Hash tables support the efficient insertion of new entries, in expected O(1) time. The time spent in searching depends on the hash function and the load of the hash table; both insertion and search approach O(1) time with well chosen values and hashes.
Typical operations on a hash table include insertion, deletion and lookup (although some hash tables are precalculated so that no insertions or deletions, only lookups are done on a live system). These operations are all performed in amortized constant time, which makes maintaining and accessing a huge hash table very efficient.
It is also possible to create a hash table statically where, for example, there is a fairly limited fixed set of input values - such as the value in a single byte (or possibly two bytes ) from which an index can be constructed directly (see section below on creating hash tables). The hash table can also be used simultaneously for tests of validity on the values that are disallowed.
An efficient collision resolution strategy is an important part of any hash table. Consider the following case, derived using the birthday paradox, of a hash table containing 1 million indices. Even if a hash function were to output random indices uniformly distributed over the array, there is a 95% chance of a collision occurring before it contains 2500 records.
There are a number of collision resolution techniques, but the most popular are open addressing and chaining.
Sometimes called simply chaining or direct chaining, this technique in its simplest form has a linked list of inserted records at each slot in the array references. Each linked list has each element that collides to the same slot. Insertion requires finding the correct slot, and appending to either end of the list in that slot; deletion requires searching the list and removal.
Chained hash tables have advantages over open addressed hash tables in that the removal operation is simple and resizing the table can be postponed for a much longer time because performance degrades more gracefully even when every slot is used. Indeed, many chaining hash tables may not require resizing at all since performance degradation is linear as the table fills. For example, a chaining hash table containing twice its recommended capacity of data would only be about twice as slow on average as the same table at its recommended capacity.
Chained hash tables inherit the disadvantages of linked lists. When storing small records, the overhead of the linked list can be significant. An additional disadvantage is that traversing a linked list has poor cache performance.
Alternative data structures can be used for chains instead of linked lists. By using a self-balancing tree, for example, the theoretical worst-case time of a hash table can be brought down to O(log n) rather than O(n). However, since each list is intended to be short, this approach is usually inefficient unless the hash table is designed to run at full capacity or there are unusually high collision rates, as might occur in input designed to cause collisions. Dynamic arrays can also be used to decrease space overhead and improve cache performance, forming a cache-conscious resolution scheme as discussed in the next section.
Some chaining implementations use an optimization where the first record of each chain is stored in the table. The purpose is to increase cache efficiency of hash table access. In order to avoid wasting large amounts of space, such hash tables would maintain a load factor of 1.0 or greater. The term direct chaining is sometimes used to describe implementations that do not use this optimization.
Open addressing hash tables store the records directly within the array. This approach is also called closed hashing. A hash collision is resolved by probing, or searching through alternate locations in the array (the probe sequence) until either the target record is found, or an unused array slot is found, which indicates that there is no such key in the table. Well known probe sequences include: linear probing : in which the interval between probes is fixed - often at 1. quadratic probing : in which the interval between probes increases proportional to the hash value (the interval thus increasing linearly and the indices are described by a quadratic function). double hashing : in which the interval between probes is computed by another hash function.
For small record sizes (a few words or less) the benefits of in-place open addressing compared to chaining are:
On the other hand, normal open addressing is a poor choice for large elements, since these elements fill entire cache lines (negating the cache advantage), and a large amount of space is wasted on large empty table slots. If the open addressing table only stores references to elements (external storage), it uses space comparable to chaining even for large records but loses its speed advantage.
Generally speaking, open addressing is better used for hash tables with small records that can be stored within the table (internal storage) and fit in a cache line. They are particularly suitable for elements of one word or less. In cases where the tables are expected to have high load factors, the records are large, or the data is variable-sized, chained hash tables often perform as well or better.
Ultimately, used sensibly, any kind of hash table algorithm is usually fast enough; and the percentage of a calculation spent in hash table code is low. Memory usage is rarely considered excessive. Therefore, in most cases the differences between these algorithms is marginal, and other considerations typically come into play.
A hybrid of chaining and open addressing, coalesced hashing links together chains of nodes within the table itself. Like open addressing, it achieves space usage and (somewhat diminished) cache advantages over chaining. Like chaining, it does not exhibit clustering effects; in fact, the table can be efficiently filled to a high density. Unlike chaining, it cannot have more elements than table slots.
If all of the keys that will be used are known ahead of time, and there are no more keys than can fit the hash table, perfect hashing 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 gives a hash table where the time to make a lookup is constant in the worst case. This is in contrast to chaining and open addressing methods, where the time for lookup is low on average, but may be arbitrarily large. A simpler alternative, which also gives worst case constant lookup time, is cuckoo hashing.
In dynamic perfect hashing, a guarantee of constant worst-case lookup times by using a second-level hash table with no collisions for each bucket is given. If there are k elements in the bucket, the bucket's hash table is given size k2, and its hash function is selected at random from a universal hash function set. If a collision ever occurs, the table is rebuilt with a different randomly-selected hash function. Because the load factor of the table is kept low (1/k), rebuilding is infrequent and amortized cost of insertions is low.
