1. Randomized Algorithms
CS648
Lecture 2
• Randomized Algorithm for Approximate Median
• Elementary Probability theory
1
2. RANDOMIZED MONTE CARLO ALGORITHM
FOR
APPROXIMATE MEDIAN
This lecture was delivered at slow pace and its flavor was that of a
tutorial.
Reason: To show that designing and analyzing a randomized
algorithm demands right insight and just elementary probability.
2
3. A simple probability exercise
3
4. 4
5. Approximate median
Definition: Given an array A[] storing n numbers and ϵ > 0, compute an
element whose rank is in the range [(1- ϵ)n/2, (1+ ϵ)n/2].
Best Deterministic Algorithm:
• “Median of Medians” algorithm for finding exact median
• Running time: O(n)
• No faster algorithm possible for approximate median
Can you give a short proof ?
5
6. ½ - Approximate median
A Randomized Algorithm
Rand-Approx-Median(A)
1. Let k  c log n;
2. S  ∅;
3. For i=1 to k
4.
x  an element selected randomly uniformly from A;
5.
S  S U {x};
6. Sort S.
7. Report the median of S.
Running time: O(log n loglog n)
6
7. Analyzing the error probability of
Rand-approx-median
n/4
Left Quarter
Elements of A arranged in
Increasing order of values
3n/4
Right Quarter
When does the algorithm err ?
To answer this question, try to characterize what
will be a bad sample S ?
7
8. Analyzing the error probability of
Rand-approx-median
n/4
Elements of A arranged in
Increasing order of values
Left Quarter
3n/4
Median of S
Right Quarter
Observation: Algorithm makes an error only if k/2 or more elements
sampled from the Right Quarter (or Left Quarter).
8
9. Analyzing the error probability of
Rand-approx-median
n/4
Elements of A arranged in
Increasing order of values
3n/4
Right Quarter
Left Quarter
¼
Exactly the same as the coin
tossing exercise we did !
9
10. Main result we discussed
10
11. ELEMENTARY PROBABILITY THEORY
(IT IS SO SIMPLE THAT YOU UNDERESTIMATE ITS ELEGANCE AND POWER)
11
12. Elementary probability theory
(Relevant for CS648)
• We shall mainly deal with discrete probability theory in this course.
• We shall take the set theoretic approach to explain probability theory.
Consider any random experiment :
o Tossing a coin 5 times.
o Throwing a dice 2 times.
o Selecting a number randomly uniformly from [1..n].
How to capture the following facts in the theory of probability ?
1. Outcome will always be from a specified set.
2. Likelihood of each possible outcome is non-negative.
3. We may be interested in a collection of outcomes.
12
13. Probability Space
Ω
13
14. Event in a Probability Space
A
Ω
14
15. Exercises
A randomized algorithm can also be viewed as a random experiment.
1. What is the sample space associated with Randomized Quick sort ?
2. What is the sample space associated with Rand-approx-median
algorithm ?
15
16. An Important Advice
In the following slides, we shall state well known equations
(highlighted in yellow boxes) from probability theory.
• You should internalize them fully.
• We shall use them crucially in this course.
• Make sincere attempts to solve exercises that follow.
16
17. Union of two Events
A
B
Ω
17
18. Union of three Events
A
B
C
Ω
18
19. Exercises
19
20. Conditional Probability
20
21. Exercises
• A man possesses five coins, two of which are double-headed, one is
double-tailed, and two are normal. He shuts his eyes, picks a coin at
random, and tosses it. What is the probability that the lower face of the
coin is a head ? He opens his eyes and sees that the coin is showing heads;
what it the probability that the lower face is a head ? He shuts his eyes
again, and tosses the coin again. What is the probability that the lower
face is a head ? He opens his eyes and sees that the coin is showing heads;
what is the probability that the lower face is a head ? He discards this
coin, picks another at random, and tosses it. What is the probability that it
shows heads ?
21
22. Partition of sample space and
an “important Equation”
B
Ω
22
Be the first to comment