A small phone book as a hash table In, a hash table ( hash map) is a which implements an, a structure that can map to. A hash table uses a to compute an index into an array of buckets or slots, from which the desired value can be found. Ideally, the hash function will assign each key to a unique bucket, but most hash table designs employ an imperfect hash function, which might cause hash where the hash function generates the same index for more than one key. Such collisions must be accommodated in some way.

In a well-dimensioned hash table, the average cost (number of ) 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 cost per operation.

The C program is successfully compiled and run on a Linux system. Both hash functions are used to and Arthur Samuel implemented a program using hashing at about I'm attempting to write a program for class that takes a dictionary file and reads it into a hash table. C Program For Separate Chaining Hash Table.

In many situations, hash tables turn out to be more efficient than or any other lookup structure. For this reason, they are widely used in many kinds of computer, particularly for associative arrays,,, and. Recent research by Google showed that using deep learning approach can optimize hash function by orders of magnitude in space usage savings, and improve retrieval speed as well. Hash collision resolved by separate chaining. In the method known as separate chaining, each bucket is independent, and has some sort of of entries with the same index. The time for hash table operations is the time to find the bucket (which is constant) plus the time for the list operation.

In a good hash table, each bucket has zero or one entries, and sometimes two or three, but rarely more than that. Therefore, structures that are efficient in time and space for these cases are preferred. Structures that are efficient for a fairly large number of entries per bucket are not needed or desirable. If these cases happen often, the hashing function needs to be fixed. [ ] Separate chaining with linked lists [ ] Chained hash tables with are popular because they require only basic data structures with simple algorithms, and can use simple hash functions that are unsuitable for other methods.

[ ] The cost of a table operation is that of scanning the entries of the selected bucket for the desired key. If the distribution of keys is, the average cost of a lookup depends only on the average number of keys per bucket—that is, it is roughly proportional to the load factor. For this reason, chained hash tables remain effective even when the number of table entries n is much higher than the number of slots. For example, a chained hash table with 1000 slots and 10,000 stored keys (load factor 10) is five to ten times slower than a 10,000-slot table (load factor 1); but still 1000 times faster than a plain sequential list. For separate-chaining, the worst-case scenario is when all entries are inserted into the same bucket, in which case the hash table is ineffective and the cost is that of searching the bucket data structure. If the latter is a linear list, the lookup procedure may have to scan all its entries, so the worst-case cost is proportional to the number n of entries in the table.

The bucket chains are often searched sequentially using the order the entries were added to the bucket. If the load factor is large and some keys are more likely to come up than others, then rearranging the chain with a may be effective. More sophisticated data structures, such as balanced search trees, are worth considering only if the load factor is large (about 10 or more), or if the hash distribution is likely to be very non-uniform, or if one must guarantee good performance even in a worst-case scenario. However, using a larger table and/or a better hash function may be even more effective in those cases.

[ ] Chained hash tables also inherit the disadvantages of linked lists. When storing small keys and values, the space overhead of the next pointer in each entry record can be significant. An additional disadvantage is that traversing a linked list has poor, making the processor cache ineffective. Separate chaining with list head cells [ ].

Hash collision by separate chaining with head records in the bucket array. Some chaining implementations store the first record of each chain in the slot array itself. The number of pointer traversals is decreased by one for most cases. The purpose is to increase cache efficiency of hash table access. The disadvantage is that an empty bucket takes the same space as a bucket with one entry. To save space, such hash tables often have about as many slots as stored entries, meaning that many slots have two or more entries. [ ] Separate chaining with other structures [ ] Instead of a list, one can use any other data structure that supports the required operations.

For example, by using a, the theoretical worst-case time of common hash table operations (insertion, deletion, lookup) can be brought down to rather than O( n). However, this introduces extra complexity into the implementation, and may cause even worse performance for smaller hash tables, where the time spent inserting into and balancing the tree is greater than the time needed to perform a on all of the elements of a list. A real world example of a hash table that uses a self-balancing binary search tree for buckets is the HashMap class in.

The variant called uses a to store all the entries that hash to the same slot. Each newly inserted entry gets appended to the end of the dynamic array that is assigned to the slot. The dynamic array is resized in an exact-fit manner, meaning it is grown only by as many bytes as needed. Alternative techniques such as growing the array by block sizes or pages were found to improve insertion performance, but at a cost in space. This variation makes more efficient use of and the (TLB), because slot entries are stored in sequential memory positions. It also dispenses with the next pointers that are required by linked lists, which saves space.

