308-203A Introduction to Computing II Lecture 8: Sorting 2

1
308-203AIntroduction to Computing IILecture 8
Sorting 2
Fall Session 2000
2
So far.
1. Bubblesort - intuitive implementation O(n2)
suboptimal 2. MergeSort - Divide-and-Conquer O
(n log n) worst-case optimal
3
The Problem with MergeSort
We do a lot of copying when we MERGE
N/2
N/2
8, 34, 51, 78, 82
2, 17, 64, 91, 123
2, 8, 17, 34, 51, 64, 78, 82, 91, 123
New Array size N
4
In Java, for example...
int merge(int a, int b) int c
new int a.length b.length // merge a
and b into c . return c
5
The Problem with MergeSort

• To merge we need O(n) extra space
• Copying bigger hidden constants
• Sorting-in-place would be preferable
• (if we can do it in optimal n log n)
• Divide-and-Conquer was good, can we do
• the same thing with an operation other than
• merge??

6
Quicksort
1. Pick pivot
64, 8, 78, 34, 82, 51, 17, 2, 91, 123
2. Partition
8, 34, 51, 17, 2 , 64, 78, 82, 91, 123
2. Recursion
2, 8, 17, 34, 51, 64, 78, 82, 91, 123
7
Quicksort
Quicksort(int A, int start, int end) int
pivot A0 int midpoint
Partition(pivot, A) Quicksort(A, start,
midpoint) Quicksort(A, midpoint1, end)
8
Running Time

• Depends significantly on choice of pivot
• Worst-case O(n2)
• Best-case O( n log n)

Proofs on separate handout
9
The Worst-Case is WORSE!

• Worst-case is O(n2) as bad as Bubblesort
• A clever trickFooling-the-Adversary
• 1. pick a random pivot
• 2. You hit a bad case only when very unlucky
• Average-case O( n log n)

Proof beyond the scope of this course
10
Randomized Quicksort
Quicksort(int A, int start, int end) int
pivot A randomInt(start, end) int
midpoint partition(pivot, A) Quicksort(A,
start, midpoint) Quicksort(A, midpoint1,
end)
11
Any questions?