# Hash Tables

Hash tables are a simple and effective method to implement dictionaries.
Average time to search for an element is O(1), while worst-case time
is O( *n*). Cormen
[2009] and Knuth
[1998] both contain excellent discussions on hashing.

## Theory

A hash table is simply an array that is addressed via a hash function. For example, in Figure
3-1, `hashTable`

is an array with 8 elements. Each element is a pointer to a linked
list of numeric data. The hash function for this example simply divides the data key by 8,
and uses the remainder as an index into the table. This yields a number from 0 to 7. Since
the range of indices for `hashTable`

is 0 to 7, we are guaranteed that the index
is valid.

**Figure 3-1: A Hash Table**

To insert a new item in the table, we hash the key to determine which list the item goes
on, and then insert the item at the beginning of the list. For example, to insert 11, we divide
11 by 8 giving a remainder of 3. Thus, 11 goes on the list starting at `hashTable[3]`

.
To find a number, we hash the number and chain down the correct list to see if it is in the
table. To delete a number, we find the number and remove the node from the linked list.

Entries in the hash table are dynamically allocated and entered on a linked
list associated with each hash table entry. This technique is known as *chaining*.
An alternative method, where all entries are stored in the hash table itself,
is known as open addressing
and may be found in the references.

If the hash function is uniform, or equally distributes the data keys among the hash table
indices, then hashing effectively subdivides the list to be searched. Worst-case behavior occurs
when all keys hash to the same index. Then we simply have a single linked list that must be
sequentially searched. Consequently, it is important to choose a good hash function. Several
methods may be used to hash key values. To illustrate the techniques, I will assume *unsigned
char* is 8-bits, *unsigned short int* is 16-bits and *unsigned long int* is 32-bits.

*Division method*(*tablesize*=*prime*). This technique was used in the preceeding example. A`hashValue`

, from 0 to (`HASH_TABLE_SIZE - 1`

), is computed by dividing the key value by the size of the hash table and taking the remainder. For example:typedef int HashIndexType; HashIndexType hash(int key) { return key % HASH_TABLE_SIZE; }

Selecting an appropriate

`HASH_TABLE_SIZE`

is important to the success of this method. For example, a`HASH_TABLE_SIZE`

divisible by two would yield even hash values for even`keys`

, and odd hash values for odd`keys`

. This is an undesirable property, as all keys would hash to even values if they happened to be even. If`HASH_TABLE_SIZE`

is a power of two, then the hash function simply selects a subset of the`key`

bits as the table index. To obtain a more random scattering,`HASH_TABLE_SIZE`

should be a prime number not too close to a power of two.*Multiplication method*(*tablesize*= 2^{n}). The multiplication method may be used for a`HASH_TABLE_SIZE`

that is a power of 2. The`key`

is multiplied by a constant, and then the necessary bits are extracted to index into the table. Knuth recommends using the the golden ratio, or (*sqrt*(5) - 1)/2, to determine the constant. Assume the hash table contains 32 (2^{5}) entries and is indexed by an unsigned char (8 bits). First construct a multiplier based on the index and golden ratio. In this example, the multiplier is 2^{8}x (*sqrt*(5) - 1)/2, or 158. This scales the golden ratio so that the first bit of the multiplier is "1".xxxxxxxx key

__xxxxxxxx__multiplier (158) xxxxxxxx x xxxxxxx xx xxxxxx xxx xxxxx xxxx xxxx xxxxx xxx xxxxxx xx__xxxxxxx x__xxxxxxxx bbbbbxxx productMultiply the key by 158 and extract the 5 most significant bits of the least significant word. These bits are indicated by "bbbbb" in the above example, and represent a thorough mixing of the multiplier and key. The following definitions may be used for the multiplication method:

