Lecture 9 Randomized Algorithms
Dive into the world of randomized algorithms with a focus on quicksort, quickselection, recursion, rejection sampling, biased coin simulation, and Monte Carlo algorithms. Explore concepts, examples, and applications in this fascinating field.
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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.
Recursion Consider the possible choices for the first pivot Let Xnbe a random variable that represents the running time of QuickSort on n numbers. ? ? ??= Pr ????? = ? ?[??|????? = ?] ?=1 1 ? ?=1 ? = ? ?? 1+ ?? + ?[?? ?] Split cost Right Part Left Part
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
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. ? ? ??= Pr ????? = ? ?[??|????? = ?] ?=1 1 ? ?=1 ? 1 ? ? ?? ? +1 = ? ?? 1+ ?? ? ?=?+1 Right Part Left Part Split cost
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 ?? ???, keep the sample otherwise throw the sample away.
Example: Biased Coin Suppose we only have a biased coin (the coin lands on heads with probability p > ). 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?
Monte Carlo Algorithm Problem: Compute the area of a circle. Monte-Carlo algorithm 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
