Algo 10-20
Chernoff Bounds
independent random variables,
Xi = 1 wp p ;
X1 . Xn
0 wp 1-p
X = sum from i = 1 to n (Xi)
mu = E[X] = np
Pr[|X - mu| > S*mu]
< 2e^(-s^2/3*mu)
for all 0 < S < 1
SET DISCREPAN
Algo 9-27
Set Cover
twelve elements
elements e1 . en
sets s1 . sm
cover collection of Sj's such that e1 in Sj
generalization of vertex cover
pick the one with most uncovered elements
greedy set covevr
Randomized Algorithms
#DNF
Input: f boolean variables x1 . 1n
3SAT clauses Cj 1 <= j <= m
Cj = Xj1 V Xj2 V Xj3
xi in cfw_Xi, or -xi
3SAT is NP-complete, hard problem
does there exist at least one trut
Algo 9-29
LP methods for approximation algorithms
primal dual scheme eg weighted vertex cover
dual fitting
randomized rounding
weighted vertex cover problem
IP
min sum v in v c(v)*xv
such that xu + xv
Chapter 13
Randomized Algorithms
The idea that a process can be random is not a modern one; we can trace the notion far back into the history of human thought and certainly see its reections in gambli
CS174 Chernoff Bounds
Lecture 10
John Canny
Chernoff bounds are another kind of tail bound. Like Markoff and Chebyshev, they bound the total amount of probability of some random variable Y that is in
CS 4540, Advanced Algorithms
Homework 1
Mon, Aug 30, 2010
Due Fri, Sept 10, 2010
Problem 1
Exercise 3, pages 782(bottom)-784(top) of Chapter 13 Randomized Algorithms by Kleinberg and
Tardos (posted in
CS 4540, Advanced Algorithms
Homework 2
Fri, Sept 10, 2010
Due Fri, Sept 17, 2010
Problem 1
Motwani and Raghavan, Problem 4.1, page 97. Note: The purpose of this problem is to familiarize
you with the
3. Coupon Collection
T = time to collect all n coupons
E[T] = nHn
Pr[ T > 2knHn]
< 1/2*k
using Markov :
Pr[X > lambda E[T]
< 1/lambda
suppose only want to collect 9/10th of the coupons
gamma + longn <