A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support variable length output.[1] The values returned by a hash function are called hash values, hash codes, hash digests, digests, or simply hashes.[2] The values are usually used to index a fixed-size table called a hash table. Use of a hash function to index a hash table is called hashing or scatter storage addressing.
Hash functions and their associated hash tables are used in data storage and retrieval applications to access data in a small and nearly constant time per retrieval. They require an amount of storage space only fractionally greater than the total space required for the data or records themselves. Hashing is a computationally and storage space-efficient form of data access that avoids the non-constant access time of ordered and unordered lists and structured trees, and the often exponential storage requirements of direct access of state spaces of large or variable-length keys.
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Hash functions are related to (and often confused with) checksums, check digits, fingerprints, lossy compression, randomization functions, error-correcting codes, and ciphers. Although the concepts overlap to some extent, each one has its own uses and requirements and is designed and optimized differently. The hash function differs from these concepts mainly in terms of data integrity.
A hash function takes a key as an input, which is associated with a datum or record and used to identify it to the data storage and retrieval application. The keys may be fixed length, like an integer, or variable length, like a name. In some cases, the key is the datum itself. The output is a hash code used to index a hash table holding the data or records, or pointers to them.
A good hash function satisfies two basic properties: 1) it should be very fast to compute; 2) it should minimize duplication of output values (collisions). Hash functions rely on generating favourable probability distributions for their effectiveness, reducing access time to nearly constant. High table loading factors, pathological key sets and poorly designed hash functions can result in access times approaching linear in the number of items in the table. Hash functions can be designed to give the best worst-case performance,[Notes 1] good performance under high table loading factors, and in special cases, perfect (collisionless) mapping of keys into hash codes. Implementation is based on parity-preserving bit operations (XOR and ADD), multiply, or divide. A necessary adjunct to the hash function is a collision-resolution method that employs an auxiliary data structure like linked lists, or systematic probing of the table to find an empty slot.
Hash functions are used in conjunction with hash tables to store and retrieve data items or data records. The hash function translates the key associated with each datum or record into a hash code, which is used to index the hash table. When an item is to be added to the table, the hash code may index an empty slot (also called a bucket), in which case the item is added to the table there. If the hash code indexes a full slot, some kind of collision resolution is required: the new item may be omitted (not added to the table), or replace the old item, or it can be added to the table in some other location by a specified procedure. That procedure depends on the structure of the hash table: In chained hashing, each slot is the head of a linked list or chain, and items that collide at the slot are added to the chain. Chains may be kept in random order and searched linearly, or in serial order, or as a self-ordering list by frequency to speed up access. In open address hashing, the table is probed starting from the occupied slot in a specified manner, usually by linear probing, quadratic probing, or double hashing until an open slot is located or the entire table is probed (overflow). Searching for the item follows the same procedure until the item is located, an open slot is found or the entire table has been searched (item not in table).
Hash functions are also used to build caches for large data sets stored in slow media. A cache is generally simpler than a hashed search table since any collision can be resolved by discarding or writing back the older of the two colliding items.[4]
A special case of hashing is known as geometric hashing or the grid method. In these applications, the set of all inputs is some sort of metric space, and the hashing function can be interpreted as a partition of that space into a grid of cells. The table is often an array with two or more indices (called a grid file, grid index, bucket grid, and similar names), and the hash function returns an index tuple. This principle is widely used in computer graphics, computational geometry and many other disciplines, to solve many proximity problems in the plane or in three-dimensional space, such as finding closest pairs in a set of points, similar shapes in a list of shapes, similar images in an image database, and so on.
This criterion only requires the value to be uniformly distributed, not random in any sense. A good randomizing function is (barring computational efficiency concerns) generally a good choice as a hash function, but the converse need not be true.
In special cases when the keys are known in advance and the key set is static, a hash function can be found that achieves absolute (or collisionless) uniformity. Such a hash function is said to be perfect. There is no algorithmic way of constructing such a function - searching for one is a factorial function of the number of keys to be mapped versus the number of table slots they're tapped into. Finding a perfect hash function over more than a very small set of keys is usually computationally infeasible; the resulting function is likely to be more computationally complex than a standard hash function and provides only a marginal advantage over a function with good statistical properties that yields a minimum number of collisions. See universal hash function.
Hash functions can have some technical properties that make it more likely that they'll have a uniform distribution when applied. One is the strict avalanche criterion: whenever a single input bit is complemented, each of the output bits changes with a 50% probability. The reason for this property is that selected subsets of the keyspace may have low variability. For the output to be uniformly distributed, a low amount of variability, even one bit, should translate into a high amount of variability (i.e. distribution over the tablespace) in the output. Each bit should change with a probability of 50% because if some bits are reluctant to change, the keys become clustered around those values. If the bits want to change too readily, the mapping is approaching a fixed XOR function of a single bit. Standard tests for this property have been described in the literature.[6] The relevance of the criterion to a multiplicative hash function is assessed here.[7]
In data storage and retrieval applications, the use of a hash function is a trade-off between search time and data storage space. If search time were unbounded, a very compact unordered linear list would be the best medium; if storage space were unbounded, a randomly accessible structure indexable by the key-value would be very large, very sparse, but very fast. A hash function takes a finite amount of time to map a potentially large keyspace to a feasible amount of storage space searchable in a bounded amount of time regardless of the number of keys. In most applications, the hash function should be computable with minimum latency and secondarily in a minimum number of instructions.
Because collisions should be infrequent, and cause a marginal delay but are otherwise harmless, it's usually preferable to choose a faster hash function over one that needs more computation but saves a few collisions.
If keys are being hashed repeatedly, and the hash function is costly, computing time can be saved by precomputing the hash codes and storing them with the keys. Matching hash codes almost certainly means the keys are identical. This technique is used for the transposition table in game-playing programs, which stores a 64-bit hashed representation of the board position.
The determinism is in the context of the reuse of the function. For example, Python adds the feature that hash functions make use of a randomized seed that is generated once when the Python process starts in addition to the input to be hashed.[9] The Python hash (SipHash) is still a valid hash function when used within a single run. But if the values are persisted (for example, written to disk) they can no longer be treated as valid hash values, since in the next run the random value might differ.
It is often desirable that the output of a hash function have fixed size (but see below). If, for example, the output is constrained to 32-bit integer values, the hash values can be used to index into an array. Such hashing is commonly used to accelerate data searches.[10] Producing fixed-length output from variable length input can be accomplished by breaking the input data into chunks of specific size. Hash functions used for data searches use some arithmetic expression that iteratively processes chunks of the input (such as the characters in a string) to produce the hash value.[10]
When the hash function is used to store values in a hash table that outlives the run of the program, and the hash table needs to be expanded or shrunk, the hash table is referred to as a dynamic hash table.
Linear hashing and spiral hashing are examples of dynamic hash functions that execute in constant time but relax the property of uniformity to achieve the minimal movement property. Extendible hashing uses a dynamic hash function that requires space proportional to n to compute the hash function, and it becomes a function of the previous keys that have been inserted. Several algorithms that preserve the uniformity property but require time proportional to n to compute the value of H(z,n) have been invented.[clarification needed] 006ab0faaa
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