CS 662 Theory of Parallel Algorithms
Chernoff Bounds
[To Lecture Notes Index]
San Diego State University  This page last updated March 21, 1996, 1996
Bernoulli Trial
 Experiment with only two possible outcomes: success and failure
p = probability of success
q = probability of failure
q + p = 1
X(n) = number of successes during n independent Bernoulli trials
P[ event ] = the probability of event occurring
We have:


Chernoff Bounds
1)
2)
Set
and using 2) we get
3)
Example
What is the probability of getting 25 or fewer heads in 100 coin tosses?
p = .5
n = 100
= .5

n    
100  0.9  45  0.77880078 
100  0.8  40  0.36787944 
100  0.7  35  0.10539922 
100  0.6  30  0.01831564 
100  0.5  25  0.00193045 
100  0.4  20  0.00012341 
100  0.3  15  4.7851E06 
100  0.2  10  1.1254E07 
100  0.1  5  1.6052E09 
Randomized Algorithms
Las Vegas type algorithm
 Always generates correct answer

 Complexity is measured in expected value or the probability that a certain
bound will be exceeded
Monte Carlo type algorithm
 The algorithm will make errors but with a small probability