thumb|upright=1.35|The collision between John Smith and Sandra Dee (both hashing to cell 873) is resolved by placing Sandra Dee at the next free location, cell 874.

Linear probing is a scheme in computer programming for resolving collisions in hash tables, data structures for maintaining a collection of key–value pairs and looking up the value associated with a given key. It was invented in 1954 by Gene Amdahl, Elaine M. McGraw, and Arthur Samuel (and, independently, by Andrey Yershov) and first analyzed in 1963 by Donald Knuth.

Along with quadratic probing and double hashing, linear probing is a form of open addressing. In these schemes, each cell of a hash table stores a single key–value pair. When the hash function causes a collision by mapping a new key to a cell of the hash table that is already occupied by another key, linear probing searches the table for the closest following free location and inserts the new key there. Lookups are performed in the same way, by searching the table sequentially starting at the position given by the hash function, until finding a cell with a matching key or an empty cell.

As write, "Hash tables are the most commonly used nontrivial data structures, and the most popular implementation on standard hardware uses linear probing, which is both fast and simple."

To search for a given key , the cells of are examined, beginning with the cell at index (where is the hash function) and continuing to the adjacent cells , , ..., until finding either an empty cell or a cell whose stored key is .

If a cell containing the key is found, the search returns the value from that cell. Otherwise, if an empty cell is found, the key cannot be in the table, because it would have been placed in that cell in preference to any later cell that has not yet been searched. In this case, the search returns as its result that the key is not present in the dictionary.

Deletion

thumb|upright=2.5|When a key–value pair is deleted, it may be necessary to move another pair backwards into its cell, to prevent searches for the moved key from finding an empty cell.

It is also possible to remove a key–value pair from the dictionary. However, it is not sufficient to do so by simply emptying its cell. This would affect searches for other keys that have a hash value earlier than the emptied cell, but that are stored in a position later than the emptied cell. The emptied cell would cause those searches to incorrectly report that the key is not present.

Instead, when a cell is emptied, it is necessary to search forward through the following cells of the table until finding either another empty cell or a key that can be moved to cell (that is, a key whose hash value is equal to or earlier than ). When an empty cell is found, then emptying cell is safe and the deletion process terminates. But, when the search finds a key that can be moved to cell , it performs this move. This has the effect of speeding up later searches for the moved key, but it also empties out another cell, later in the same block of occupied cells. The search for a movable key continues for the new emptied cell, in the same way, until it terminates by reaching a cell that was already empty. In this process of moving keys to earlier cells, each key is examined only once. Therefore, the time to complete the whole process is proportional to the length of the block of occupied cells containing the deleted key, matching the running time of the other hash table operations.

Analysis

Using linear probing, dictionary operations can be implemented in constant expected time. In other words, insert, remove and search operations can be implemented in O(1), as long as the load factor of the hash table is a constant strictly less than one.

In more detail, the time for any particular operation (a search, insertion, or deletion) is proportional to the length of the contiguous block of occupied cells at which the operation starts. If all starting cells are equally likely, in a hash table with cells, then a maximal block of occupied cells will have probability of containing the starting location of a search, and will take time whenever it is the starting location. Therefore, the expected time for an operation can be calculated as the product of these two terms, , summed over all of the maximal blocks of contiguous cells in the table. A similar sum of squared block lengths gives the expected time bound for a random hash function (rather than for a random starting location into a specific state of the hash table), by summing over all the blocks that could exist (rather than the ones that actually exist in a given state of the table), and multiplying the term for each potential block by the probability that the block is actually occupied. That is,

defining to be the event that there is a maximal contiguous block of occupied cells of length beginning at index , the expected time per operation is

:<math>E[T] = O(1) + \sum_{i=1}^N \sum_{k=1}^n O(k^2/N) \operatorname{Pr}[\operatorname{Block}(i,k)].</math>

This formula can be simplified by replacing by a simpler necessary condition , the event that

at least elements have hash values that lie within a block of cells of length . After this replacement, the value within the sum no longer depends on , and the factor cancels the terms of the outer summation. These simplifications lead to the bound

:<math>E[T] \le O(1) + \sum_{k=1}^n O(k^2) \operatorname{Pr}[\operatorname{Full}(k)].</math>

But by the multiplicative form of the Chernoff bound, when the load factor is bounded away from one, the probability that a block of length contains at least hashed values is exponentially small as a function of ,

causing this sum to be bounded by a constant independent of&nbsp;. It is also possible to perform the same analysis using Stirling's approximation instead of the Chernoff bound to estimate the probability that a block contains exactly hashed values.

