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Space Efficient Hash Tables With Worst Case Constant Access Time
 In STACS
, 2003
"... We generalize Cuckoo Hashing [23] to dary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ffl) n memory cells, for any constant ffl ? 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln ffl ) probes ..."
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Cited by 46 (4 self)
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We generalize Cuckoo Hashing [23] to dary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ffl) n memory cells, for any constant ffl ? 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln ffl ) probes and the expected amortized insertion time is constant. This is the first dictionary that has worst case constant access time and expected constant update time, works with (1 + ffl) n space, and supports satellite information. Experiments indicate that d = 4 choices suffice for ffl 0:03. We also describe variants of the data structure that allow the use of hash functions that can be evaluted in constant time.
External perfect hashing for very large key sets
 In Proceedings of the 16th ACM Conference on Information and Knowledge Management (CIKM’07
, 2007
"... A perfect hash function (PHF) h: S → [0, m − 1] for a key set S ⊆ U of size n, where m ≥ n and U is a key universe, is an injective function that maps the keys of S to unique values. A minimal perfect hash function (MPHF) is a PHF with m = n, the smallest possible range. Minimal perfect hash functio ..."
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Cited by 14 (2 self)
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A perfect hash function (PHF) h: S → [0, m − 1] for a key set S ⊆ U of size n, where m ≥ n and U is a key universe, is an injective function that maps the keys of S to unique values. A minimal perfect hash function (MPHF) is a PHF with m = n, the smallest possible range. Minimal perfect hash functions are widely used for memory efficient storage and fast retrieval of items from static sets. In this paper we present a distributed and parallel version of a simple, highly scalable and nearspace optimal perfect hashing algorithm for very large key sets, recently presented in [4]. The sequential implementation of the algorithm constructs a MPHF for a set of 1.024 billion URLs of average length 64 bytes collected from the Web in approximately 50 minutes using a commodity PC. The parallel implementation proposed here presents the following performance using 14 commodity PCs: (i) it constructs a MPHF for the same set of 1.024 billion URLs in approximately 4 minutes; (ii) it constructs a MPHF for a set of 14.336 billion 16byte random integers in approximately 50 minutes with a performance degradation of 20%; (iii) one version of the parallel algorithm distributes the description of the MPHF among the participating machines and its evaluation is done in a distributed way, faster than the centralized function.
Simple and spaceefficient minimal perfect hash functions
 In Proc. of the 10th Intl. Workshop on Data Structures and Algorithms
, 2007
"... Abstract. A perfect hash function (PHF) h: U → [0, m − 1] for a key set S is a function that maps the keys of S to unique values. The minimum amount of space to represent a PHF for a given set S is known to be approximately 1.44n 2 /m bits, where n = S. In this paper we present new algorithms for ..."
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Cited by 14 (7 self)
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Abstract. A perfect hash function (PHF) h: U → [0, m − 1] for a key set S is a function that maps the keys of S to unique values. The minimum amount of space to represent a PHF for a given set S is known to be approximately 1.44n 2 /m bits, where n = S. In this paper we present new algorithms for construction and evaluation of PHFs of a given set (for m = n and m = 1.23n), with the following properties: 1. Evaluation of a PHF requires constant time. 2. The algorithms are simple to describe and implement, and run in linear time. 3. The amount of space needed to represent the PHFs is around a factor 2 from the information theoretical minimum. No previously known algorithm has these properties. To our knowledge, any algorithm in the literature with the third property either: – Requires exponential time for construction and evaluation, or – Uses nearoptimal space only asymptotically, for extremely large n.
A practical minimal perfect hashing method
 In Proc. of the 4th International Workshop on Efficient and Experimental Algorithms (WEA’05
, 2005
"... Abstract. We propose a novel algorithm based on random graphs to construct minimal perfect hash functions h. For a set of n keys, our algorithm outputs h in expected time O(n). The evaluation of h(x) requires two memory accesses for any key x and the description of h takes up 1.15n words. This impro ..."
