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CSCE-221 Selection

CSCE-221 Selection. Emil Thomas 03/07/19. Based on Slides by Prof. Scott Schafer. 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.

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CSCE-221 Selection

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  1. CSCE-221Selection Emil Thomas 03/07/19 Based on Slides by Prof. Scott Schafer

  2. The Selection Problem • Given an integer k and n elements x1, x2, …, xn, taken from a total order, find the kth smallest element in this set. • Also called order statistics, ith order statistic is ith smallest element • Minimum - k=1 - 1st order statistic • Maximum - k=n - nth order statistic • Median - k=n/2 • etc

  3. The Selection Problem • Naïve solution - SORT! • 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

  4. 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 Note: Partition same as Quicksort Quick-Select x x L G E k > |L|+|E| k’ = k - |L| - |E| k < |L| |L| < k < |L|+|E| (done)

  5. In-Place Partitioning • Perform the partition using two indices to split S into two subarrays L and E&G (L:Less, E&G: Equal&Greater) • Repeat until l and r cross: • Scan l to the right until finding an element > x. • Scan r to the left until finding an element < x. • Swap elements at indices l and r l r 3 2 5 1 0 7 3 5 9 2 7 9 8 9 7 9 6 (pivot = 6) l r 3 2 5 1 0 7 3 5 9 2 7 9 8 9 7 9 6

  6. In-Place Partition PseudoCode //This Algo uses the last element as pivot Algorithm q = inPlacePartition(S,first,last) Input: sequence Array S, Starting index first, and ending index last Output: Returns an index q such that S[first .. q] <= S[q + 1, …last] Code: pivot S[last] l first – 1 //l is incremented below r last // start from end leaving pivot while (TRUE) { do {l++} while(S[l] < pivot) do {r--} while (pivot < =S[r]); if (l >= r) { return r; } swap(S, l, r); } • q = inPlacePartition(S, first, last) returns the index q such that • L= S[first … q] • E&G = S[q+1 … last]

  7. Quick-Select InLabDemo&Discussion

  8. Quick-Select Visualization • An execution of quick-select can be visualized by a recursion path • Each node represents a recursive call of quick-select, and stores k and the remaining sequence k=5, S=(7 4 9 2 6 5 1 8 3) k=2, S=(7 4 9 6 5 8) k=2, S=(7 4 6 5) k=1, S=(7 6 5) Expected running time is O(n) 5

  9. Quick-Select PseudoCode //Input: Array S[first .. last] and the rank(k) //Output: returns the kth smallest element in S Quick_select(S, first, last, k) { If (first == last) //Base case :One element array return S[first]; q = inPlacePartition(S,first,last); L_Size = q – first + 1 //Size of the L array If (k <= L_Size) { return Quick_select(S, first, q, k); //The L array } else return Quick_select(S, q+1, last, k - L_size) //E&G array } }

  10. Exercise & Survey • http://people.tamu.edu/~yanzp/courses/csce221/notes/QuickSelectSortLABA.docx • Survey: on eCampus

  11. References • http://faculty.cs.tamu.edu/schaefer/teaching/221_Spring2017/Lectures/Sorting.ppt • https://en.wikipedia.org/wiki/Quickselect

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