsch and Mitzenmacher (Allerton '07) proposed a de-amortization of cuckoo hashing using various queueing techniques that preserve i…
Cuckoo hashing is an efficient technique for creating large hash tables with high space utilization and guaranteed constant access times. There, each item can be placed in a location given by any one out of k different hash functions. In this paper we investigate further the random walk heuristic for inserting in an online fashion new items into the hash table. Provided that k > 2 and that the number of items in the table is below (but arbitrarily close) to the theoretically achievable load threshold, we show a polylogarithmic bound for the maximum insertion time that holds with high probabili…
Most hash tables have an insertion time of $O(1)$, possibly qualified as expected and/or amortised. While insertions into cuckoo hash tables indeed seem to take $O(1)$ expected time in practice, only polylogarithmic guarantees are proven in all but the simplest of practically relevant cases. Given the widespread use of cuckoo hashing to implement compact dictionaries and Bloom filter alternatives, closing this gap is an important open problem for theoreticians. In this paper, we show that random walk insertions into cuckoo hash tables take $O(1)$ expected amortised time when any number $k \geq…
It is shown that for cuckoo hashing with a stash as proposed by Kirsch, Mitzenmacher, and Wieder (2008) families of very simple hash functions can be used, maintaining the favorable performance guarantees: with stash size $s$ the probability of a rehash is $O(1/n^{s+1})$, and the evaluation time is $O(s)$. Instead of the full randomness needed for the analysis of Kirsch et al. and of Kutzelnigg (2010) (resp. $Θ(\log n)$-wise independence for standard cuckoo hashing) the new approach even works with 2-wise independent hash families as building blocks. Both construction and analysis build upon t…
A Perfect Hash Function (PHF) is a hash function that has no collisions on a given input set. PHFs can be used for space efficient storage of data in an array, or for determining a compact representative of each object in the set. In this paper, we present the PHF construction algorithm SicHash - Small Irregular Cuckoo Tables for Perfect Hashing. At its core, SicHash uses a known technique: It places objects in a cuckoo hash table and then stores the final hash function choice of each object in a retrieval data structure. We combine the idea with irregular cuckoo hashing, where each object has…
Efficient $d$-ary Cuckoo Hashing at High Load Factors by Bubbling Up
本文提出冒泡上浮杜鹃哈希，在高负载因子下用接近最优的哈希函数数量同时实现快速插入和 O(1) 正查询时间。
Cuckoo Hashing in Cryptography: Optimal Parameters, Robustness and Applications
本文提出使用更多哈希函数的布谷鸟哈希构造，显著降低查询开销并证明最优性，同时引入鲁棒布谷鸟哈希应对恶意输入，并应用于改进批量码和PIR构造。
Cuckoo hashing with a stash is a robust multiple choice hashing scheme with high memory utilization that can be used in many network device applications. Unfortunately, for memory loads beyond 0.5, little is known on its performance. In this paper, we analyze its average performance over such loads. We tackle this problem by recasting the problem as an analysis of the expected maximum matching size of a given random bipartite graph. We provide exact results for any finite system, and also deduce asymptotic results as the memory size increases. We further consider other variants of this problem…
Cuckoo hashing is a common hashing technique, guaranteeing constant-time lookups in the worst case. Adding a stash was proposed by Kirsch, Mitzenmacher, and Wieder at SICOMP 2010, as a way to reduce the probability of failure (i.e., the probability that a valid Cuckoo assignment fails to exist). It has since become a standard technique in areas such as cryptography, where a negligible probability of failure is often required. We focus on an extension of Cuckoo hashing that allows multiple items per bucket, which improves the load factor. That extension was also analyzed by Kirsch \emph{et al.}…
A dictionary (or map) is a key-value store that requires all keys be unique, and a multimap is a key-value store that allows for multiple values to be associated with the same key. We design hashing-based indexing schemes for dictionaries and multimaps that achieve worst-case optimal performance for lookups and updates, with a small or negligible probability the data structure will require a rehash operation, depending on whether we are working in the the external-memory (I/O) model or one of the well-known versions of the Random Access Machine (RAM) model. One of the main features of our cons…
The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There,we are given a table with $n$ locations, and we assume that each location can hold one item. Each item to be inserted chooses randomly k>1 locations and has to be placed in any one of them. How much load can cuckoo hashing handle before collisions prevent the successful assignment of the available items to the chosen locations? Practical evaluations of this method have shown that one can allocate a numbe…