Although each bucket requires quadratic space, if the keys inserted into the (first-level) hash table are uniformly distributed, the structure as a whole occupies expected O(n) space, since bucket sizes are small with high probability.
Perhaps the simplest solution to a collision is to replace the value that is already in the slot with the new value, or slightly less commonly, drop the record that is to be inserted. In later searches, this may result in a search not finding a record which has been inserted. This technique is particularly useful for implementing caching.
An even more space-efficient solution which is similar to this is use a bit array (an array of one-bit fields) for our table. Initially all bits are set to zero, and when we insert a key, we set the corresponding bit to one. False negatives cannot occur, but false positives can, since if the search finds a 1 bit, it will claim that the value was found, even if it was just another value that hashed into the same array slot by coincidence. In reality, such a hash table is merely a specific type of Bloom filter.
One interesting variation on double-hashing collision resolution is that of Robin Hood hashing. The idea is that a key already inserted may be displaced by a new key if its probe count is larger than the key at the current position. The net effect of this is that it reduces worst case search times in the table. This is similar to Knuth's ordered hash tables except the criteria for bumping a key does not depend on a direct relationship between the keys.
A chained hash table is a simple and flexible data structure but it does not make good use of CPU cache. This is especially the case when variable-length strings are used as keys. When a collision occurs, a linked list is traversed which can attract a surplus of cache-misses. In addition, linked lists will waste an excessive amount of space due to pointers and memory allocation overheads. We can address these issues by replacing linked lists with dynamic arrays, forming a cache-conscious hash table known as an array hash.
When a string is hashed to a slot, it is appended to the end of the dynamic array that is assigned to that slot. In this manner, nodes and pointers are eliminated and strings are accessed in a sequential manner, promoting good use of cache.
With a good hash function, a hash table can typically contain about 70%–80% as many elements as it does table slots and still perform well. Depending on the collision resolution mechanism, performance can begin to suffer either gradually or dramatically as more elements are added. To deal with this, when the load factor exceeds some threshold, it is necessary to allocate a new, larger table, and add all the contents of the original table to this new table. In Java's HashMap class, for example, the default load factor threshold is 0.75.
This can be a very expensive operation, and the necessity for it is one of the hash table's disadvantages. In fact, some naive methods for doing this, such as enlarging the table by one each time you add a new element, reduce performance so drastically as to make the hash table useless. However, if the table is enlarged by some fixed percent, such as 10% or 100%, it can be shown using amortized analysis that these resizings are so infrequent that the average time per insertion remains constant-time. To see why this is true, suppose a hash table using chaining begins at the minimum size of 1 and is doubled each time it fills above 100%. If in the end it contains n elements, then the total add operations performed for all the resizings is:
Because the costs of the resizings form a geometric series, the total cost is O(n). But it is necessary also to perform n operations to add the n elements in the first place, so the total time to add n elements with resizing is O(n), an amortized time of O(1) per element.
On the other hand, some hash table implementations, notably in real-time systems, cannot pay the price of enlarging the hash table all at once, because it may interrupt time-critical operations. One simple approach is to initially allocate the table with enough space for the expected number of elements and forbid the addition of too many elements. Another useful but more memory-intensive technique is to perform the resizing gradually:
To ensure that the old table will be completely copied over before the new table itself needs to be enlarged, it's necessary to increase the size of the table by a factor of at least (k + 1)/k during the resizing.
Linear hashing is a hash table algorithm that permits incremental hash table expansion. It is implemented using a single hash table, but with two possible look-up functions.
Another way to decrease the cost of table resizing is to choose a hash function in such a way that the hashes of most values do not change when the table is resized. This approach, called consistent hashing, is prevalent in disk-based and distributed hashes, where resizing is prohibitively costly.
Compared to other associative array data structures, hash tables are most useful when large numbers of records are to be stored, especially if the size of the data set can be predicted.
Hash tables may be used as in-memory data structures. Hash tables may also be adopted for use with persistent data structures; database indices sometimes use disk-based data structures based on hash tables, although balanced trees are more popular.
Choosing a good hash function is tricky. The literature is replete with poor choices, at least when measured by modern standards. For example, the very popular multiplicative hash advocated by Donald Knuth in The Art of Computer Programming (see reference below) has particularly poor clustering behavior. However, since poor hashing merely degrades hash table performance for particular input key distributions, such problems commonly go undetected.
The literature is similarly sparse on the criteria for choosing a hash function. Unlike most other fundamental algorithms and data structures, there is no universal consensus on what makes a "good" hash function. The remainder of this section is organized by three criteria: simplicity, speed, and strength. In addition, it surveys algorithms known to perform well by these criteria.