Despite frequent array resizing, space overheads incurred by the operating system such as memory fragmentation were found to be small. [ ] An elaboration on this approach is the so-called, where a bucket that contains k entries is organized as a perfect hash table with k 2 slots. While it uses more memory ( n 2 slots for n entries, in the worst case and n × k slots in the average case), this variant has guaranteed constant worst-case lookup time, and low amortized time for insertion. It is also possible to use a for each bucket, achieving constant time for all operations with high probability. Open addressing [ ]. Hash collision resolved by open addressing with linear probing (interval=1). Note that 'Ted Baker' has a unique hash, but nevertheless collided with 'Sandra Dee', that had previously collided with 'John Smith'.

In another strategy, called open addressing, all entry records are stored in the bucket array itself. When a new entry has to be inserted, the buckets are examined, starting with the hashed-to slot and proceeding in some probe sequence, until an unoccupied slot is found.

When searching for an entry, the buckets are scanned in the same 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. The name 'open addressing' refers to the fact that the location ('address') of the item is not determined by its hash value.

This graph compares the average number of cache misses required to look up elements in tables with chaining and linear probing. As the table passes the 80%-full mark, linear probing's performance drastically degrades. Open addressing avoids the time overhead of allocating each new entry record, and can be implemented even in the absence of a memory allocator.

It also avoids the extra indirection required to access the first entry of each bucket (that is, usually the only one). It also has better, particularly with linear probing. With small record sizes, these factors can yield better performance than chaining, particularly for lookups.

Hash tables with open addressing are also easier to, because they do not use pointers. [ ] On the other hand, normal open addressing is a poor choice for large elements, because these elements fill entire 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 or less. 3m Mpro110 Driver For Windows 7 here. If the table is expected to have a high load factor, 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 are marginal, and other considerations typically come into play.

[ ] Coalesced hashing [ ] A hybrid of chaining and open addressing, 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.

Cuckoo hashing [ ] Another alternative open-addressing solution is, which ensures constant lookup time in the worst case, and constant amortized time for insertions and deletions. It uses two or more hash functions, which means any key/value pair could be in two or more locations. For lookup, the first hash function is used; if the key/value is not found, then the second hash function is used, and so on. If a collision happens during insertion, then the key is re-hashed with the second hash function to map it to another bucket. If all hash functions are used and there is still a collision, then the key it collided with is removed to make space for the new key, and the old key is re-hashed with one of the other hash functions, which maps it to another bucket.

If that location also results in a collision, then the process repeats until there is no collision or the process traverses all the buckets, at which point the table is resized. By combining multiple hash functions with multiple cells per bucket, very high space utilization can be achieved. [ ] Hopscotch hashing [ ] Another alternative open-addressing solution is, which combines the approaches of and, yet seems in general to avoid their limitations. In particular it works well even when the load factor grows beyond 0.9. The algorithm is well suited for implementing a resizable. The hopscotch hashing algorithm works by defining a neighborhood of buckets near the original hashed bucket, where a given entry is always found.

Thus, search is limited to the number of entries in this neighborhood, which is logarithmic in the worst case, constant on average, and with proper alignment of the neighborhood typically requires one cache miss. When inserting an entry, one first attempts to add it to a bucket in the neighborhood. However, if all buckets in this neighborhood are occupied, the algorithm traverses buckets in sequence until an open slot (an unoccupied bucket) is found (as in linear probing). At that point, since the empty bucket is outside the neighborhood, items are repeatedly displaced in a sequence of hops. (This is similar to cuckoo hashing, but with the difference that in this case the empty slot is being moved into the neighborhood, instead of items being moved out with the hope of eventually finding an empty slot.) Each hop brings the open slot closer to the original neighborhood, without invalidating the neighborhood property of any of the buckets along the way. In the end, the open slot has been moved into the neighborhood, and the entry being inserted can be added to it. [ ] Robin Hood hashing [ ] One interesting variation on double-hashing collision resolution is Robin Hood hashing.

The idea is that a new key may displace a key already inserted, if its probe count is larger than that of 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 ordered hash tables except that the criterion for bumping a key does not depend on a direct relationship between the keys. Since both the worst case and the variation in the number of probes is reduced dramatically, an interesting variation is to probe the table starting at the expected successful probe value and then expand from that position in both directions. External Robin Hood hashing is an extension of this algorithm where the table is stored in an external file and each table position corresponds to a fixed-sized page or bucket with B records.