Cuckoo hashing [4] is a multiple choice hashing scheme in which each item can be placed in multiple locations, and collisions are resolved by moving items to their alternative locations. In the classical implementation of two-way cuckoo hashing, the memory is partitioned into contiguous disjoint fixed-size buckets. Each item is hashed to two buckets, and may be stored in any of the positions within those buckets. Ref. [2] analyzed a variation in which the buckets are contiguous and overlap. However, many systems retrieve data from secondary storage in same-size blocks called pages. Fetching a …
The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constant-time operations in the worst case with high probability, and in terms of space consumption there are known constructions that use essentially optimal space. However, although the first analysis of a dynamic dictionary dates back more than 45 years ago (when Knuth analyzed linear probing in 1963), the trade-off between these aspects of performance is still not completely understood. In thi…
We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k >= 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds alpha_k such that, as n goes to infinity, if n/m <= alpha for some alpha < alpha_k then a hash table can be constructed successfully with high probability, and if n/m >= …
Cuckoo hashing guarantees constant-time lookups regardless of table density, making it a viable candidate for high-density tables. Cuckoo hashing insertions perform poorly at high table densities, however. In this paper, we mitigate this problem through the introduction of novel kick-out eviction algorithms. Experimentally, our algorithms reduce the number of bins viewed per insertion for high-density tables by as much as a factor of ten. We also introduce an optimistic concurrency scheme for transactional multi-writer cuckoo hash tables (not using hardware transactional memory). For delete-li…
Dietzfelbinger and Weidling [DW07] proposed a natural variation of cuckoo hashing where each of $cn$ objects is assigned $k = 2$ intervals of size $\ell$ in a linear (or cyclic) hash table of size $n$ and both start 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 $\ell$. In particular, the load threshold is higher, i.e. the load $c$ that can be achieved with h…
Compact Parallel Hash Tables on the GPU
本文研究GPU上紧凑哈希表设计，提出紧凑桶式布谷鸟哈希和紧凑冰山哈希两种方案，并通过CUDA实现与基准测试证明紧凑性可显著提升查找和插入吞吐量，同时保持动态操作性能。
Fast Insertion for Bucketized Cuckoo Hashing
研究布谷鸟哈希的快速插入问题，提出新插入过程，能将哈希表填充到接近最优阈值的负载因子，有poly(ε^-1)插入时间界和强摊销保证，传统算法无法匹配，且查询算法能大概率猜出键所在桶，提升查询效率。
Hash tables are ubiquitous in computer science for efficient access to large datasets. However, there is always a need for approaches that offer compact memory utilisation without substantial degradation of lookup performance. Cuckoo hashing is an efficient technique of creating hash tables with high space utilisation and offer a guaranteed constant access time. We are given $n$ locations and $m$ items. Each item has to be placed in one of the $k\ge2$ locations chosen by $k$ random hash functions. By allowing more than one choice for a single item, cuckoo hashing resembles multiple choice allo…
We consider space efficient hash tables that can grow and shrink dynamically and are always highly space efficient, i.e., their space consumption is always close to the lower bound even while growing and when taking into account storage that is only needed temporarily. None of the traditionally used hash tables have this property. We show how known approaches like linear probing and bucket cuckoo hashing can be adapted to this scenario by subdividing them into many subtables or using virtual memory overcommitting. However, these rather straightforward solutions suffer from slow amortized inser…
In the past few years, General Purpose Graphics Processors (GPUs) have been used to significantly speed up numerous applications. One of the areas in which GPUs have recently led to a significant speed-up is model checking. In model checking, state spaces, i.e., large directed graphs, are explored to verify whether models satisfy desirable properties. GPUexplore is a GPU-based model checker that uses a hash table to efficiently keep track of already explored states. As a large number of states is discovered and stored during such an exploration, the hash table should be able to quickly handle …
We study the the following question in Random Graphs. We are given two disjoint sets $L,R$ with $|L|=n=αm$ and $|R|=m$. We construct a random graph $G$ by allowing each $x\in L$ to choose $d$ random neighbours in $R$. The question discussed is as to the size $μ(G)$ of the largest matching in $G$. When considered in the context of Cuckoo Hashing, one key question is as to when is $μ(G)=n$ whp? We answer this question exactly when $d$ is at least four. We also establish a precise threshold for when Phase 1 of the Karp-Sipser Greedy matching algorithm suffices to compute a maximum matching whp.
