Randomized Algorithms Randomized Algorithms CS648 Lecture 2 Randomized Algorithm for Approximate Median Elementary Probability theory Lecture 2 Randomized Algorithm for Approximate Median Elementary Probability theory 1

RANDOMIZED MONTE CARLO ALGORITHM FOR APPROXIMATE MEDIAN 2 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.

A simple probability exercise 3

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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 5 Can you give a short proof ?

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

Analyzing the error probability of Rand-approx-median 7 Elements of A arranged in Increasing order of values n/4n/4 3n/4 Right QuarterLeft Quarter When does the algorithm err ? To answer this question, try to characterize what will be a bad sample S ?

Analyzing the error probability of Rand-approx-median Observation: Algorithm makes an error only if k/2 or more elements sampled from the Right Quarter (or Left Quarter). 8 n/4n/4 Left QuarterRight Quarter Elements of A arranged in Increasing order of values 3n/4 Median of S

Analyzing the error probability of Rand-approx-median 9 Elements of A arranged in Increasing order of values n/4n/4 3n/4 Right QuarterLeft Quarter Exactly the same as the coin tossing exercise we did ! ¼

Main result we discussed 10

ELEMENTARY PROBABILITY THEORY (IT IS SO SIMPLE THAT YOU UNDERESTIMATE ITS ELEGANCE AND POWER) 11

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

Probability Space 13 Ω

Event in a Probability Space 14 A Ω

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

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

Union of two Events 17 A B Ω

Union of three Events 18 A BC Ω

Exercises 19

Conditional Probability 20

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

Partition of sample space and an “important Equation” 22 Ω B

Exercises 23

Independent Events 24 P(A ∩ B) = P(A) · P(B)

Exercises 25