Computer programming is the process of designing, writing, testing, debugging, and maintaining the source code of computer programs. This source code is written in one or more programming languages. The purpose of programming is to create a program that performs specific operations or exhibits a...
In computer science, a hash table or hash map is a data structure that uses a hash function to map identifying values, known as keys , to their associated values . Thus, a hash table implements an associative array...
s.
It is an open addressing method to handle overflows after a collision takes place in some bucket of a hash table.
Quadratic probing operates by taking the original hash value and adding successive values of an arbitrary quadratic polynomial
Quadratic polynomial
In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...
to the starting value.
The form of the equation is . The function used might even be if and are taken as zero.
In this case, suppose a cell H is reached but is occupied, then the next sequence of cells to be examined would be .
Linear Probing, instead, would examine the sequence . This would result in primary clustering and the larger the cluster grows, lesser will be the search efficiency for those items. Quadratic probing can be a more efficient algorithm in a closed hash table, since it better avoids the clustering problem that can occur with linear probing, although it is not immune. It also provides good memory caching because it preserves some locality of reference
Locality of reference
In computer science, locality of reference, also known as the principle of locality, is the phenomenon of the same value or related storage locations being frequently accessed. There are two basic types of reference locality. Temporal locality refers to the reuse of specific data and/or resources...
Linear probing is a scheme in computer programming for resolving hash collisions of values of hash functions by sequentially searching the hash table for a free location. This is accomplished using two values - one as a starting value and one as an interval between successive values in modular...
has greater locality and, thus, better cache performance.
Quadratic probing is used in the Berkeley Fast File System to allocate free blocks. The allocation routine chooses a new cylinder-group when the current is nearly full using quadratic probing, because of the speed it shows in finding unused cylinder-groups.
A hash function is any algorithm or subroutine that maps large data sets to smaller data sets, called keys. For example, a single integer can serve as an index to an array...
that maps an element to an integer in , where is the size of the table. Let the th probe position for a value be given by the function , where ≠ 0. If , then degrades to a linear probe
Linear probing
Linear probing is a scheme in computer programming for resolving hash collisions of values of hash functions by sequentially searching the hash table for a free location. This is accomplished using two values - one as a starting value and one as an interval between successive values in modular...
In computer science, a hash table or hash map is a data structure that uses a hash function to map identifying values, known as keys , to their associated values . Thus, a hash table implements an associative array...
, the values of and remain constant.
Examples:
If , then the probe sequence will be
For , a good choice for the constants are 1/2, as the values for in are all distinct. This leads to a probe sequence of where the values increase by .
For prime , most choices of and will make distinct for in . Such choices include 1/2, 1, and . Because there are only about m/2 distinct probes for a given element, it is difficult to guarantee that insertions will succeed when the load factor is > 1/2.
Quadratic Probing Insertion
The problem, here, is to insert a key at an available key space in a given Hash Table using quadratic probing.
Algorithm to Insert key in Hash Table
1. Get the key k
2. Set counter j = 0
3. Compute hash function h[k] = k % SIZE
4. If hashtable[h[k]] is empty
(4.1) Insert key k at hashtable[h[k]]
(4.2) Stop
Else
(4.3) The key space at hashtable[h[k]] is occupied, so we need to find the next available key space
(4.4) Increment j
(4.5) Compute new hash function h[k] = ( k + j * j ) % SIZE
(4.6) Repeat Step 4 till j is more than SIZE of hash table
5. The hash table is full
6. Stop
C function for Key Insertion
int quadratic_probing_insert(int *hashtable, int key, int *empty)
{
/* hashtable[] is an integer hash table; empty[] is another array which indicates whether the key space is occupied;
If an empty key space is found, the function returns the index of the bucket where the key is inserted, otherwise it
returns (-1) if no empty key space is found */
Key space at position h[k] is empty : Insert the key at the position.
Key space at position h[k] is occupied: Compute the next hash function h[k].
Consider a hash table initially containing some elements.