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Cited by 10 (6 self)
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Abstract. We propose a novel algorithm based on random graphs to construct minimal perfect hash functions h. For a set of n keys, our algorithm outputs h in expected time O(n). The evaluation of h(x) requires two memory accesses for any key x and the description of h takes up 1.15n words. This improves the space requirement to 55 % of a previous minimal perfect hashing scheme due to Czech, Havas and Majewski. A simple heuristic further reduces the space requirement to 0.93n words, at the expense of a slightly worse constant in the time complexity. Large scale experimental results are presented. 1
Practical perfect hashing in nearly optimal space
 Information Systems
"... A hash function is a mapping from a key universe U to a range of integers, i.e., h: U↦→{0, 1,...,m−1}, where m is the range’s size. A perfect hash function for some set S ⊆ U is a hash function that is onetoone on S, where m≥S. A minimal perfect hash function for some set S ⊆ U is a perfect hash ..."
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Cited by 2 (1 self)
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A hash function is a mapping from a key universe U to a range of integers, i.e., h: U↦→{0, 1,...,m−1}, where m is the range’s size. A perfect hash function for some set S ⊆ U is a hash function that is onetoone on S, where m≥S. A minimal perfect hash function for some set S ⊆ U is a perfect hash function with a range of minimum size, i.e., m=S. This paper presents a construction for (minimal) perfect hash functions that combines theoretical analysis, practical performance, expected linear construction time and nearly optimal space consumption for the data structure. For n keys and m=n the space consumption ranges from 2.62n to 3.3n bits, and for m=1.23n it ranges from 1.95n to 2.7n bits. This is within a small constant factor from the theoretical lower bounds of 1.44n bits for m=n and 0.89n bits for m=1.23n. We combine several theoretical results into a practical solution that has turned perfect hashing into a very compact data structure to solve the membership problem when the key set S is static and known in advance. By taking into account the memory hierarchy we can construct (minimal) perfect hash functions for over a billion keys in 46 minutes using a commodity PC. An open source implementation of the algorithms is available
Maintaining External Memory Efficient Hash Tables
"... Abstract. In typical applications of hashing algorithms the amount of data to be stored is often too large to fit into internal memory. In this case it is desirable to find the data with as few as possible nonconsecutive or at least nonoblivious probes into external memory. Extending a static sche ..."
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Abstract. In typical applications of hashing algorithms the amount of data to be stored is often too large to fit into internal memory. In this case it is desirable to find the data with as few as possible nonconsecutive or at least nonoblivious probes into external memory. Extending a static scheme of Pagh [11] we obtain new randomized algorithms for maintaining hash tables, where a hash function can be evaluated in constant time and by probing only one external memory cell or O(1) consecutive external memory cells. We describe a dynamic version of Pagh’s hashing scheme achieving 100 % table utilization but requiring (2 + ɛ) · n log n space for the hash function encoding as well as (3 + ɛ) · n log n space for the auxiliary data structure. Update operations are possible in expected constant amortized time. Then we show how to reduce the space for the hash function encoding and the auxiliary data structure to O(n log log n). We achieve 100 % utilization in the static version (and thus a minimal perfect hash function) and 1 − ɛ utilization in the dynamic case. 1
ABSTRACT An Approach for Minimal Perfect Hash Functions for Very Large Databases
"... We propose a novel external memory based algorithm for constructing minimal perfect hash functions h for huge sets of keys. For a set of n keys, our algorithm outputs h in time O(n). The algorithm needs a small vector of one byte entries in main memory to construct h. The evaluation of h(x) requires ..."
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We propose a novel external memory based algorithm for constructing minimal perfect hash functions h for huge sets of keys. For a set of n keys, our algorithm outputs h in time O(n). The algorithm needs a small vector of one byte entries in main memory to construct h. The evaluation of h(x) requires three memory accesses for any key x. The description of h takes a constant number of up to 9 bits for each key, which is optimal and close to the theoretical lower bound, i.e., around 2 bits per key. In our experiments, we used a collection of 1 billion URLs collected from the web, each URL 64 characters long on average. For this collection, our algorithm (i) finds a minimal perfect hash function in approximately 3 hours using a commodity PC, (ii) needs just 5.45 megabytes of internal memory to generate h and (iii) takes 8.1 bits per key for the description of h.