This document discusses different hashing collision resolution techniques including linear probing, quadratic probing, and double hashing. Linear probing resolves collisions by probing the next slot in the hash table, but can result in clustering. Quadratic probing resolves collisions by probing slots at i^2 distance from the original slot, improving on linear probing by reducing clustering. Double hashing uses two hash functions to better distribute elements in the table and avoids clustering.

1. Hashing
Group A
Indian Institute of Information Technology Vadodara
May 8, 2018
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2. Overview
1 First Section
Subsection Example
2 Second Section
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3. Paragraphs of Text
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ligula non lorem vulputate fermentum accumsan neque mollis.
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libero. Aliquam vel dui orci, a porta odio. Nullam id suscipit ipsum.
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tincidunt libero, quis commodo erat ullamcorper id.
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4. Collision Resolution Technique
Linear Probing.
Quadratic Probing
Double Hashing
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5. Linear Probing
Linearly probe for the next slot.
LP(K, i) = [hash(K) + i] mod m, ∀i ∈ 0, 1, 2, ..., m − 1
hash(K)= K mod m
K = key, m = size of address space.
i values runs in hash address space.
Example: Keys = 55,98 ,50, 62, 79, 79, 45, 85, 99, 29
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6. Linear Probing
LP(55,0) = [(55 mod m)+0] mod m =(55 mod 10) mod 10
= 5 mod 10 =5
LP(98,0) = [(98 mod 10)+0] mod 10 = 8 mod 10 = 8
LP(50,0) = 0
LP(62,0) = 2
LP(79,0) = 9
LP(45,0) = 5 (collision 1)
LP(45,1) = 6
LP(85,0) = 5 (collision 2)
LP(85,1) = 6 (collision 3)
LP(85,2) = 7
LP(99,0) = 9 (collision 4)
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7. Linear Probing
LP(99,0) = 9 (collision 4)
LP(99,1) = 0 (collision 5)
LP(99,2) = 1
LP(29,0) = 9
LP(29,1) = 0 (collision 6)
LP(29,2) = 1 (collision 7)
LP(29,3) = 2 (collision 8)
LP(29,4) = 3
Total collision is 8
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8. Quadratic Probing
QP(K, i) = [(hash(K) + C1.i + C2.i2)] mod m
C1 = 1, C2 = 1
We look for i2th slot in ith iteration.
Example: Keys = 55,98 ,50, 62, 79, 79, 45, 85, 99, 29
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9. Quadratic Probing
QP(55,0) = [55 mod 10] =5
QP(98,0) = 8
QP(50,0) = 0
QP(62,0) = 2
QP(79,0) = 9
QP(45,0) = 5 (collision 1)
QP(45,1) = [5 + 1 ∗ 1 + 1 ∗ 12]mod 10 = 7 mod 10 =7
QP(85,0) = 5 (collision 2)
QP(85,1) = [5 + 1 ∗ 1 + 1 ∗ 12]mod 10 = 7 mod 10 =7 (collision 3)
QP(85,2) = [5 + 1 ∗ 2 + 1 ∗ 22]mod 10 = 11 mod 10 =1
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10. Quadratic Probing
QP(99,0) = 9 (collision 4)
QP(99,1) = [9 + 1 ∗ 1 + 1 ∗ 12]mod 10 = 1
QP(99,2) = 5 (collision 6)
QP(99,3) = 1 (collision 7)
QP(99,4) = 9 (collision 8)
QP(99,5) = 9 (collision 9)
QP(99,6) = 1 (collision 10)
QP(99,7) = 5 (collision 11)
QP(99,8) = 1 (collision 12)
QP(99,9) = 9 (collision 13)
QP(29,0) — (10 collision)
Total Collision 23
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11. Double Hashing
DH(K, i) = [(hash1(K) + i ∗ hash2(K)] mod m.
∀i ∈ 0, 1, 2, ........m − 1.
hash2(K) =1+[k mod (m-2)]
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12. Double Hashing
DH(55,0) = [55mod10 + 0 ∗ [1 + (55mod8)] mod 10
[5 + 0 ∗ (1 + 7)]mod 10
5 mod 10 = 5
DH(98,0) = 8
DH(50,0) = 0
DH(62,0) = 2
DH(79,0) = 9
DH(45,0) = 5
DH(45,1) = [5 + 1 ∗ [1 + (45mod8)]]mod10]
(5 + 6) mod 10 =1
DH(85,0) = 5
DH(85,1) = [5 + 1 ∗ [1 + 5]] mod 10 =1
DH(85,2) = 7
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13. Double Hashing
DH(99,0) = 9
DH(99,1) = 3
DH(29,0) —- 10 collision
Thus total collision =14
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14. Comparison
Linear probing
Best cache performance.
Suffers from clustering.
Easy to compute.
Quadratic Probing
Cache performance and clustering
Double Hashing
Poor cache performance but no clustering
More computation time because of two hash functions.
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15. Comparison
Block 1
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Integer lectus nisl,
ultricies in feugiat rutrum, porttitor sit amet augue. Aliquam ut tortor
mauris. Sed volutpat ante purus, quis accumsan dolor.
Block 2
Pellentesque sed tellus purus. Class aptent taciti sociosqu ad litora
torquent per conubia nostra, per inceptos himenaeos. Vestibulum quis
magna at risus dictum tempor eu vitae velit.
Block 3
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venenatis rutrum, ipsum neque consectetur orci, sed blandit justo nisi ac
lacus.
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16. Multiple Columns
Heading
1 Statement
2 Explanation
3 Example
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consectetur adipiscing elit. Integer
lectus nisl, ultricies in feugiat rutrum,
porttitor sit amet augue. Aliquam ut
tortor mauris. Sed volutpat ante
purus, quis accumsan dolor.
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17. Table
Treatments Response 1 Response 2
Treatment 1 0.0003262 0.562
Treatment 2 0.0015681 0.910
Treatment 3 0.0009271 0.296
Table: Table caption
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18. Theorem
Theorem (Mass–energy equivalence)
E = mc2
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19. Verbatim
Example (Theorem Slide Code)
begin{frame}
frametitle{Theorem}
begin{theorem}[Mass--energy equivalence]
$E = mc^2$
end{theorem}
end{frame}
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20. Figure
Uncomment the code on this slide to include your own image from the
same directory as the template .TeX file.
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21. Citation
An example of the cite command to cite within the presentation:
This statement requires citation [Smith, 2012].
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22. References
John Smith (2012)
Title of the publication
Journal Name 12(3), 45 – 678.
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23. The End
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