We study randomness properties of graphs and hypergraphs generated by simple hash functions. Several hashing applications can be analyzed by studying the structure of $d$-uniform random ($d$-partite) hypergraphs obtained from a set $S$ of $n$ keys and $d$ randomly chosen hash functions $h_1,\dots,h_d$ by associating each key $x\in S$ with a hyperedge $\{h_1(x),\dots, h_d(x)\}$. Often it is assumed that $h_1,\dots,h_d$ exhibit a high degree of independence. We present a simple construction of a hash class whose hash functions have small constant evaluation time and can be stored in sublinear sp…
Randomized algorithms are often enjoyed for their simplicity, but the hash functions used to yield the desired theoretical guarantees are often neither simple nor practical. Here we show that the simplest possible tabulation hashing provides unexpectedly strong guarantees. The scheme itself dates back to Carter and Wegman (STOC'77). Keys are viewed as consisting of c characters. We initialize c tables T_1, ..., T_c mapping characters to random hash codes. A key x=(x_1, ..., x_q) is hashed to T_1[x_1] xor ... xor T_c[x_c]. While this scheme is not even 4-independent, we show that it provides ma…
Bloom filters, cuckoo filters, and other approximate set membership sketches have a wide range of applications. Oftentimes, expensive operations can be skipped if an item is not in a data set. These filters provide an inexpensive, memory efficient way to test if an item is in a set and avoid unnecessary operations. Existing sketches only allow membership testing for single set. However, in some applications such as join processing, the relevant set is not fixed and is determined by a set of predicates. We propose the Conditional Cuckoo Filter, a simple modification of the cuckoo filter that al…
We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.
Many applications of approximate membership query data structures, or filters, require only an incremental filter that supports insertions but not deletions. However, the design space of incremental filters is missing a "sweet spot" filter that combines space efficiency, fast queries, and fast insertions. Incremental filters, such as the Bloom and blocked Bloom filter, are not space efficient. Dynamic filters (i.e., supporting deletions), such as the cuckoo or vector quotient filter, are space efficient but do not exhibit consistently fast insertions and queries. In this paper, we propose the …
CavityRank: Zero-Extra-Byte Residual Routing for Cuckoo Filters
CavityRank 为布谷鸟过滤器设计的零额外字节残差路由，通过利用压缩桶资源编码秩，在插入性能上优于随机踢出与深度10 BFS，缩小了预言机残差间隙，处于实用设计区间。
Robin Hood hashing is a variation on open addressing hashing designed to reduce the maximum search time as well as the variance in the search time for elements in the hash table. While the case of insertions only using Robin Hood hashing is well understood, the behavior with deletions has remained open. Here we show that Robin Hood hashing can be analyzed under the framework of finite-level finite-dimensional jump Markov chains. This framework allows us to re-derive some past results for the insertion-only case with some new insight, as well as provide a new analysis for a standard deletion mo…
We present a GPU-based Locality Sensitive Hashing (LSH) algorithm to speed up beam search for sequence models. We utilize the winner-take-all (WTA) hash, which is based on relative ranking order of hidden dimensions and thus resilient to perturbations in numerical values. Our algorithm is designed by fully considering the underling architecture of CUDA-enabled GPUs (Algorithm/Architecture Co-design): 1) A parallel Cuckoo hash table is applied for LSH code lookup (guaranteed O(1) lookup time); 2) Candidate lists are shared across beams to maximize the parallelism; 3) Top frequent words are merg…
Randomized algorithms are often enjoyed for their simplicity, but the hash functions employed to yield the desired probabilistic guarantees are often too complicated to be practical. Here we survey recent results on how simple hashing schemes based on tabulation provide unexpectedly strong guarantees. Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as consisting of $c$ characters and we have precomputed character tables $h_1,...,h_c$ mapping characters to random hash values. A key $x=(x_1,...,x_c)$ is hashed to $h_1[x_1] \oplus h_2[x_2].....\oplus h_c[x_c]$. This scheme…
ShockHash: Towards Optimal-Space Minimal Perfect Hashing Beyond Brute-Force
ShockHash 利用布谷鸟哈希与伪森林图，将最小完美哈希的种子尝试次数从 e^n 降至 (e/2)^n，空间接近理论下界，比暴力搜索快近 2^n 倍。
Hash tables are one of the most fundamental data structures in computer science, in both theory and practice. They are especially useful in external memory, where their query performance approaches the ideal cost of just one disk access. Knuth gave an elegant analysis showing that with some simple collision resolution strategies such as linear probing or chaining, the expected average number of disk I/Os of a lookup is merely $1+1/2^{Ω(b)}$, where each I/O can read a disk block containing $b$ items. Inserting a new item into the hash table also costs $1+1/2^{Ω(b)}$ I/Os, which is again almost …
In this paper, the author proposes a series of multilevel double hashing schemes called cascade hash tables. They use several levels of hash tables. In each table, we use the common double hashing scheme. Higher level hash tables work as fail-safes of lower level hash tables. By this strategy, it could effectively reduce collisions in hash insertion. Thus it gains a constant worst case lookup time with a relatively high load factor(70%-85%) in random experiments. Different parameters of cascade hash tables are tested.
