1. Lecture 9 Randomized Algorithms

2. Quicksort Goal: Sort a list of numbers a[] = {4, 2, 8, 6, 3, 1, 7, 5}
Algorithm:
Pick a random pivot number (say 3)
Partition the array into numbers smaller and larger than the pivot ({2, 1}, {4, 8, 6, 7, 5})
Recursively sort the two parts.

3. Recursion Consider the possible choices for the first pivot
Let Xn be a random variable that represents the running time of QuickSort on n numbers.
𝔼 𝑋𝑛 = 𝑖=1 𝑛 Pr 𝑝𝑖𝑣𝑜𝑡=𝑖 𝔼[𝑋𝑛|𝑝𝑖𝑣𝑜𝑡=𝑖] = 1 𝑛 𝑖=1 𝑛 𝔼 𝑋 𝑖−1 +𝐴𝑛+𝔼[ 𝑋 𝑛−𝑖 ]
Left Part
Split cost
Right Part

4. QuickSelection
Goal: Given an array of numbers Find the k-th smallest number.
Example:
a[] = {4, 2, 8, 6, 3, 1, 7, 5} k = 3
Output = 3

5. Recursion Consider the possible choices for the first pivot
Let Xn be a random variable that represents the running time of QuickSelect on n numbers.
𝔼 𝑋𝑛 = 𝑖=1 𝑛 Pr 𝑝𝑖𝑣𝑜𝑡=𝑖 𝔼[𝑋𝑛|𝑝𝑖𝑣𝑜𝑡=𝑖] = 1 𝑛 𝑖=1 𝑘−1 𝔼 𝑋 𝑛−𝑖 𝑛 𝑖=𝑘+1 𝑛 𝔼 𝑋 𝑖−1 +𝐴𝑛
Right Part
Left Part
Split cost

6. Rejection Sampling
Problem: Given a random variable X, Pr[X=i] = pi want to generate a random variable Y, such that Pr[Y=i] = qi
Rejection sampling: Sample X, then with probability 𝑞𝑖 𝐶𝑝𝑖 , keep the sample otherwise throw the sample away.

7. Example: Biased Coin
Suppose we only have a biased coin (the coin lands on heads with probability p > ½). How do we simulate a fair coin?
[Von Neumann] Toss the coin twice, if the result is HT, claim Heads; if the result is TH, claim Tails; if the result is HH or TT, retry.
Question: How many coin tosses do we need to do until we get the result?

8. Monte Carlo Algorithm Problem: Compute the area of a circle.
Count = 0
FOR i = 1 to n
Generate x, y from [-1,1]
IF x2+y2 <= 1 THEN
Count = Count + 1
RETURN 4.0*Count/n