2-choice hashing [ ] employs two different hash functions, h 1( x) and h 2( x), for the hash table. Both hash functions are used to compute two table locations. When an object is inserted in the table, then it is placed in the table location that contains fewer objects (with the default being the h 1( x) table location if there is equality in bucket size).

2-choice hashing employs the principle of the power of two choices. Dynamic resizing [ ] The good functioning of a hash table depends on the fact that the table size is proportional to the number of entries. With a fixed size, and the common structures, it is similar to linear search, except with a better constant factor. In some cases, the number of entries may be definitely known in advance, for example keywords in a language. More commonly, this is not known for sure, if only due to later changes in code and data. It is one serious, although common, mistake to not provide any way for the table to resize. A general-purpose hash table 'class' will almost always have some way to resize, and it is good practice even for simple 'custom' tables.

An implementation should check the load factor, and do something if it becomes too large (this needs to be done only on inserts, since that is the only thing that would increase it). To keep the load factor under a certain limit, e.g., under 3/4, many table implementations expand the table when items are inserted. For example, in HashMap class the default load factor threshold for table expansion is 3/4 and in 's dict, table size is resized when load factor is greater than 2/3.

Since buckets are usually implemented on top of a and any constant proportion for resizing greater than 1 will keep the load factor under the desired limit, the exact choice of the constant is determined by the same as for dynamic arrays. Resizing is accompanied by a full or incremental table rehash whereby existing items are mapped to new bucket locations. To limit the proportion of memory wasted due to empty buckets, some implementations also shrink the size of the table—followed by a rehash—when items are deleted.

From the point of space–time tradeoffs, this operation is similar to the deallocation in dynamic arrays. Resizing by copying all entries [ ] A common approach is to automatically trigger a complete resizing when the load factor exceeds some threshold r max. Then a new larger table is, each entry is removed from the old table, and inserted into the new table. When all entries have been removed from the old table then the old table is returned to the free storage pool. Symmetrically, when the load factor falls below a second threshold r min, all entries are moved to a new smaller table. For hash tables that shrink and grow frequently, the resizing downward can be skipped entirely. In this case, the table size is proportional to the maximum number of entries that ever were in the hash table at one time, rather than the current number.

The disadvantage is that memory usage will be higher, and thus cache behavior may be worse. For best control, a 'shrink-to-fit' operation can be provided that does this only on request. If the table size increases or decreases by a fixed percentage at each expansion, the total cost of these resizings, over all insert and delete operations, is still a constant, independent of the number of entries n and of the number m of operations performed. For example, consider a table that was created with the minimum possible size and is doubled each time the load ratio exceeds some threshold. If m elements are inserted into that table, the total number of extra re-insertions that occur in all dynamic resizings of the table is at most m − 1. In other words, dynamic resizing roughly doubles the cost of each insert or delete operation. Alternatives to all-at-once rehashing [ ] Some hash table implementations, notably in, cannot pay the price of enlarging the hash table all at once, because it may interrupt time-critical operations.

If one cannot avoid dynamic resizing, a solution is to perform the resizing gradually: Disk-based hash tables almost always use some alternative to all-at-once rehashing, since the cost of rebuilding the entire table on disk would be too high. Incremental resizing [ ] One alternative to enlarging the table all at once is to perform the rehashing gradually: • During the resize, allocate the new hash table, but keep the old table unchanged. • In each lookup or delete operation, check both tables. • Perform insertion operations only in the new table. • At each insertion also move r elements from the old table to the new table. • When all elements are removed from the old table, deallocate it.

To ensure that the old table is completely copied over before the new table itself needs to be enlarged, it is necessary to increase the size of the table by a factor of at least ( r + 1)/ r during resizing. Monotonic keys [ ] If it is known that key values will always increase (or decrease), then a variation of can be achieved by keeping a list of the single most recent key value at each hash table resize operation. Upon lookup, keys that fall in the ranges defined by these list entries are directed to the appropriate hash function—and indeed hash table—both of which can be different for each range.

Since it is common to grow the overall number of entries by doubling, there will only be ranges to check, and binary search time for the redirection would be O(log(log( N))). As with consistent hashing, this approach guarantees that any key's hash, once issued, will never change, even when the hash table is later grown. 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 lookup functions. Hashing for distributed hash tables [ ] 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. Such hash functions are prevalent in disk-based and, where rehashing is prohibitively costly.

