On the GPU, hash table operation speed is determined in large part by cache line efficiency, and state-of-the-art hashing schemes thus divide tables into cache line-sized buckets. This raises the question whether performance can be further improved by increasing the number of entries that fit in such buckets. Known compact hashing techniques have not yet been adapted to the massively parallel setting, nor have they been evaluated on the GPU. We consider a compact version of bucketed cuckoo hashing, and a version of compact iceberg hashing suitable for the GPU. We discuss the tables from a theoretical perspective, and provide an open source implementation of both schemes in CUDA for comparative benchmarking. In terms of performance, the state-of-the-art cuckoo hashing benefits from compactness on lookups and insertions (most experiments show at least 10-20% increase in throughput), and the iceberg table benefits significantly, to the point of being comparable to compact cuckoo hashing--while supporting performant dynamic operation.
We revisit the problem of building static hash tables on the GPU and design and build three bucketed hash tables that use different probing schemes. Our implementations are lock-free and offer efficient memory access patterns; thus, only the probing scheme is the factor affecting the performance of the hash table's different operations. Our results show that a bucketed cuckoo hash table that uses three hash functions (BCHT) outperforms alternative methods that use power-of-two choices, iceberg hashing, and a cuckoo hash table that uses a bucket size one. At high load factors as high as 0.99, B…
Deletion from open-address hash table is not so easy as deletion from chained hash table, because in open-address table we can't simply mark a slot containing deleted key as empty. Search for keys may become incorrect. The classical method to implement deletion is to mark slots in hash table by three values: "free", "busy", "deleted". That method is easy to implement, but there are some disadvantages. In this article we consider alternative method of deletion keys, where we avoid using the mark "deleted". The article contains the implementation of the method in Java.
This paper considers the basic question of how strong of a probabilistic guarantee can a hash table, storing $n$ $(1 + Θ(1)) \log n$-bit key/value pairs, offer? Past work on this question has been bottlenecked by limitations of the known families of hash functions: The only hash tables to achieve failure probabilities less than $1 / 2^{\polylog n}$ require access to fully-random hash functions -- if the same hash tables are implemented using the known explicit families of hash functions, their failure probabilities become $1 / \poly(n)$. To get around these obstacles, we show how to construct …
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…
Concurrent hash tables are one of the most important concurrent data structures with numerous applications. Since hash table accesses can dominate the execution time of the overall application, we need implementations that achieve good speedup. Unfortunately, currently available concurrent hashing libraries turn out to be far away from this requirement in particular when contention on some elements occurs. Our starting point for better performing data structures is a fast and simple lock-free concurrent hash table based on linear probing that is limited to word-sized key-value types and does n…
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…
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…
This paper proposes round-hashing, which is suitable for data storage on distributed servers and for implementing external-memory tables in which each lookup retrieves at most a single block of external memory, using a stash. For data storage, round-hashing is like consistent hashing as it avoids a full rehashing of the keys when new servers are added. Experiments show that the speed to serve requests is tenfold or more than the state of the art. In distributed data storage, this guarantees better throughput for serving requests and, moreover, greatly reduces decision times for which data shou…
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.
In this article we discuss a data structure, which combines advantages of two different ways for representing graphs: adjacency matrix and collection of adjacency lists. This data structure can fast add and search edges (advantages of adjacency matrix), use linear amount of memory, let to obtain adjacency list for certain vertex (advantages of collection of adjacency lists). Basic knowledge of linked lists and hash tables is required to understand this article. The article contains examples of implementation on Java.
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 …
This article presents the Quotient Hash Table (QHT) a new data structure for duplicate detection in unbounded streams. QHTs stem from a corrected analysis of streaming quotient filters (SQFs), resulting in a 33\% reduction in memory usage for equal performance. We provide a new and thorough analysis of both algorithms, with results of interest to other existing constructions. We also introduce an optimised version of our new data structure dubbed Queued QHT with Duplicates (QQHTD). Finally we discuss the effect of adversarial inputs for hash-based duplicate filters similar to QHT.
