Randomized Algorithms Randomized Algorithms CS648 Lecture 20 Probabilistic Method (part 1) Lecture 20 Probabilistic Method (part 1) 1

PROBABILISTIC METHOD 2

Probabilistic methods Methods that use Probability theory Randomized algorithm to prove deterministic combinatorial results 3

PROBLEM 1 HOW MANY MIN CUTS ? 4

Min-Cut 5

Algorithm for min-cut 6

Analysis of Algorithm for min-cut 7

Number of min-cuts 8

PROBLEM 2 HOW MANY ACUTE TRIANGLES ? 9

How many acute triangles 10

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Two stage sampling 14

Number of acute triangles 15

PROBLEM 3 SUM FREE SUBSET OF LARGE SIZE 16

Large subset that is sum-free 17

Large subset that is sum-free 18 To prove it, use the fact that mapping is 1-1 and uniform. and Linearity of expectation.

Large subset that is sum-free 19

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PROBLEM 4 LARGE CUT IN A GRAPH 22

Large cut in a graph 23

Large cut in a graph 24

Large cut in a graph 25

Large cut in a graph 26