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Hash Table

In computer science, a hash table or hash map is a data structure that uses a hash function to efficiently map certain identifiers or keys (e.g., person names) to associated values (e.g., their telephone numbers). The hash function is used to transform the key into the index (the hash) of an array element (the slot or bucket) where the corresponding value is to be sought.

Ideally the hash function should map each possible key to a different slot index, but this ideal is rarely achievable in practice. Most hash table designs assume that hash collisions — pairs of different keys with the same hash values — are normal occurrences and must be accommodated in some way.

In a well-dimensioned hash table, the average cost (number of instructions) for each lookup is independent of the number of elements stored in the table. Many hash table designs also allow arbitrary insertions and deletions of key-value pairs, at constant average (indeed,amortized) cost per operation.

In many situations, hash tables turn out to be more efficient than search trees or any other table lookup structure. For this reason, they are widely used in many kinds of computer software, particularly for associative arrays, database indexing, caches and sets.

Hash tables should not be confused with the hash lists and hash trees used in cryptography and data transmission.
 
A small phone book as a hash table.
 
Hash function
 
At the heart of the hash table algorithm is a simple array of items; this is often simply called the hash table. Hash table algorithms calculate an index from the data item's key and use this index to place the data into the array. The implementation of this calculation is the hash function, f: index = f(key,arrayLength)
The hash function calculates an index within the array from the data key. arrayLength is the size of the array.
 
Choosing a good hash function
A good hash function and implementation algorithm are essential for good hash table performance, but may be difficult to achieve. Poor hashing usually degrades hash table performance by a constant factor but hashing is often only a small part of the overall computation.
A basic requirement is that the function should provide a uniform distribution of hash values. A non-uniform distribution increases the number of collisions, and the cost of resolving them. Uniformity is sometimes difficult to ensure by design, but may be evaluated empirically by the chi-squared test.
The distribution needs to be uniform only for the table sizes s that will occur in the application. In particular, if one uses dynamic resizing with exact doubling and halving of s, the hash function needs to be uniform only when s is a power of two. On the other hand, some hashing algorithms provide uniform hashes only when s is a prime number.

For open addressing schemes, the hash function should also avoid clustering, the mapping of two or more keys to consecutive slots. Such clustering may cause the lookup cost to skyrocket, even if the load factor is low and collisions are infrequent. The very popular multiplicative hash advocated by Donald Knuth in The Art of Computer Programming is claimed to have particularly poor clustering behavior.

Cryptographic hash functions are believed to provide good hash functions for any table size s, either by modulo reduction or by bit masking. They may also be appropriate if there is a risk of malicious users trying to sabotage a network service by submitting requests designed to generate a large number of collisions in the server's hash tables. However, these presumed qualities are hardly worth their much larger computational cost and algorithmic complexity, and the risk of sabotage can be avoided by cheaper methods (such as applying a secret salt to the data, or using a universal hash function).

Some authors claim that good hash functions should have the avalanche effect; that is, a single-bit change in the input key should affect, on average, half the bits in the output. Some popular hash functions fail to pass this test.
 
Perfect hash function

If all of the keys that will be used are known ahead of time, a perfect hash function can be used to create a perfect hash table, in which there will be no collisions. If minimal perfect hashing is used, every location in the hash table can be used as well.

Perfect hashing allows for constant time lookups in the worst case. This is in contrast to most chaining and open addressing methods, where the time for lookup is low on average, but may be very large (proportional to the number of entries) for some sets of keys.

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