ShockHash: Near Optimal-Space Minimal Perfect Hashing Beyond Brute-Force
本文提出 ShockHash，利用高度过载布谷鸟哈希表与伪森林结构，以近最优空间和远低于暴力搜索的代价构造 MPHF，并在 RecSplit 等框架中实现显著加速。
While separate chaining is a common strategy for resolving collisions in a hash table taught in most textbooks, compact hashing is a less common technique for saving space when hashing integers whose domain is relatively small with respect to the problem size. It is widely believed that hash tables waste a considerable amount of memory, as they either leave allocated space untouched (open addressing) or store additional pointers (separate chaining). For the former, Cleary introduced the compact hashing technique that stores only a part of a key to save space. However, as can be seen by the lin…
First introduced in 1954, linear probing is one of the oldest data structures in computer science, and due to its unrivaled data locality, it continues to be one of the fastest hash tables in practice. It is widely believed and taught, however, that linear probing should never be used at high load factors; this is because primary-clustering effects cause insertions at load factor $1 - 1 /x$ to take expected time $Θ(x^2)$ (rather than the ideal $Θ(x)$). The dangers of primary clustering, first discovered by Knuth in 1963, have been taught to generations of computer scientists, and have influenc…
The power of two choices is a classic paradigm for load balancing when assigning $m$ balls to $n$ bins. When placing a ball, we pick two bins according to two hash functions $h_0$ and $h_1$, and place the ball in the least loaded bin. Assuming fully random hash functions, when $m=O(n)$, Azar et al.~[STOC'94] proved that the maximum load is $\lg \lg n + O(1)$ with high probability. In this paper, we investigate the power of two choices when the hash functions $h_0$ and $h_1$ are implemented with simple tabulation, which is a very efficient hash function evaluated in constant time. Following the…
Hash tables are a ubiquitous class of dictionary data structures. However, standard hash table implementations do not translate well into the external memory model, because they do not incorporate locality for insertions. Iacono and Patracsu established an update/query tradeoff curve for external hash tables: a hash table that performs insertions in $O(λ/B)$ amortized IOs requires $Ω(\log_λN)$ expected IOs for queries, where $N$ is the number of items that can be stored in the data structure, $B$ is the size of a memory transfer, $M$ is the size of memory, and $λ$ is a tuning parameter. They p…
PtrHash: Minimal Perfect Hashing at RAM Throughput
PtrHash通过优化内存和吞吐量，实现高效构建和查询的最小完美哈希函数，适用于大规模数据处理。
For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Theta(log n) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglog n) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(loglog n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for …
In multiple-choice data structures each element $x$ in a set $S$ of $m$ keys is associated with a random set $e(x) \subseteq [n]$ of buckets with capacity $\ell \geq 1$ by hash functions. This setting is captured by the hypergraph $H = ([n],\{e(x) \mid x \in S\})$. Accomodating each key in an associated bucket amounts to finding an $\ell$-orientation of $H$ assigning to each hyperedge an incident vertex such that each vertex is assigned at most $\ell$ hyperedges. If each subhypergraph of $H$ has minimum degree at most $\ell$, then an $\ell$-orientation can be found greedily and $H$ is called $…
One of the basic tasks in bioinformatics is localizing a short subsequence $S$, read while sequencing, in a long reference sequence $R$, like the human geneome. A natural rapid approach would be finding a hash value for $S$ and compare it with a prepared database of hash values for each of length $|S|$ subsequences of $R$. The problem with such approach is that it would only spot a perfect match, while in reality there are lots of small changes: substitutions, deletions and insertions. This issue could be repaired if having a hash function designed to tolerate some small distortion accordingly…
Tight Bounds for Classical Open Addressing
本研究提出rainbow hashing经典开放寻址哈希表，证明其时间-负载因子权衡最优，且扩展动态调整大小无额外时间成本，解决了此前三类操作同时达o(ε⁻¹)时间界的开放问题。
Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of $1.883$ when the table was full. Furthermore, by using a non-standard mean-centered search algorithm, this would imply that searches could be performed in expected constant time even in a full table. In spite of the time elapsed…
Suppose that we are to place $m$ balls into $n$ bins sequentially using the $d$-choice paradigm: For each ball we are given a choice of $d$ bins, according to $d$ hash functions $h_1,\dots,h_d$ and we place the ball in the least loaded of these bins breaking ties arbitrarily. Our interest is in the number of balls in the fullest bin after all $m$ balls have been placed. Azar et al. [STOC'94] proved that when $m=O(n)$ and when the hash functions are fully random the maximum load is at most $\frac{\lg \lg n }{\lg d}+O(1)$ whp (i.e. with probability $1-O(n^{-γ})$ for any choice of $γ$). In this p…
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