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Randomized Algorithms
CS648

1
Overview

To design efficient algorithm

A new tool to design an efficient algorithm

2
RECAP OF THE PREVIOUS LECTURE

3
Graph and Multi-graph

A multi-graph may have:
• More than one edge between a pair of vertices
• No self loop

4
Min-Cut

5
SOME BASIC FACTS

6
Min-Cut

How about cuts in multi-graph ?

7
8
9
10
11
12
Algorithm for min-cut

13
Algorithm for min-cut

14
Algorithm for min-cut

15
REVISITING RECURRENCES
In the following slides, we shall revisit common recurrences. Try to solve these
recurrences in a simple manner instead of using Master’s theorem. This will help
you develop a useful insight into recurrences. This insight will help you fine-tune
the previous inefficient algorithm and eventually lead to design (and analysis) of
a more efficient algorithm for min-cut.

16
Common recurrences

17
Common recurrences

18
FASTER MIN-CUT ALGORITHM

19
Revisiting algorithm for min-cut

We shall modify this algorithm to improve its success probability.
But we shall not allow any significant blow up in the running time.
20
Algorithm for min-cut

21
Key observations about
Min-Cut algorithm

recursively

22
Revised algorithm for min-cut

23
Faster algorithm for min-cut

24
25
Solving the recurrence

26
Solving the recurrence

27
Solving the recurrence

28
Solving the recurrence

proof

29
This lecture introduced many concepts which can be used for
design and analysis of efficient randomized algorithms. Try to
reflect upon these concept…

30

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Lecture 19-cs648