We introduce PaCHash, a hash table that stores its objects contiguously in an array without intervening space, even if the objects have variable size. In particular, each object can be compressed using standard compression techniques. A small search data structure allows locating the objects in constant expected time. PaCHash is most naturally described as a static external hash table where it needs a constant number of bits of internal memory per block of external memory. Here, in some sense, PaCHash beats a lower bound on the space consumption of k-perfect hashing. An implementation for fast…
We describe a consistent hashing algorithm which performs multiple lookups per key in a hash table of nodes. It requires no additional storage beyond the hash table, and achieves a peak-to-average load ratio of 1 + epsilon with just 1 + 1/epsilon lookups per key.
We describe a variant of linear probing hash tables that never moves elements and thus supports referential integrity, i.e., pointers to elements remain valid while this element is in the hash table. This is achieved by the folklore method of marking some table entries as formerly occupied (tombstones). The innovation is that the number of tombstones is minimized. Experiments indicate that this allows an unbounded number of operations with bounded overhead compared to linear probing without tombstones (and without referential integrity).
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…
In this work, attempt is made to select three good hash functions which uniformly distribute hash values that permute their internal states and allow the input bits to generate different output bits. These functions are used in different levels of hash tables that are coded in Java Programming Language and a quite number of data records serve as primary data for testing the performances. The result shows that the two-level hash tables with three different hash functions give a superior performance over one-level hash table with two hash functions or zero-level hash table with one function in t…
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 …
Hash tables are an essential data-structure for numerous networking applications (e.g., connection tracking, firewalls, network address translators). Among these, cuckoo hash tables provide excellent performance by allowing lookups to be processed with very few memory accesses (2 to 3 per lookup). Yet, for large tables, cuckoo hash tables remain memory bound and each memory access impacts performance. In this paper, we propose algorithmic improvements to cuckoo hash tables allowing to eliminate some unnecessary memory accesses; these changes are conducted without altering the properties of the…
Hash tables are one of the most fundamental data structures for effectively storing and accessing sparse data, with widespread usage in domains ranging from computer graphics to machine learning. This study surveys the state-of-the-art research on data-parallel hashing techniques for emerging massively-parallel, many-core GPU architectures. Key factors affecting the performance of different hashing schemes are discovered and used to suggest best practices and pinpoint areas for further research.
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…
Hash tables are ubiquitous and used in a wide range of applications for efficient probing of large and unsorted data. If designed properly, hash-tables can enable efficients look ups in a constant number of operations or commonly referred to as O(1) operations. As data sizes continue to grow and data becomes less structured (as is common for big-data applications), the need for efficient and scalable hash table also grows. In this paper we introduce HashGraph, a new scalable approach for building hash tables that uses concepts taken from sparse graph representations--hence the name HashGraph. …
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 >= …
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 …
Modern hash table designs strive to minimize space while maximizing speed. The most important factor in speed is the number of cache lines accessed during updates and queries. This is especially important on PMEM, which is slower than DRAM and in which writes are more expensive than reads. This paper proposes two stronger design objectives: stability and low-associativity. A stable hash table doesn't move items around, and a hash table has low associativity if there are only a few locations where an item can be stored. Low associativity ensures that queries need to examine only a few memory lo…
Most cloud services and distributed applications rely on hashing algorithms that allow dynamic scaling of a robust and efficient hash table. Examples include AWS, Google Cloud and BitTorrent. Consistent and rendezvous hashing are algorithms that minimize key remapping as the hash table resizes. While memory errors in large-scale cloud deployments are common, neither algorithm offers both efficiency and robustness. Hyperdimensional Computing is an emerging computational model that has inherent efficiency, robustness and is well suited for vector or hardware acceleration. We propose Hyperdimensi…
Hash tables are used in a plethora of applications, including database operations, DNA sequencing, string searching, and many more. As such, there are many parallelized hash tables targeting multicore, distributed, and accelerator-based systems. We present in this work a multi-GPU hash table implementation that can process keys at a throughput comparable to that of distributed hash tables. Distributed CPU hash tables have received significantly more attention than GPU-based hash tables. We show that a single node with multiple GPUs offers roughly the same performance as a 500-1,000-core CPU-ba…
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…
We extend a result of Goldreich and Ron about estimating the collision probability of a hash function. Their estimate has a polynomial tail. We prove that when the load factor is greater than a certain constant, the estimator has a gaussian tail. As an application we find an estimate of an upper bound for the average search time in hashing with chaining, for a particular user (we allow the overall key distribution to be different from the key distribution of a particular user). The estimator has a gaussian tail.
