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Selection - Quick-Sort 8:56 AM The Selection Problem Given...

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Selection 1 Selection Selection 2 The Selection Problem Given an integer k and n elements x 1 , x 2 , …, x n , taken from a total order, find the k-th smallest element in this set. Of course, we can sort the set in O(n log n) time and then index the k-th element. Can we solve the selection problem faster? 7 4 9 6 2 2 4 6 7 9 k=3 Selection 3 Quick-Select (§10.7) Quick-select is a randomized selection algorithm based on the prune-and-search paradigm: Prune : pick a random element x (called pivot ) and partition S into L elements less than x E elements equal x G elements greater than x Search : depending on k, either answer is in E , or we need to recur on either L or G x x L G E k < | L | |L| < k < | L |+| E| (done) k > | L |+| E| k’ = k - |L| - |E| Selection 4 Partition We partition an input sequence as in the quick-sort algorithm: We remove, in turn, each element y from S and We insert y into L , E or G , depending on the result of the comparison with the pivot x Each insertion and removal is at the beginning or at the end of a sequence, and
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