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10_rand_II - Chapter 10 Randomized Algorithms II By Sariel...

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Chapter 10 Randomized Algorithms II By Sariel Har-Peled , September 28, 2007 Version: 0.1 10.1 QuickSort with High Probability One can think about QuickSort as playing a game in rounds. Every round, QuickSort picks a pivot, splits the problem into two subproblems, and continue playing the game recursively on both subproblems. If we track a single element in the input, we see a sequence of rounds that involve this element. The game ends, when this element find itself alone in the round (i.e., the subproblem is to sort a single element). Thus, to show that QuickSort takes O ( n log n ) time, it is enough to show, that every element in the input, participates in at most 32 ln n rounds with high enough probability. Indeed, let X i be the event that the i th element participates in more than 32 ln n rounds. Let C QS be the number of comparisons performed by QuickSort . A comparison between a pivot and an element will be always charged to the element. And as such, the number of compar- isons overall performed by QuickSort is bounded by i r i , where r i is the number of rounds the i th element participated in (the last round where it was a pivot is ignored). We have that α = Pr C QS 32 n ln n Pr i X i n i = 1 Pr [ X i ] . Here, we used the union rule , that states that for any two events A and B , we have that Pr [ A B ] Pr [ A ] + Pr [ B ]. Assume, for the time being, that Pr [ X i ] 1 / n 3 . This implies that α n i = 1 Pr [ X i ] n i = 1 1 / n 3 = 1 n 2 . Namely, QuickSort performs at most 32 n ln n comparisons with high probability. It follows, that QuickSort runs in O ( n log n ) time, with high probability, since the running time of QuickSort is proportional to the number of comparisons it performs. This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
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