The document discusses linear probing as an open addressing technique for hash tables. It describes linear probing as placing an identifier in the next empty slot after its home bucket if a collision occurs. The document provides an example to demonstrate linear probing. It also discusses linear probing with chaining, where a chain of collided elements is maintained through pointers rather than replacing elements. Pseudocode for insertion and searching in linear probing tables is provided.

1. Sanjivani Rural Education Society’s
Sanjivani College of Engineering, Kopargaon-423 603
(An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune)
NACC ‘A’ Grade Accredited, ISO 9001:2015 Certified
Department of Computer Engineering
(NBA Accredited)
Subject- Data Structures-I(CO203)
Unit III- Closed Hashing

2. • Open Addressing/Closed Hashing:
1. Separate chaining requires additional memory space for pointers.
2. With this method, a hash collision is resolved by probing, or searching through
alternative locations in the array until an empty bucket is found.
3. As all the elements are stored inside the table, a large memory space is
needed for open addressing.
DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 2

3. • There are three different popular methods for open addressing techniques. These
methods are −
• Linear Probing
• Quadratic Probing
• Double Hashing
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4. Linear Probing
• In this method, hash address of an identifier x is obtained
• If collision occurs, the identifier is place in the next empty slot after its home
bucket
• If slot hash(x) % S is full, then we try (hash(x) + 1) % S
• If (hash(x) + 1) % S is also full, then we try (hash(x) + 2) % S
• If (hash(x) + 2) % S is also full, then we try (hash(x) + 3) % S
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5. DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 5

6. • Table size=11
• Hash function: h(x)= x mod 11
• Insert keys:20,30,2,13,25,24,10,9
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h(20)= 20 % 11=9
h(30)= 30 % 11=8
h(2)= 2 % 11 =2
h(13)= 13 % 11=2
h(25)= 25 % 11=3
h(24)= 24 % 11=2
h(10)= 10 % 11=10
h(9)= 9 % 11=9
0
1
2
3
4
5
6
7
8
9
10
2
30
20
0
1
2
3
4
5
6
7
8
9
10
2
13
30
20
0
1
2
3
4
5
6
7
8
9
10
2
13
25
30
20
0
1
2
3
4
5
6
7
8
9
10
2
13
25
24
30
20
Insert 13 Insert 25 Insert 24
0
1
2
3
4
5
6
7
8
9
10
2
13
25
24
30
20
10
Insert 10
0
1
2
3
4
5
6
7
8
9
10
9
2
13
25
24
30
20
10
Insert 9

7. DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 7
• Let us see the following example to get better idea. If we have some elements like
{15, 47, 23, 34, 85, 97, 65, 89, 70}. And our hash function is h(x) = x mod 7.

8. • Advantages :
1. Simple to implement.
2. Best case time complexity=O(1)
• Disadvantages :
1. If an element is not found at computed location. Searching goes in linear way.
2. Hash table may become Full.
3. Make clustering: many consecutive elements form groups.
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9. #define MAX 10
int insert_lb(int key)
{
j=key % MAX;
for(i=0;i < MAX ; i++)
{
if(ht[j]== -1)
{
ht[j]=key;
DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 9
return(j);
}
j=(j+1)%MAX;
}
return(-1);
}
Procedure to insert key in hash table

10. • Procedure to search key in hash table
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int search_lb()
{
cout<<“Enter the key to be search”;
cin>>key;
j= key % MAX;
for(i=0;i<MAX;i++)
{
if(ht[j]==key)
return(j);
else
j=(j+1)%MAX;
}
return -1;
}

11. Linear Probing with Chaining
• In this method, another field get added into the table.
• In this field, the address of the colliding data can be stored with the first colliding
element in the chain table, without replacement.
• Records can be easily located.
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12. • For example, consider elements:
131, 3, 4, 21, 61, 6, 71, 8, 9
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0
1
2
3
4
5
6
7
8
9
-1 -1
131 -1
-1 -1
3 -1
4 -1
-1 -1
-1 -1
-1 -1
-1 -1
-1 -1
h(131)= 131%10=1
h(3)= 3%10=3
h(4)= 4%10=4
h(21)= 21%10=1
h(61)= 61%10=1
h(6)= 6%10=6
h(71)= 71%10=1
h(8)= 8%10=8
h(9)= 9%10=9
0
1
2
3
4
5
6
7
8
9
-1 -1
131 2
21 5
3 -1
4 -1
61 -1
-1 -1
-1 -1
-1 -1
-1 -1
Insert 21,61
0
1
2
3
4
5
6
7
8
9
-1 -1
131 2
21 5
3 -1
4 -1
61 -1
6 -1
-1 -1
-1 -1
-1 -1
Insert 6
0
1
2
3
4
5
6
7
8
9
-1 -1
131 2
21 5
3 -1
4 -1
61 7
6 -1
71 -1
-1 -1
-1 -1
Insert 71
0
1
2
3
4
5
6
7
8
9
-1 -1
131 2
21 5
3 -1
4 -1
61 7
6 -1
71 -1
8 -1
9 -1
Insert 8,9

