In computer science, a family of hash functions is said to be
Contents
Background
The goal of hashing is usually to map keys from some large domain (universe)
The solution to these problems is to pick a function randomly from a large family of hash functions. The randomness in choosing the hash function can be used to guarantee some desired random behavior of the hash codes of any keys of interest. The first definition along these lines was universal hashing, which guarantees a low collision probability for any two designated keys. The concept of
Definitions
The strictest definition, introduced by Wegman and Carter under the name "strongly universal
This definition is equivalent to the following two conditions:
- for any fixed
x ∈ U , ash is drawn randomly fromH ,h ( x ) is uniformly distributed in[ m ] . - for any fixed, distinct keys
x 1 , … , x k ∈ U , ash is drawn randomly fromH ,h ( x 1 ) , … , h ( x k ) are independent random variables.
Often it is inconvenient to achieve the perfect joint probability of
Observe that, even if
Polynomials with random coefficients
The original technique for constructing k-independent hash functions, given by Carter and Wegman, was to select a large prime number p, choose k random numbers modulo p, and use these numbers as the coefficients of a polynomial of degree k whose values modulo p are used as the value of the hash function. All polynomials of the given degree modulo p are equally likely, and any polynomial is uniquely determined by any k-tuple of argument-value pairs with distinct arguments, from which it follows that any k-tuple of distinct arguments is equally likely to be mapped to any k-tuple of hash values.
Tabulation hashing
Tabulation hashing is a technique for mapping keys to hash values by partitioning each key into bytes, using each byte as the index into a table of random numbers (with a different table for each byte position), and combining the results of these table lookups by a bitwise exclusive or operation. Thus, it requires more randomness in its initialization than the polynomial method, but avoids possibly-slow multiplication operations. It is 3-independent but not 4-independent. Variations of tabulation hashing can achieve higher degrees of independence by performing table lookups based on overlapping combinations of bits from the input key, or by applying simple tabulation hashing iteratively.
Independence needed by different hashing methods
The notion of k-independence can be used to differentiate between different hashing methods, according to the level of independence required to guarantee constant expected time per operation.
For instance, hash chaining takes constant expected time even with a 2-independent hash function, because the expected time to perform a search for a given key is bounded by the expected number of collisions that key is involved in. By linearity of expectation, this expected number equals the sum, over all other keys in the hash table, of the probability that the given key and the other key collide. Because the terms of this sum only involve probabilistic events involving two keys, 2-independence is sufficient to ensure that this sum has the same value that it would for a truly random hash function.
Double hashing is another method of hashing that requires a low degree of independence. It is a form of open addressing that uses two hash functions: one to determine the start of a probe sequence, and the other to determine the step size between positions in the probe sequence. As long as both of these are 2-independent, this method gives constant expected time per operation.
On the other hand, linear probing, a simpler form of open addressing where the step size is always one, requires 5-independence. It can be guaranteed to work in constant expected time per operation with a 5-independent hash function, and there exist 4-independent hash functions for which it takes logarithmic time per operation.