Quick Sort

1 Background

Quicksort was contributed by Hoare in 1962. Hundreds of subsequent papers!
Over the decades of teaching, we "know" that only a few students understand Quicksort. Why? Bad teaching? Poor student?

For an array of size n, the best we can do is O(n lg n). Quicksort and Heapsort are both O(n lg n).
Our interest is not in the perf analysis. Nor in the performance itself.

Signature: Quicksort(lo, hi).
Array to be sorted: A[] of integers. Low index 1, high index n. Global. Quicksort(1, n). n >= 1.
Notation: A[lo .. hi] stands for the set of elements A[lo], .., A[hi].
Pre:: true;;
Post::
1. hi >= lo and sorted(A, lo, hi) and permuted(A.old, A, lo, hi), or
2. hi < lo and A[] is unchanged, i.e., equal(A.old, A, lo, hi).

Quicksort(lo, hi). Focus only on A[lo] .. A[hi]. Assume hi >= lo. Leave other elements of A[] untouched.
Pick a value z := A[x], lo >= x >= hi. (The choice of z has an impact on performance.)
Permute the elements of A[] so that any A[lo .. p] <= z < any A[p+1 .. hi].
1. permuted(A.old, A, lo, hi),
2. p may equal lo - 1 (so A[lo .. p] is empty),
3. p may equal hi (so A[p+1 .. hi] is empty),
Quizz: At the end of the above, is A[x] equal to z?
Then do
1. Quicksort(1, p)
2. Quicksort(p+1, n)
All of this is elegantly discussed in (cite) [Foley and Hoare 1971].

http://algs4.cs.princeton.edu/lectures/23Quicksort.pdf From the book by Sedgewick and Wayne.
http://gauss.ececs.uc.edu/Courses/C321/html/quicksort.java.html
Exercise: Suppose we give you the primitive operation: exchange(i, j) that excahnges the i-th element of the array with the j-th element. If we use this op exclusively, the permutation property is guaranteed. Rewrite Quicksort using exchange(). No array[x] := e are permitted.

Hoare, Charles AR. "Quicksort." The Computer Journal 5.1 (1962): 10-16. https://comjnl.oxfordjournals.org/content/5/1/10.full.pdf. Hoare is a Turning Award winner. Required Reading.
Foley, Michael, and Charles Antony Richard Hoare. "Proof of a recursive program: Quicksort." The Computer Journal 14.4 (1971): 391-395. Recommended Reading.
Sedgewick, R. Quicksort Ph. D. Dissertation, Advisor: Donald Knuth, Computer Science Dept., Stanford University (May 1975) Report STAN-CS-75-492, 1975. Reference.
Sedgewick, Robert, and Jon Bentley. "Quicksort is optimal." Knuthfest, Stanford University, Stanford (2002). https://www.cs.princeton.edu/~rs/talks/QuicksortIsOptimal.pdf Recommended Reading.
Bentley, Jon L., and M. Douglas McIlroy. "Engineering a sort function." Software: Practice and Experience 23.11 (1993): 1249-1265.
http://bigocheatsheet.com/ [Big-Oh Cheatsheet] Know Thy Complexities! Includes "Array Sorting Algorithms". Highly Recommended Visit.
http://algs4.cs.princeton.edu/lectures/23Quicksort.pdf From the book by Sedgewick and Wayne. Highly Recommended Visit.