With double hashing, for a key $x$, one generates two hash values $f(x)$ and $g(x)$, and then uses combinations $(f(x) +i g(x)) \bmod n$ 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 coupli…
This paper describes an approach for obtaining direct access to the attacked squares of sliding pieces without resorting to rotated bitboards. The technique involves creating four hash tables using the built in hash arrays from an interpreted, high level language. The rank, file, and diagonal occupancy are first isolated by masking the desired portion of the board. The attacked squares are then directly retrieved from the hash tables. Maintaining incrementally updated rotated bitboards becomes unnecessary as does all the updating, mapping and shifting required to access the attacked squares. F…
In recent years, information retrieval algorithms have taken center stage for extracting important data in ever larger datasets. Advances in hardware technology have lead to the increasingly wide spread use of flash storage devices. Such devices have clear benefits over traditional hard drives in terms of latency of access, bandwidth and random access capabilities particularly when reading data. There are however some interesting trade-offs to consider when leveraging the advanced features of such devices. On a relative scale writing to such devices can be expensive. This is because typical fl…
We present an efficient lock-free algorithm for parallel accessible hash tables with open addressing, which promises more robust performance and reliability than conventional lock-based implementations. ``Lock-free'' means that it is guaranteed that always at least one process completes its operation within a bounded number of steps. For a single processor architecture our solution is as efficient as sequential hash tables. On a multiprocessor architecture this is also the case when all processors have comparable speeds. The algorithm allows processors that have widely different speeds or come…
Department of Computer Science, University of Southern California（南加州大学计算机科学系）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
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…
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…
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…
We consider the hash function $h(x) = ((ax+b) \bmod p) \bmod n$ where $a,b$ are chosen uniformly at random from $\{0,1,\ldots,p-1\}$. We prove that when we use $h(x)$ in hashing with chaining to insert $n$ elements into a table of size $n$ the expected length of the longest chain is $\tilde{O}\!\left(n^{1/3}\right)$. The proof also generalises to give the same bound when we use the multiply-shift hash function by Dietzfelbinger et al. [Journal of Algorithms 1997].
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.
Consider an $m$-bit query $q$ to a bitwise trie $T$. A wildcard $*$ is an unspecified bit in $q$ for which the query asks the membership for both cases $*=0$ and $*=1$. It is common that such partial-match queries with wildcards are issued in tries. With uniformly random occurrences of $w$ wildcards in $q$ assumed, the obvious upper bound on the average number of traversal steps in $T$ is $2^w m$. We show that the average does not exceed \[ \frac{m+1}{w+1} \left( 2^{w+2} - 2 w - 4 \right) + m = O \left( \frac{2^w m}{w} \right), \] and equals the value exactly when $T$ includes all the $m$-bit …
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…
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…
In this paper, we analyze hashing from a worst-case perspective. To this end, we study a new property of hash families that is strongly related to d-perfect hashing, namely c-ideality. On the one hand, this notion generalizes the definition of perfect hashing, which has been studied extensively; on the other hand, it provides a direct link to the notion of c-approximativity. We focus on the usually neglected case where the average load αis at least 1 and prove upper and lower parametrized bounds on the minimal size of c-ideal hash families. As an aside, we show how c-ideality helps to analyze …
An important question that arises in the study of high dimensional vector representations learned from data is: given a set $\mathcal{D}$ of vectors and a query $q$, estimate the number of points within a specified distance threshold of $q$. We develop two estimators, LSH Count and Multi-Probe Count that use locality sensitive hashing to preprocess the data to accurately and efficiently estimate the answers to such questions via importance sampling. A key innovation is the ability to maintain a small number of hash tables via preprocessing data structures and algorithms that sample from multip…
