Open Addressing | Linear Probing | Collision

Collision Resolution Techniques-

 

Before you go through this article, make sure that you have gone through the previous article on Collision Resolution Techniques.

 

We have discussed-

  • Hashing is a well-known searching technique.
  • Collision occurs when hash value of the new key maps to an occupied bucket of the hash table.
  • Collision resolution techniques are classified as-

 

 

In this article, we will discuss about Open Addressing.

 

Open Addressing-

 

In open addressing,

  • Unlike separate chaining, all the keys are stored inside the hash table.
  • No key is stored outside the hash table.

 

Techniques used for open addressing are-

  • Linear Probing
  • Quadratic Probing
  • Double Hashing

 

Operations in Open Addressing-

 

Let us discuss how operations are performed in open addressing-

 

Insert Operation-

 

  • Hash function is used to compute the hash value for a key to be inserted.
  • Hash value is then used as an index to store the key in the hash table.

 

In case of collision,

  • Probing is performed until an empty bucket is found.
  • Once an empty bucket is found, the key is inserted.
  • Probing is performed in accordance with the technique used for open addressing.

 

Search Operation-

 

To search any particular key,

  • Its hash value is obtained using the hash function used.
  • Using the hash value, that bucket of the hash table is checked.
  • If the required key is found, the key is searched.
  • Otherwise, the subsequent buckets are checked until the required key or an empty bucket is found.
  • The empty bucket indicates that the key is not present in the hash table.

 

Delete Operation-

 

  • The key is first searched and then deleted.
  • After deleting the key, that particular bucket is marked as “deleted”.

 

NOTE-

 

  • During insertion, the buckets marked as “deleted” are treated like any other empty bucket.
  • During searching, the search is not terminated on encountering the bucket marked as “deleted”.
  • The search terminates only after the required key or an empty bucket is found.

 

Open Addressing Techniques-

 

Techniques used for open addressing are-

 

1. Linear Probing-

 

In linear probing,

  • When collision occurs, we linearly probe for the next bucket.
  • We keep probing until an empty bucket is found.

 

Advantage-

 

  • It is easy to compute.

 

Disadvantage-

 

  • The main problem with linear probing is clustering.
  • Many consecutive elements form groups.
  • Then, it takes time to search an element or to find an empty bucket.

 

Time Complexity-

 

Worst time to search an element in linear probing is O (table size).

 

This is because-

  • Even if there is only one element present and all other elements are deleted.
  • Then, “deleted” markers present in the hash table makes search the entire table.

 

2. Quadratic Probing-

 

In quadratic probing,

  • When collision occurs, we probe for i2‘th bucket in ith iteration.
  • We keep probing until an empty bucket is found.

 

3. Double Hashing-

 

In double hashing,

  • We use another hash function hash2(x) and look for i * hash2(x) bucket in ith iteration.
  • It requires more computation time as two hash functions need to be computed.

 

Comparison of Open Addressing Techniques-

 

Linear Probing
Quadratic Probing
Double Hashing
Primary Clustering
YesNoNo
Secondary Clustering
YesYesNo
Number of Probe Sequence
(m = size of table)
mmm2
Cache performance
BestLies between the twoPoor

 

Conclusions-

 

  • Linear Probing has the best cache performance but suffers from clustering.
  • Quadratic probing lies between the two in terms of cache performance and clustering.
  • Double caching has poor cache performance but no clustering.

 

Load Factor (α)-

 

Load factor (α) is defined as-

 

 

In open addressing, the value of load factor always lie between 0 and 1.

 

This is because-

  • In open addressing, all the keys are stored inside the hash table.
  • So, size of the table is always greater or at least equal to the number of keys stored in the table.

 

PRACTICE PROBLEM BASED ON OPEN ADDRESSING-

 

Problem-

 

Using the hash function ‘key mod 7’, insert the following sequence of keys in the hash table-

50, 700, 76, 85, 92, 73 and 101

 

Use linear probing technique for collision resolution.

 

Solution-

 

The given sequence of keys will be inserted in the hash table as-

 

Step-01:

 

  • Draw an empty hash table.
  • For the given hash function, the possible range of hash values is [0, 6].
  • So, draw an empty hash table consisting of 7 buckets as-

 

 

Step-02:

 

  • Insert the given keys in the hash table one by one.
  • The first key to be inserted in the hash table = 50.
  • Bucket of the hash table to which key 50 maps = 50 mod 7 = 1.
  • So, key 50 will be inserted in bucket-1 of the hash table as-

 

 

Step-03:

 

  • The next key to be inserted in the hash table = 700.
  • Bucket of the hash table to which key 700 maps = 700 mod 7 = 0.
  • So, key 700 will be inserted in bucket-0 of the hash table as-

 

 

Step-04:

 

  • The next key to be inserted in the hash table = 76.
  • Bucket of the hash table to which key 76 maps = 76 mod 7 = 6.
  • So, key 76 will be inserted in bucket-6 of the hash table as-

 

 

Step-05:

 

  • The next key to be inserted in the hash table = 85.
  • Bucket of the hash table to which key 85 maps = 85 mod 7 = 1.
  • Since bucket-1 is already occupied, so collision occurs.
  • To handle the collision, linear probing technique keeps probing linearly until an empty bucket is found.
  • The first empty bucket is bucket-2.
  • So, key 85 will be inserted in bucket-2 of the hash table as-

 

 

Step-06:

 

  • The next key to be inserted in the hash table = 92.
  • Bucket of the hash table to which key 92 maps = 92 mod 7 = 1.
  • Since bucket-1 is already occupied, so collision occurs.
  • To handle the collision, linear probing technique keeps probing linearly until an empty bucket is found.
  • The first empty bucket is bucket-3.
  • So, key 92 will be inserted in bucket-3 of the hash table as-

 

 

Step-07:

 

  • The next key to be inserted in the hash table = 73.
  • Bucket of the hash table to which key 73 maps = 73 mod 7 = 3.
  • Since bucket-3 is already occupied, so collision occurs.
  • To handle the collision, linear probing technique keeps probing linearly until an empty bucket is found.
  • The first empty bucket is bucket-4.
  • So, key 73 will be inserted in bucket-4 of the hash table as-

 

 

Step-08:

 

  • The next key to be inserted in the hash table = 101.
  • Bucket of the hash table to which key 101 maps = 101 mod 7 = 3.
  • Since bucket-3 is already occupied, so collision occurs.
  • To handle the collision, linear probing technique keeps probing linearly until an empty bucket is found.
  • The first empty bucket is bucket-5.
  • So, key 101 will be inserted in bucket-5 of the hash table as-

 

 

To gain better understanding about Open Addressing,

Watch this Video Lecture

 

Next Article- Separate Chaining Vs Open Addressing

 

Get more notes and other study material of Data Structures.

Watch video lectures by visiting our YouTube channel LearnVidFun.

Summary
Open Addressing | Linear Probing | Collision
Article Name
Open Addressing | Linear Probing | Collision
Description
Open Addressing is a collision resolution technique used for handling collisions in hashing. Techniques Used- Linear Probing, Quadratic Probing, Double Hashing. In linear probing, the next bucket is linearly probed.
Author
Publisher Name
Gate Vidyalay
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