QS simple Background

Added complexity.

Author: LindaStern

src/algorithms/explanations/QSExp.md

@@ -5,7 +5,10 @@
 Quicksort is a divide and conquer algorithm. It first rearranges the input
 array into two smaller sub-arrays: the (relatively) low elements and the
 (relatively) high elements. It then recursively sorts each of the sub-arrays.
-The steps for Quicksort are:
+
+### Sorting using Basic Quicksort
+
+The steps for basic Quicksort, used in this animation are:
 
 * Pick the rightmost element of the array, called a pivot.
 
@@ -15,12 +18,29 @@ The steps for Quicksort are:
 
 The base case of the recursion is arrays of size one or zero, which are in order by definition, so they never need to be sorted.
 
+
+### Complexity
+
+Time complexity:
+<code>
+   Average case     <i>O(nlogn)</i>
+   Worst case       <i>O(n<sup>2</sup>)</i>
+   Best case        <i>O(nlogn)</i>
+
+Note: Worst case in simple quicksort occurs when a file is already sorted, since the partition is highly asymmetrical. Improvements such as median-of-three quicksort make a significant improvement, although worst case behaviour is still possible.  
+</code>
+
+Space complexity is O(1), that is, no extra space is required.
+
+
+
+### Development of Quicksort
+
 The first version of quicksort was published by Tony Hoare in 1961 and
 quicksort remains the *fastest* sorting algorithm on average (subject to
 various caveats).  The pivot selection and partitioning steps can be
 done in *many* different ways and the choice of specific implementation
 details and computer hardware can significantly affect the algorithm's
 performance. In 1975, Robert Sedgewick completed a Ph.D. thesis on this
 single algorithm.  Our presentation here is influenced by the original
-Hoare version and some of Sedgewick's adaptations.
-
+Hoare version and some of Sedgewick's adaptations.