1. CSCE-221 Selection 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? 
k=3

4. Quick-Select
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 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
(pivot = 6) l r

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=( )
k=2, S=( )
k=2, S=( )
k=1, S=( )
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
Survey: on eCampus

11. References