What does it mean for a value to be randomly selected How can we make use of randomness Monte Carlo Algorithms Don t always give the correct answer The runtime can be described consistently Las Vegas Algorithms They always give the correct answers Their runtime is not consistent Algorithm Select a value at random call it p Partition the list around p See if it was the median same number in each side of the partitioning If it is great If it wasn t oh well try again Question 1 Does this work Question 2 Is it a good algorithm Algorithm Select a value at random call it p Partition around p See if it was the median same number in each side of the partitioning If it wasn t then we have still found the xth smallest value in the list the value of x will be based on the size of the partitions If x is before the median take the right side and find the n 2 x th smallest Otherwise take the left side and find the n 2 th smallest Note If this ends up being a good idea we d end up coding general selection Question 1 Does this work Question 2 Is it a good algorithm How do we analyze the runtime of something like this Partitioning takes n 1 comparisons Random number generation may take some time we ll ignore this for now The recursion may or may not be needed and we don t know exactly how many values will be passed into that recursion T n n T The best case is easy we find it on the first shot and it s n 1 comparisons What about worst case and average case We will assume unique values in the list We ll round things and say the partitioning takes n comparisons We will look at worst expected runtime We ll compute assuming we have to look in the larger of the two sub lists which is true for median finding We won t worry about floor ceiling issues in this initial exploration
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