Simplicity and speed are readily measured objectively (by number of lines of code and CPU benchmarks, for example), but strength is a more slippery concept. Obviously, a cryptographic hash function such as SHA-1 would satisfy the relatively lax strength requirements needed for hash tables, but their slowness and complexity makes them unappealing. However, using cryptographic hash functions can protect against collision attacks when the hash table modulus and its factors can not be kept secret from the attacker, or alternatively, by applying a secret salt. However, for these specialized cases, a universal hash function can be used instead of one static hash.
In the absence of a standard measure for hash function strength, the current state of the art is to employ a battery of statistical tests to measure whether the hash function can be readily distinguished from a random function. Arguably the most important test is to determine whether the hash function displays the avalanche effect, which essentially states that any single-bit change in the input key should affect, on average, half the bits in the output. Bret Mulvey advocates testing the strict avalanche condition in particular, which states that, for any single-bit change, each of the output bits should change with probability one-half, independent of the other bits in the key. Purely additive hash functions such as CRC fail this stronger condition miserably.
Clearly, a strong hash function should have a uniform distribution of hash values. Bret Mulvey proposes the use of a chi-squared test for uniformity, based on power of two hash table sizes ranging from 21 to 216. This test is considerably more sensitive than many others proposed for measuring hash functions, and finds problems in many popular hash functions.
Fortunately, there are good hash functions that satisfy all these criteria. The simplest class all consume one byte of the input key per iteration of the inner loop. Within this class, simplicity and speed are closely related, as fast algorithms simply don't have time to perform complex calculations.
A mathematical byte-by-byte implementation that performs particularly well is the Jenkins One-at-a-time hash, adapted here from an article by Bob Jenkins, its creator.
The avalanche behavior of this hash is shown on the right. The image was made using Bret Mulvey's AvalancheTest in his Hash.cs toolset
Each of the 24 rows corresponds to a single bit in the 3-byte input key, and each of the 32 columns corresponds to a bit in the output hash. Colors are chosen by how well the input key bit affects the given output hash bit: a green square indicates good mixing behavior, a yellow square weak mixing behavior, and red would indicate no mixing. Only a few bits in the last byte of the input key are weakly mixed to a minority of bits in the output hash, a performance vastly better than a number of widely used hash functions.
Many commonly used hash functions perform poorly when subjected to such rigorous avalanche testing. The widely favored FNV hash, for example, shows many bits with no mixing at all, especially for short keys. See the evaluation of FNV by Bret Mulvey for a more thorough analysis.
If speed is more important than simplicity, then the class of hash functions which consume multibyte chunks per iteration may be of interest. One of the most sophisticated is "lookup3" by Bob Jenkins, which consumes input in 12 byte (96 bit) chunks. Note, though, that any speed improvement from the use of this hash is only likely to be useful for large keys, and that the increased complexity may also have speed consequences such as preventing an optimizing compiler from inlining the hash function. Bret Mulvey analyzed an earlier version lookup2, and found it to have excellent avalanche behavior.
One desirable property of a hash function is that conversion from the hash value (typically 32 bits) to a bucket index for a particular-size hash table can be done simply by masking, preserving only the lower k bits for a table of size 2k (an operation equivalent to computing the hash value modulo the table size). This property enables the technique of incremental doubling of the size of the hash table - each bucket in the old table maps to only two in the new table. Because of its use of XOR-folding, the FNV hash does not have this property. Some older hashes are even worse, requiring table sizes to be a prime number rather than a power of two, again computing the bucket index as the hash value modulo the table size. In general, such a requirement is a sign of a fundamentally weak function; using a prime table size is a poor substitute for using a stronger function.
Although hash table lookups use constant time on average, the time spent can be significant. Evaluating a good hash function can be a slow operation. In particular, if simple array indexing can be used instead, this is usually faster.
Hash tables in general exhibit poor locality of reference—that is, the data to be accessed is distributed seemingly at random in memory. Because hash tables cause access patterns that jump around, this can trigger microprocessor cache misses that cause long delays. Compact data structures such as arrays, searched with linear search, may be faster if the table is relatively small and keys are cheap to compare, such as with simple integer keys. According to Moore's Law, cache sizes are growing exponentially and so what is considered "small" may be increasing. The optimal performance point varies from system to system.
More significantly, hash tables are more difficult and error-prone to write and use. Hash tables require the design of an effective hash function for each key type, which in some situations is more difficult and time-consuming to design and debug than the simple comparison function required for a self-balancing binary search tree. In open-addressed hash tables it is fairly easy to create a poor hash function.
Additionally, in some applications, a black hat with knowledge of the hash function may be able to supply information to a hash which creates worst-case behavior by causing excessive collisions, resulting in very poor performance (i.e., a denial of service attack). In critical applications, either universal hashing can be used or a data structure with better worst-case guarantees may be preferable. For details, see Crosby and Wallach's Denial of Service via Algorithmic Complexity Attacks.
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