Sorting: Advanced Techniques

1
Sorting Advanced Techniques
2
Outline

• Divide-and-conquer sorting algorithms
• Mergesort
• Quicksort
• For each algorithm
• Idea
• Example
• Implementation
• Running time for each algorithm

3
Mergesort Basic Idea

• Divide and Conquer approach
• Idea
• Merging two sorted array takes O(n) time
• Split an array into two takes O(1) time

4
Mergesort Merge Implementation

• Implement operation to merge two sorted arrays
  into one sorted array
• public static void merge(int A, int B, int
  C)

Assume A and B are sorted and C A B
5
Mergesort Algorithm

• If the number of items to sort is 0 or 1, return.
• Recursively sort the first and second half
  separately.
• Merge the two sorted halves into a sorted group.

6
Mergesort Example
split
7
Mergesort Example
split
merge
8
Mergesort Example
merge
9
Mergesort Implementation

• MergeSort implementation (and driver method)
• void mergeSort(int array)
• mergeSort(array, 0, a.length-1)
• void mergeSort(int a, int left, int right)
• if(left lt right)
• int centre (left right)/2
• mergeSort(a, left, centre)
• mergeSort(a, center1, right)
• merge(a, left, center1, right)

How to merge the two subarrays of A without any
temporary space?
10
Mergesort Merge Implementation

• Implement operation to merge two sorted
  subarrays
• public static void merge(int A, int l, int c,
  int r)

11
Mergesort Analysis

• Running Time O(n log n)
• Why?

12
Quicksort Basic Idea

• Divide and Conquer approach
• Quicksort(S) algorithm
• If the number of items in S is 0 or 1, return.
• Pick any element v ? S. This element is called
  the pivot.
• Partition S v into two disjoint groups
• L x ? S v x ? v and
• R x ? S v x ? v
• Return the result of Quicksort(L), followed by v,
  followed by Quicksort(R).

13
Quicksort Select Pivot
-1
58
4
2
42
3
43
40
0
1
3
65
14
Quicksort Partition
65
2
42
0
3
40
4
43
-1
58
3
1
15
Quicksort Recursive Sort Merge
43
65
58
42
2
1
3
0
-1
3
4
16
Quicksort Partition Algorithm 1
17
Quicksort Partition Algorithm 1
18
Quicksort Partition Algorithm 1
19
Quicksort Partition Algorithm 2
move pivot out of the way
while gt pivot right--
while lt pivot left
20
Quicksort Partition Algorithm 2
CROSSING
move pivot back
Quicksort recursively
Quicksort recursively
21
Quicksort Implementation

• static void QuickSort(int a, int low, int high)
  
• if(high lt low) return // base case
• pivot choosePivot(a) // select best
  pivot
• int ilow, jhigh-1
• swap(a,pivot,aj) // move pivot out of
  the way
• while(i lt j)
• // find large element starting from left
• while(ilthigh ailtpivot) i
• // find small element starting from right
• while(jgtlow ajgtpivot) j--
• // if the indexes have not crossed, swap
• if(igtj) swap(a, i, j)
• swap(a,i,high-1) // restore pivot

22
Quicksort Analysis

• Partitioning takes
• O(n)
• Merging takes
• O(1)
• So, for each recursive call, the algorithm takes
  O(n)
• How many recursive calls does a quick sort need?

23
Quicksort Choosing The Pivot

• Ideal pivot
• Median element
• Common pivot
• First element
• Element at the middle
• Median of three

24
Further Reading

• http//telaga.cs.ui.ac.id/WebKuliah/IKI10100/resou
  rces/animation/
• Chapter 8 Sorting Algorithm