13. From the example, you can see that the chain is maintained from the number who
demands for location 1.
First number 131 comes, we place it at index 1. Next comes 21, but collision occurs
so by linear probing we will place 21 at index 2, and chain is maintained by writing
2 in chain table at index 1.
Similarly next comes 61, by linear probing we can place 61 at index 5 and chain will
maintained at index 2.
Thus any element which gives hash key as 1 will be stored by linear probing at
empty location but a chain is maintained so that traversing the hash table will be
efficient.
DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 13

14. • Insert following keys into hash table with linear probing with chaining without
replacement method. Use H(x)= x % 10 function
1. 0,1,4,7,64,89,11,33,55 Use H(x)= x % 10 function
2. 10,12,20,18,15 Use H(x)= x % 8 function
3. 11,32,41,54,33,1 Use H(x)= x % 10 function
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15. DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 15
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9

16. • Disadvantages:
1. Main purpose of hashing is not achieved, because all records are not
mapped at correct position or mapped at wrong position/address.
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17. DEPARTMENT OF COMPUTER ENGINEERING, Sanjivani COE, Kopargaon 17
void insertwor(int b[][2])
{
int r,s,t,a;
printf("n Enter Data");
scanf("%d",&a);
s= a % MAX;
if(b[s][0]==-1)
b[s][0]=a;
else
{
while(1)
{
printf("nTrue Collision Occurs");
getch();
if(b[s][1]== -1)
{
t=s;
while(b[s][0]!=-1)
{
s=(s+1)%MAX;
if(s==t)
{
printf("Overflow");
break;
}
}

18. if(s!=t)
{
b[s][0]=a;
b[t][1]=s;
}
break;
} //if end
else
s=b[s][1];
}//while end
}//else end
}
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19. void displaywor(int b[][2])
{
int i,r,s,t,a;
printf("nINDEXtDATAtCHAIN");
for(i=0;i<max;i++)
{
if(b[i][0]!=-1)
{
printf("n%d",i);
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printf("t%d",b[i][0]);
//if(b[i][1]!=-1)
printf("t%d",b[i][1]);
}
}
getch();
}

20. void searchwor(int b[][2])
{
int r,s,a;
printf("Enter Data Which U Want To Search");
scanf("%d",&a);
s=a%MAX;
while(1)
{
if(b[s][0]==a)
break;
s=b[s][1];
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if(s==-1)
{
printf("Not Found");
break;
}
}//end of while
else
{
printf("nIndex Is %d:",s);
printf("nData Is %d:",b[s][0]);
printf("n Index Which Is Chained From It:%d",b[s][1]);
}
}

21. Linear Probing with Replacement
• In this method, if another identifier ‘Y’ is occupying the position of an identifier
‘X’, X replaces it and then ‘Y’ is relocated to a new position.
• e.g. 11,32,41,54,33
• First 3 will get placed at address 1,2,3,4. but when 33 need to be placed it
position is occupied by 41.
• Therefore 33 replaces 41 and 41 inserted in new empty bucket i.e. 5 and chain
from element 11 at position 1 is modified
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22. 11,32,41,54,33,22
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0
1
2
3
4
5
6
7
8
9
-1 -1
11 3
32 -1
41 -1
54 -1
-1 -1
-1 -1
-1 -1
-1 -1
-1 -1
0
1
2
3
4
5
6
7
8
9
-1 -1
11 5
32 -1
33 -1
54 -1
41 -1
-1 -1
-1 -1
-1 -1
-1 -1
Insert 33
0
1
2
3
4
5
6
7
8
9
-1 -1
11 5
32 6
33 -1
54 -1
41 -1
22 -1
-1 -1
-1 -1
-1 -1
Insert 22

23. Insert following data 12,1,4,3,7,8,10,2,5,14 using linear probing with chaining with
replacement. Calculate average cost or no. of comparisons
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0
1
2
3
4
5
6
7
8
9
10 -1
1 -1
12 6
3 -1
4 9
5 -1
2 -1
7 -1
8 -1
14 -1
Avg. Cost= (1+1+1+1+1+1+1+2+1+2)/no. of buckets
=12/10
=1.2

24. • Pseudo-code to insert Data in Linear probing with replacement
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25. • The load factor for an open addressing technique should be 0.5. For separate
chaining technique, the load factor is 1.
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