In terms of the load factor , the expected time to perform a successful search on a random key is , and the expected time to perform an unsuccessful search (or the insertion of a new key) is .

For constant load factors, with high probability, the longest probe sequence (among the probe sequences for all keys stored in the table) has logarithmic length.

Primary Clustering

Linear probing hash tables suffer from a problem known as primary clustering, in which elements to group together into long contiguous runs.

Choice of hash function

Because linear probing is especially sensitive to unevenly distributed hash values,

The hash value that this class associates with each object, its identityHashCode, is guaranteed to remain fixed for the lifetime of an object but is otherwise arbitrary. Because the identityHashCode is constructed only once per object, and is not required to be related to the object's address or value, its construction may involve slower computations such as the call to a random or pseudorandom number generator. For instance, Java 8 uses an Xorshift pseudorandom number generator to construct these values.

For most applications of hashing, it is necessary to compute the hash function for each value every time that it is hashed, rather than once when its object is created. In such applications, random or pseudorandom numbers cannot be used as hash values, because then different objects with the same value would have different hashes. And cryptographic hash functions (which are designed to be computationally indistinguishable from truly random functions) are usually too slow to be used in hash tables. Instead, other methods for constructing hash functions have been devised. These methods compute the hash function quickly, and can be proven to work well with linear probing. In particular, linear probing has been analyzed from the framework of -independent hashing, a class of hash functions that are initialized from a small random seed and that are equally likely to map any -tuple of distinct keys to any -tuple of indexes. The parameter can be thought of as a measure of hash function quality: the larger is, the more time it will take to compute the hash function but it will behave more similarly to completely random functions.

For linear probing, 5-independence is enough to guarantee constant expected time per operation,

while some 4-independent hash functions perform badly, taking up to logarithmic time per operation.

Another method of constructing hash functions with both high quality and practical speed is tabulation hashing. In this method, the hash value for a key is computed by using each byte of the key as an index into a table of random numbers (with a different table for each byte position). The numbers from those table cells are then combined by a bitwise exclusive or operation. Hash functions constructed this way are only 3-independent. Nevertheless, linear probing using these hash functions takes constant expected time per operation. Both tabulation hashing and standard methods for generating 5-independent hash functions are limited to keys that have a fixed number of bits. To handle strings or other types of variable-length keys, it is possible to compose a simpler universal hashing technique that maps the keys to intermediate values and a higher quality (5-independent or tabulation) hash function that maps the intermediate values to hash table indices.

In an experimental comparison, Richter et al. found that the Multiply-Shift family of hash functions (defined as <math>h_z(x) = (x \cdot z \bmod 2^w) \div 2^{w-d}</math>) was "the fastest hash function when integrated with all hashing schemes, i.e., producing the highest throughputs and also of good quality" whereas tabulation hashing produced "the lowest throughput".

They point out that each table look-up require several cycles, being more expensive than simple arithmetic operations. They also found MurmurHash to be superior than tabulation hashing: "By studying the results provided by Mult and Murmur, we think that the trade-off for by tabulation (...) is less attractive in practice".

History

The idea of an associative array that allows data to be accessed by its value rather than by its address dates back to the mid-1940s in the work of Konrad Zuse and Vannevar Bush, but hash tables were not described until 1953, in an IBM memorandum by Hans Peter Luhn. Luhn used a different collision resolution method, chaining, rather than linear probing.

summarizes the early history of linear probing. It was the first open addressing method, and was originally synonymous with open addressing. According to Knuth, it was first used by Gene Amdahl, Elaine M. McGraw (née Boehme), and Arthur Samuel in 1954, in an assembler program for the IBM 701 computer. Another early publication of this method was by Soviet researcher Andrey Ershov, in 1958.

The first theoretical analysis of linear probing, showing that it takes constant expected time per operation with random hash functions, was given by Knuth.

Knuth's analysis including a variation of linear probing known as graveyard hashing that achieves the ideal bound of <math>\Theta(1 + \varepsilon^{-1})

</math>.

Other later developments include a more detailed analysis of the probability distribution of the running time, amortized analyses for workloads with both insertions and deletions,