/* 8-bit index */ typedef unsigned char HashIndexType; static const HashIndexType M = 158; /* 16-bit index */ typedef unsigned short int HashIndexType; static const HashIndexType M = 40503; /* 32-bit index */ typedef unsigned long int HashIndexType; static const HashIndexType M = 2654435769; /* w=bitwidth(HashIndexType), size of table=2**n */ static const int S = w - n; HashIndexType hashValue = (HashIndexType)(M * key) >> S;

For example, if

`HASH_TABLE_SIZE`

is 1024 (2^{10}), then a 16-bit index is sufficient and`S`

would be assigned a value of 16 - 10 = 6. Thus, we have:typedef unsigned short int HashIndexType; HashIndexType hash(int key) { static const HashIndexType M = 40503; static const int S = 6; return (HashIndexType)(M * key) >> S; }

*Variable string addition method*(*tablesize*= 256). To hash a variable-length string, each character is added, modulo 256, to a total. A`hashValue`

, range 0-255, is computed.unsigned char hash(char *str) { unsigned char h = 0; while (*str) h += *str++; return h; }

*Variable string exclusive-or method*(*tablesize*= 256). This method is similar to the addition method, but successfully distinguishes similar words and anagrams. To obtain a hash value in the range 0-255, all bytes in the string are exclusive-or'd together. However, in the process of doing each exclusive-or, a random component is introduced.unsigned char rand8[256]; unsigned char hash(char *str) { unsigned char h = 0; while (*str) h = rand8[h ^ *str++]; return h; }

`Rand8`

is a table of 256 8-bit unique random numbers. The exact ordering is not critical. The exclusive-or method has its basis in cryptography, and is quite effective (Pearson [1990]).*Variable string exclusive-or method*(*tablesize*<= 65536). If we hash the string twice, we may derive a hash value for an arbitrary table size up to 65536. The second time the string is hashed, one is added to the first character. Then the two 8-bit hash values are concatenated together to form a 16-bit hash value.unsigned char rand8[256]; unsigned short int hash(char *str) { unsigned short int h; unsigned char h1, h2; if (*str == 0) return 0; h1 = *str; h2 = *str + 1; str++; while (*str) { h1 = rand8[h1 ^ *str]; h2 = rand8[h2 ^ *str]; str++; } /* h is in range 0..65535 */ h = ((unsigned short int)h1 << 8)|(unsigned short int)h2; /* use division method to scale */ return h % HASH_TABLE_SIZE }

Assuming*n*data items, the hash table size should be large enough to accommodate a
reasonable number of entries. As seen in Table 3-1, a small table size substantially increases
the average time to find a key. A hash table may be viewed as a collection of linked lists.
As the table becomes larger, the number of lists increases, and the average number of nodes
on each list decreases. If the table size is 1, then the table is really a single linked list
of length *n*. Assuming a perfect hash function, a table size of 2 has two lists of length *n*/2. If the table size is 100, then we have 100 lists of length *n*/100. This considerably
reduces the length of the list to be searched. There is considerable leeway in the choice of
table size.

size | time | size | time | |
---|---|---|---|---|

1 | 869 | 128 | 9 | |

2 | 432 | 256 | 6 | |

4 | 214 | 512 | 4 | |

8 | 106 | 1024 | 4 | |

16 | 54 | 2048 | 3 | |

32 | 28 | 4096 | 3 | |

64 | 15 | 8192 | 3 |

## Implementation in C

An ANSI-C implementation of a hash table is included. Typedefs `recType`

, `keyType`

and comparison operator `compEQ`

should
be altered to reflect the data stored in the table. The `hashTableSize`

must be
determined and the `hashTable`

allocated. The division method was used in the `hash`

function. Function `insert`

allocates a new node and inserts it in the table. Function `delete`

deletes and frees a node from the table. Function `find`

searches
the table for a particular value.

## Implementation in Visual Basic

The hash table algorithm has been implemented as objects, using a module for the algorithm, and a class for the nodes. It has also been implemented as a class, using arrays. The array implementation is recommended.