The problem of designing a hash such that most values do not change when the table is resized is known as the problem. The four most popular approaches are,, the algorithm, and distance. Performance analysis [ ] In the simplest model, the hash function is completely unspecified and the table does not resize. For the best possible choice of hash function, a table of size k with open addressing has no collisions and holds up to k elements, with a single comparison for successful lookup, and a table of size k with chaining and n keys has the minimum max(0, n − k) collisions and O(1 + n/ k) comparisons for lookup. For the worst choice of hash function, every insertion causes a collision, and hash tables degenerate to linear search, with Ω( n) amortized comparisons per insertion and up to n comparisons for a successful lookup. Adding rehashing to this model is straightforward.

As in a, geometric resizing by a factor of b implies that only n/ b i keys are inserted i or more times, so that the total number of insertions is bounded above by bn/( b − 1), which is O( n). By using rehashing to maintain n. Main article: Hash tables are commonly used to implement many types of in-memory tables. They are used to implement (arrays whose indices are arbitrary or other complicated objects), especially in like,,, and. When storing a new item into a and a hash collision occurs, the multimap unconditionally stores both items. When storing a new item into a typical associative array and a hash collision occurs, but the actual keys themselves are different, the associative array likewise stores both items.

However, if the key of the new item exactly matches the key of an old item, the associative array typically erases the old item and overwrites it with the new item, so every item in the table has a unique key. Database indexing [ ] Hash tables may also be used as -based data structures and (such as in ) although are more popular in these applications. In multi-node database systems, hash tables are commonly used to distribute rows amongst nodes, reducing network traffic for hash joins.

Main article: Hash tables can be used to implement, auxiliary data tables that are used to speed up the access to data that is primarily stored in slower media. In this application, hash collisions can be handled by discarding one of the two colliding entries—usually erasing the old item that is currently stored in the table and overwriting it with the new item, so every item in the table has a unique hash value. Sets [ ] Besides recovering the entry that has a given key, many hash table implementations can also tell whether such an entry exists or not.

Those structures can therefore be used to implement a, which merely records whether a given key belongs to a specified set of keys. In this case, the structure can be simplified by eliminating all parts that have to do with the entry values.

Hashing can be used to implement both static and dynamic sets. Object representation [ ] Several dynamic languages, such as,,,, and, use hash tables to implement objects.

In this representation, the keys are the names of the members and methods of the object, and the values are pointers to the corresponding member or method. Unique data representation [ ].

Main article: Hash tables can be used by some programs to avoid creating multiple character strings with the same contents. For that purpose, all strings in use by the program are stored in a single string pool implemented as a hash table, which is checked whenever a new string has to be created. This technique was introduced in interpreters under the name, and can be used with many other kinds of data ( in a symbolic algebra system, records in a database, files in a file system, binary decision diagrams, etc.). Transposition table [ ]. Main article: Implementations [ ] In programming languages [ ] Many programming languages provide hash table functionality, either as built-in associative arrays or as standard modules.

In, for example, the class provides hash tables for keys and values of arbitrary type. The programming language (including the variant which is used on ) includes the HashSet, HashMap, LinkedHashSet, and LinkedHashMap collections. In 5 and 7, the Zend 2 engine and the Zend 3 engine (respectively) use one of the hash functions from to generate the hash values used in managing the mappings of data pointers stored in a hash table. In the PHP source code, it is labelled as DJBX33A (Daniel J. Bernstein, Times 33 with Addition).

's built-in hash table implementation, in the form of the dict type, as well as 's hash type (%) are used internally to implement namespaces and therefore need to pay more attention to security, i.e., collision attacks. Python also use hashes internally, for fast lookup (though they store only keys, not values). In the, support for hash tables is provided via the non-generic Hashtable and generic Dictionary classes, which store key-value pairs, and the generic HashSet class, which stores only values. In 's standard library, the generic HashMap and HashSet structs use linear probing with Robin Hood bucket stealing.

History [ ] The idea of hashing arose independently in different places. In January 1953, H. Luhn wrote an internal IBM memorandum that used hashing with chaining.,,, and implemented a program using hashing at about the same time. Open addressing with linear probing (relatively prime stepping) is credited to Amdahl, but (in Russia) had the same idea. See also [ ] • • • • • • • • Related data structures [ ] There are several data structures that use hash functions but cannot be considered special cases of hash tables: •, memory efficient data-structure designed for constant-time approximate lookups; uses hash function(s) and can be seen as an approximate hash table. • (DHT), a resilient dynamic table spread over several nodes of a network. •, a structure, similar to the, but where each key is hashed first.

References [ ].