Randomized Algorithms
CS648
Lecture 22
Chebyshev Inequality
Method of Bounded Difference
1
Chernoff Bound
Theorem : Suppose be independent Bernoulli random variables with
parameters , that is, takes value 1 with probability and 0 with probability .
Let

Randomized Algorithms
CS648
Lecture 21
Random Walk and Electric Networks
1
OVERVIEW AND MOTIVATION
2
What do we know about Random walk till now?
We have discussed uniform random walk on
A line.
A complete graph.
Two complete graphs joined by an edge (M

Randomized Algorithms
CS648
Lecture 23
Probabilistic methods - II
1
3-SAT PROBLEM
Its solution can be viewed as an application of Probabilistic method.
2
3-SAT Problem
Notations:
A Boolean variable: a variable that can take value true or false.
A term:

Randomized Algorithms
CS648
Lecture 25
Derandomization using conditional expectation
A probability gem
1
DERANDOMIZATION USING
CONDITIONAL EXPECTATION
2
Problem 1: Large cut in a graph
Problem: Let be an undirected graph on vertices and edges. Compute a

Randomized Algorithms
CS648
Lecture 24
Random bit complexity
Derandomization
1
Random bit complexity
Input
A Randomized Algorithm
(for Min-Cut, QuickSort, RIC,)
Random Bit generator
Definition : The total number of random bits taken from the Random Bit

Randomized Algorithms
CS648
Lecture 19
algorithm for Min-cut in a graph
1
Overview
1. Recap of the previous lecture
time Monte Carlo algorithm for min-cut
2. Knowledge of recurrence
To design efficient algorithm
3. time Monte Carlo algorithm for min-cu

Randomized Algorithms
CS648
Lecture 18
Approximate Distance Oracles
Algorithm for Min-cut : part 1
1
APPROXIMATE DISTANCE ORACLES
2
All-Pairs Shortest Paths
A graph on
Notations and Terminologies :
vertices
edges
A path from to : a sequence , , where

Randomized Algorithms
CS648
Lecture 17
Miscellaneous applications of Backward analysis
1
MINIMUM SPANNING TREE
2
Minimum spanning tree
d
17
h
v
19
6
3
22
1
b
a
10
7
16
12
u
4
3
9
2
c
11
5
x
y
18
15
13
3
Minimum spanning tree
d
17
h
v
19
6
3
22
1
b
a
16
10

Randomized Algorithms
CS648
Lecture 14
Expected duration of a randomized experiment
Part II
1
REVISITING
SOME DISCRETE MATHEMATICS
2
Recurrence 1
For any ,
Question: What is the smallest value of such that for a given ?
Answer: ?
For any ,
for some
Ques

Randomized Algorithms
CS648
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
M

Randomized Algorithms
CS648
Lecture 16
Randomized Incremental Construction
(Backward analysis)
1
PROBLEM 1
FIND-MIN PROBLEM
2
Find-Min algorithm
A
1
2
: no. of times is updated.
Find-Min(A[1.])
cfw_
A[1];
For to do
cfw_ if (A[] )
A[] ;
return ;
Pr

Randomized Algorithms
CS648
Lecture 15
Randomized Incremental Construction
(building the background)
1
Partition Theorem
A set of events , defined over a probability space (,P) is said to induce a
partition of if
=
=
for all
B
Partition Theorem:
P(B) =

Randomized Algorithms
CS648
Lecture 10
Random Sampling
part-II
(To find a subset with desired property)
1
Overview
There is a huge list (1 million) of blood donors.
Unfortunately the blood group information is missing at present.
We need a donor with bloo

Randomized Algorithms
CS648
Lecture 9
Random Sampling
part-I
(Approximating a parameter)
1
Overview of the Lecture
Randomization Framework for estimation of a parameter
1. Number of balls from a bag
2. Size of transitive closure of a directed graph
. An I

Randomized Algorithms
CS648
Lecture 12
Hashing - II
1
RECAP OF LAST LECTURE
Problem Definition
called universe
and
Examples:
,
Aim
Given a set , build a data structure storing s.t. we can answer in O(1) time :
Does ? for any given .
Hashing
Hash
tab

Randomized Algorithms
CS648
Lecture 11
Hashing - I
1
Problem Definition
called universe
and
Examples:
,
Aim
Maintain a data structure for storing to support the search query :
Does ? for any given .
Solutions
Solutions with worst case guarantees
Soluti

Randomized Algorithms
CS648
Lecture 13
Expected duration of a randomized experiment
Part I
1
COUPON COLLECTOR PROBLEM
2
Coupon Collector Problem
is a bag containing distinct coupons.
There
Each coupon has a unique label from [].
Experiment:
Repeat
1. Sel

Randomized Algorithms
CS648
Lecture 4
Linearity of Expectation with applications
(Most important tool for analyzing randomized algorithms)
1
RECAP FROM THE LAST LECTURE
2
Random variable
Definition: A random variable defined over a probability space (,P)

Randomized Algorithms
CS648
Lecture 7
Two applications of Union Theorem
Balls into Bin experiment : Maximum load
Randomized Quick Sort: Concentration of the running time
1
Union theorem
Theorem: Suppose there is an event defined over a probability spac

Randomized Algorithms
CS648
Lecture 6
Reviewing the last 3 lectures
Application of Fingerprinting Techniques
1-dimensional Pattern matching
Preparation for the next lecture.
1
Randomized Algorithms
discussed till now
Randomized algorithm for Approximat

Randomized Algorithms
CS648
Lecture 3
Two fundamental problems
Balls into bins
Randomized Quick Sort
Random Variable and Expected value
1
BALLS INTO BINS
CALCULATING PROBABILITY OF SOME INTERESTING EVENTS
2
Balls into Bins
1
1
2
3 4
2
3
5
i
m-1 m
n

Randomized Algorithms
CS648
Lecture 8
Tools for bounding deviation of a random variable
Markovs Inequality
Chernoff Bound
1
Markovs Inequality and Chernoff bound were stated and proved in this
lecture class in an interactive manner providing all intuiti

CS648 : Randomized Algorithms
Semester I, 2013-14, CSE, IIT Kanpur
Why does Quick Sort behave as expected almost always ?
It is well known that the average running time of quick sort is O(n log n) time. We analysed randomized
quick sort earlier and showed

Randomized Algorithms
CS648
Lecture 2
Randomized Algorithm for Approximate Median
Elementary Probability theory
1
RANDOMIZED MONTE CARLO ALGORITHM
FOR
APPROXIMATE MEDIAN
This lecture was delivered at slow pace and its flavor was that of a
tutorial.
Reas

Randomized Algorithms
CS648
Lecture 1
1
Overview of the lecture
What is a randomized algorithm ?
Motivation
The structure of the course
2
What is a randomized algorithm ?
3
Deterministic Algorithm
Output
Input
Algorithm
The output as well as the runnin