Suppose we want to insert a key 10 in the hash table.
h[k] = 10 % 8 = 2
Slot 2 being occupied the hash function will search for new available key space.
h[k] = ( k + j * j ) % SIZE
h[k] = ( 2 + 1 * 1 ) % 8 = 3
Slot 3 is also occupied, so the hash function will search for next available key space.
h[k] = ( 2 + 2 * 2 ) % 8 = 6
Slot 6 is empty, so key will be inserted here.
Algorithm to Search Element in Hash Table
1. Get the key k to be searched
2. Set counter j = 0
3. Compute hash function h[k] = k % SIZE
4. If the key space at hashtable[h[k]] is occupied
(4.1) Compare the element at hashtable[h[k]] with the key k.
(4.2) If they are equal
(4.2.1) The key is found at the bucket h[k]
(4.2.2) Stop
Else
(4.3) The element might be placed at the next location given by the quadratic function
(4.4) Increment j
(4.5) Compute new hash function h[k] = ( k + j * j ) % SIZE
(4.6) Repeat Step 4 till j is greater than SIZE of hash table
5. The key was not found in the hash table
6. Stop
C function for Key Searching
int quadratic_probing_search(int *hashtable, int key, int *empty)
{
/* If the key is found in the hash table, the function returns the index of the hashtable where the key is inserted, otherwise it
returns (-1) if the key is not found */
int j = 0, hk;
hk = key % SIZE;
while(j < SIZE)
{
if((empty[hk] 0) && (hashtable[hk]
For linear probing it is a bad idea to let the hash table get nearly full, because performance is degraded as the hash table gets filled.
In the case of quadratic probing, the situation is even more drastic. There is no guarantee of finding an empty cell once the table gets more than half full, or even before the table gets half full if the table size is not prime. This is because at most half of the table can be used as alternative locations to resolve collisions.
If the hash table size is b (a prime greater than 3), it can be proven that the first alternative locations including the initial location h(k) are all distinct and unique.
Suppose, we assume two of the alternative locations to be given by and , where 0 ≤ x, y ≤ (b / 2).
If these two locations point to the same key space, but x ≠ y. Then the following would have to be true,
As b (table size) is a prime greater than 3, either (x - y) or (x + y) has to be equal to zero.
Sine x and y are unique, (x - y) cannot be zero.
Also, since 0 ≤ x, y ≤ (b / 2) , (x + y) cannot be zero.
Thus, by contradiction, it can be said that the first (b / 2) alternative locations after h(k) are unique.
So an empty key space can always be found as long as at most (b / 2) locations are filled, i.e., the hash table is not more than half full.
See also
Not to be confused with wireless packet collision.In computer science, a collision or clash is a situation that occurs when two distinct pieces of data have the same hash value, checksum, fingerprint, or cryptographic digest....
Double hashing is a computer programming technique used in hash tables to resolve hash collisions, cases when two different values to be searched for produce the same hash key...
Linear probing is a scheme in computer programming for resolving hash collisions of values of hash functions by sequentially searching the hash table for a free location. This is accomplished using two values - one as a starting value and one as an interval between successive values in modular...
A hash function is any algorithm or subroutine that maps large data sets to smaller data sets, called keys. For example, a single integer can serve as an index to an array...
. The function used might even be 
if 
and 
are taken as zero.
.
. This would result in primary clustering and the larger the cluster grows, lesser will be the search efficiency for those items. Quadratic probing can be a more efficient algorithm in a closed hash table, since it better avoids the clustering problem that can occur with linear probing, although it is not immune. It also provides good memory caching because it preserves some locality of reference
be a hash function
to an integer in 
, where 
is the size of the table. Let the 
th probe position for a value 
be given by the function 
, where 
≠ 0. If 
, then 
degrades to a linear probe
and 
remain constant.
alternative locations including the initial location h(k) are all distinct and unique.
and 
, where 0 ≤ x, y ≤ (b / 2).
, then the probe sequence will be 
, a good choice for the constants are 
1/2, as the values 
for 
in 
are all distinct. This leads to a probe sequence of 
where the values increase by 
.
, most choices of 
and 
will make 
distinct for 
in 
. Such choices include 
1/2, 
1, and 
. Because there are only about m/2 distinct probes for a given element, it is difficult to guarantee that insertions will succeed when the load factor is > 1/2.