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Linear probing with constant independence
 In STOC ’07: Proceedings of the thirtyninth annual ACM symposium on Theory of computing
, 2007
"... Hashing with linear probing dates back to the 1950s, and is among the most studied algorithms. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analyses rely either on complicated and space ..."
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Cited by 19 (2 self)
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Hashing with linear probing dates back to the 1950s, and is among the most studied algorithms. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analyses rely either on complicated and space consuming hash functions, or on the unrealistic assumption of free access to a truly random hash function. Already Carter and Wegman, in their seminal paper on universal hashing, raised the question of extending their analysis to linear probing. However, we show in this paper that linear probing using a pairwise independent family may have expected logarithmic cost per operation. On the positive side, we show that 5wise independence is enough to ensure constant expected time per operation. This resolves the question of finding a space and time efficient hash function that provably ensures good performance for linear probing.
On the kindependence required by linear probing and minwise independence
 In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP
, 2010
"... )independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for exp ..."
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)independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for expected constanttime performance, matching an upper bound of [Pagh et al. STOC’07]. For (1 + ε)approximate minwise independence, we show that Ω(lg 1 ε 1
String hashing for linear probing
 In Proc. 20th SODA
, 2009
"... Linear probing is one of the most popular implementations of dynamic hash tables storing all keys in a single array. When we get a key, we first hash it to a location. Next we probe consecutive locations until the key or an empty location is found. At STOC’07, Pagh et al. presented data sets where t ..."
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Linear probing is one of the most popular implementations of dynamic hash tables storing all keys in a single array. When we get a key, we first hash it to a location. Next we probe consecutive locations until the key or an empty location is found. At STOC’07, Pagh et al. presented data sets where the standard implementation of 2universal hashing leads to an expected number of Ω(log n) probes. They also showed that with 5universal hashing, the expected number of probes is constant. Unfortunately, we do not have 5universal hashing for, say, variable length strings. When we want to do such complex hashing from a complex domain, the generic standard solution is that we first do collision free hashing (w.h.p.) into a simpler intermediate domain, and second do the complicated hash function on this intermediate domain. Our contribution is that for an expected constant number of linear probes, it is suffices that each key has O(1) expected collisions with the first hash function, as long as the second hash function is 5universal. This means that the intermediate domain can be n times smaller, and such a smaller intermediate domain typically means that the overall hash function can be made simpler and at least twice as fast. The same doubling of hashing speed for O(1) expected probes follows for most domains bigger than 32bit integers, e.g., 64bit integers and fixed length strings. In addition, we study how the overhead from linear probing diminishes as the array gets larger, and what happens if strings are stored directly as intervals of the array. These cases were not considered by Pagh et al. 1
Double Hashing is Computable and Randomizable with Universal Hash Functions
"... Universal hash functions that exhibit c log nwise independence are shown to give a performance in double hashing and virtually any reasonable generalization of double hashing that has an expected probe count of 1/(1alpha) + epsilon for the insertion of the alpha nth item into a table of size n, f ..."
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Cited by 3 (1 self)
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Universal hash functions that exhibit c log nwise independence are shown to give a performance in double hashing and virtually any reasonable generalization of double hashing that has an expected probe count of 1/(1alpha) + epsilon for the insertion of the alpha nth item into a table of size n, for any fixed alpha 0. This performance is within epsilon of optimal. These results are derived from a novel formulation that overestimates the expected probe count by underestimating the presence of partial items already inserted into the hash table, and from a sharp analysis of the underlying stochastic structures formed by colliding items.
LINEAR PROBING WITH 5WISE INDEPENDENCE ∗
"... Abstract. Hashing with linear probing dates back to the 1950s, and is among the most studied algorithms for storing (key,value) pairs. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analy ..."
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Abstract. Hashing with linear probing dates back to the 1950s, and is among the most studied algorithms for storing (key,value) pairs. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analyses rely either on complicated and space consuming hash functions, or on the unrealistic assumption of free access to a hash function with random and independent function values. Carter and Wegman, in their seminal paper on universal hashing, raised the question of extending their analysis to linear probing. However, we show in this paper that linear probing using a 2wise independent hash function may have expected logarithmic cost per operation. Recently, Pǎtra¸scu and Thorup have shown that also 3 and 4wise independent hash functions may give rise to logarithmic expected query time. On the positive side, we show that 5wise independence is enough to ensure constant expected time per operation. This resolves the question of finding a space and time efficient hash function that provably ensures good performance for hashing with linear probing.