**Prerequisite:** Hashing | Set 1 (Introduction)

__What is a Hash Function?__

A function that converts a given big phone number to a small practical integer value. The mapped integer value is used as an index in the hash table. In simple terms, a hash function maps a big number or string to a small integer that can be used as the index in the hash table.

__What is meant by Good Hash Function?__

A good hash function should have the following properties:

- Efficiently computable.
- Should uniformly distribute the keys (Each table position equally likely for each key)

**For example:** For phone numbers, a bad hash function is to take the first three digits. A better function is considered the last three digits. Please note that this may not be the best hash function. There may be better ways.

In practice, we can often employ * heuristic techniques* to create a hash function that performs well. Qualitative information about the distribution of the keys may be useful in this design process. In general, a hash function should depend on every single bit of the key, so that two keys that differ in only one bit or one group of bits (regardless of whether the group is at the beginning, end, or middle of the key or present throughout the key) hash into different values. Thus, a hash function that simply extracts a portion of a key is not suitable. Similarly, if two keys are simply digited or character permutations of each other

*(such as 139 and 319)*, they should also hash into different values.

The two heuristic methods are* hashing by division* and

*which are as follows:*

**hashing by multiplication****The mod method:**- In this method for creating hash functions, we map a key into one of the slots of table by taking the remainder of key divided by table_size. That is, the hash function is

- In this method for creating hash functions, we map a key into one of the slots of table by taking the remainder of key divided by table_size. That is, the hash function is

h(key) = keymodtable_size i.e. key % table_size

- Since it requires only a single division operation, hashing by division is quite fast.
- When using the division method, we usually avoid certain values of table_size like table_size should not be a power of a number suppose
**r**, since if, then h(key) is just the p lowest-order bits of key. Unless we know that all low-order p-bit patterns are equally likely, we are better off designing the hash function to depend on all the bits of the key.**table_size = r^p** - It has been found that the best results with the division method are achieved when the table size is prime. However, even if table_size is prime, an additional restriction is called for. If
**r**is the number of possible character codes on an computer, and if**table_size**is a prime such that r % table_size equal 1, then hash function**h(key) = key % table_size**is simply the*sum of the binary representation of the characters*in the key mod table_size. - Suppose
**r**= 256 and**table_size**= 17, in which r % table_size i.e. 256 % 17 = 1. - So for
**key = 37596**, its hash is

37596 % 17 = 12

- But for
**key = 573**, its hash function is also

573 % 17 = 12

- Hence it can be seen that by this hash function, many keys can have the same hash. This is called Collision.
- A prime not too close to an exact power of 2 is often good choice for table_size.

**The multiplication method:**- In multiplication method, we multiply the key
**k**by a constant real number**c**in the range**0 < c < 1**and extract the*fractional part of*.**k * c** - Then we multiply this value by table_size
**m**and take the floor of the result. It can be represented as

- In multiplication method, we multiply the key

h(k) = floor (m * (k * c mod 1)) or h(k) = floor (m * frac (k * c))

- where the function
**floor(x)**, available in standard library*math.h*, yields the integer part of the real number x, and**frac(x)**yields the fractional part.**[frac(x) = x – floor(x)]**

- An advantage of the multiplication method is that the value of
*m is not critical*, we typically choose it to be a power of 2 (**m = 2**for some integer^{p}**p**), since we can then easily implement the function on most computers - Suppose that the word size of the machine is
**w**bits and that key fits into a single word. - We restrict
**c**to be a fraction of the form**s / (2**, where s is an integer in the range^{w})**0 < s < 2**.^{w}

- Referring to figure, we first multiply key by the w-bit integer
**s = c * 2**. The result is a 2w-bit value^{w}

r1 * 2where^{w}+ r0r1= high-order word of the productr0= lower order word of the product

- Although this method works with any value of the constant
**c**, it works better with some values than the others.

c ~ (sqrt (5) – 1) / 2 = 0.618033988 . . .

- is likely to work reasonably well.
- Suppose
**k**= 123456,**p**= 14, **m**= 2^14 = 16384, and**w**= 32.- Adapting
**Knuth’s suggestion**,**c**to be fraction of the form**s / 2^32**. - Then
**key * s**= 327706022297664 = (76300 * 2^32) + 17612864, - So
**r1**= 76300 and**r0**= 176122864. - The 14 most significant bits of r0 yield the value
**h(key) = 67.**

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