Randomized Algorithms CS 648 Lecture 12 Hashing II

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Randomized Algorithms CS 648 Lecture 12 Hashing - II 1

Randomized Algorithms CS 648 Lecture 12 Hashing - II 1

RECAP OF LAST LECTURE

RECAP OF LAST LECTURE

Problem Definition •

Problem Definition •

Hashing •

Hashing •

Collision •

Collision •

Universal Hash Family • This definition appears strange in the beginning! But we shall

Universal Hash Family • This definition appears strange in the beginning! But we shall soon see that there is a very natural way to arrive at this definition.

HASHING WITH OPTIMAL SPACE AND WORST CASE O(1) SEARCH TIME

HASHING WITH OPTIMAL SPACE AND WORST CASE O(1) SEARCH TIME

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

Optimal space hashing with worst case O(1) search time •

WHY SUCH A DEFINITION FOR UNIVERSAL HASH FAMILY ?

WHY SUCH A DEFINITION FOR UNIVERSAL HASH FAMILY ?

Why does hashing work so well in Practice ? • similar question while Quick

Why does hashing work so well in Practice ? • similar question while Quick Sort Randomized Quick Sort

Universal Hash Family •

Universal Hash Family •

A SIMPLE AND COMPACT UNIVERSAL HASH FAMILY

A SIMPLE AND COMPACT UNIVERSAL HASH FAMILY

The starting point •

The starting point •

mod operation • 12 20 12 11

mod operation • 12 20 12 11

mod operation • 1 2 3 4 5 6 3 6 2 5 1

mod operation • 1 2 3 4 5 6 3 6 2 5 1 4

mod operation • 1 2 3 4 5 6 3 4 6 1 2

mod operation • 1 2 3 4 5 6 3 4 6 1 2 5 5 2 1 6 4 3 Not possible

mod operation •

mod operation •

• Students must realize that it is a necessary condition and not sufficient

• Students must realize that it is a necessary condition and not sufficient condition for collision. To get an idea, study the example given at the last slide of this lecture.

Example • 1 2 3 4 5 6 2 4 6 1 3 5

Example • 1 2 3 4 5 6 2 4 6 1 3 5 3 6 2 5 1 4 4 1 5 2 6 3 5 3 1 6 4 2 6 5 4 3 2 1

Homework: •

Homework: •

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Lecture 8 Randomized Algorithms Randomized Algorithm Algorithms thatLecture 8 Randomized Algorithms Randomized Algorithm Algorithms that
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INDEX 1 Hashing 2 Hashing 3 1 HashingINDEX 1 Hashing 2 Hashing 3 1 Hashing
Hashing Hashing 2 Hashing Again a dynamic setHashing Hashing 2 Hashing Again a dynamic set