Week 11 Sorting Algorithms
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Week 11 Sorting Algorithms
Sorting
Sorting Algorithms
• A sorting algorithm is an algorithm that puts elements of a list in a certain order.
• We need sorting algorithm in lots of areas: Science Engineering Business……
https://en.wikipedia.org/wiki/Sorting_algorithm
Quadratic Sorting Algorithms
• We are given n records to sort. • There are a number of simple sorting
algorithms whose worst and average case performance is quadratic O(n2):– Selection sort– Insertion sort– Bubble sort
Sorting an Array of IntegersSorting an Array of Integers
• Example: we are given an array of six integers that we want to sort from smallest to largest
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Selection sort
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Start by finding the smallest entry.
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Swap the smallest entry with the first entry.
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Swap the smallest entry with the first entry.
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Part of the array is now sorted.
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Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Find the smallest element in the unsorted side.
Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• Swap with the front of the unsorted side.
Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• We have increased the size of the sorted side by one element.
Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• The process continues...
Sorted side Unsorted side
Smallestfrom
unsorted
Smallestfrom
unsorted
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• The process continues...
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Swap
with
front
Swap
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• The process continues...
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is bigger
Sorted sideis bigger
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• The process keeps adding one more number to the sorted side.
• The sorted side has the smallest numbers, arranged from small to large.
Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• We can stop when the unsorted side has just one number, since that number must be the largest number. [0] [1] [2] [3] [4] [5]
Sorted side Unsorted side
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The Selection Sort AlgorithmThe Selection Sort Algorithm
• The array is now sorted.
• We repeatedly selected the smallest element, and moved this element to the front of the unsorted side.
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Selection sort
• animation
template <class Item>void selection_sort(Item data[ ], size_t n){ size_t i, j, smallest; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 0; i < n-1 ; ++i) {
// find smallest in unsorted part of arraysmallest = i;for(j = i+1; j < n; ++j) if(data[smallest] > data[j]) smallest = j;
// put it at front of unsorted part of array (swap)temp = data[i];data[i] = data[smallest];data[smallest] = temp;
}}
Selection Time Sort Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 1 to n-1
find smallest key in unsorted part of array
swap smallest item to front of unsorted array
decrease size of unsorted array by 1
Selection Time Sort Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 1 to n-1 O(n)
find smallest key in unsorted part of array O(n)swap smallest item to front of unsorted arraydecrease size of unsorted array by 1
• Selection sort analysis: O(n2)
template <class Item>void selection_sort(Item data[ ], size_t n){ size_t i, j, smallest; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 0; i < n-1 ; ++i) {
// find smallest in unsorted part of arraysmallest = i;for(j = i+1; j < n; ++j) if(data[smallest] > data[j]) smallest = j;
// put it at front of unsorted part of array (swap)temp = data[i];data[i] = data[smallest];data[smallest] = temp;
}}
Outer loop:O(n)
template <class Item>void selection_sort(Item data[ ], size_t n){ size_t i, j, smallest; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 0; i < n-1 ; ++i) {
// find smallest in unsorted part of arraysmallest = i;for(j = i+1; j < n; ++j) if(data[smallest] > data[j]) smallest = j;
// put it at front of unsorted part of array (swap)temp = data[i];data[i] = data[smallest];data[smallest] = temp;
}}
Outer loop:O(n)
Inner loop:O(n)
Insertion sort
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
• The Insertion Sort algorithm also views the array as having a sorted side and an unsorted side.
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
• The sorted side starts with just the first element, which is not necessarily the smallest element.
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Sorted side Unsorted side
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
• The sorted side grows by taking the front element from the unsorted side...
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
• ...and inserting it in the place that keeps the sorted side arranged from small to large.
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
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The Insertion Sort AlgorithmThe Insertion Sort Algorithm
• Sometimes we are lucky and the new inserted item doesn't need to move at all.
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The Insertionsort AlgorithmThe Insertionsort Algorithm
• Sometimes we are lucky twice in a row.
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How to Insert One ElementHow to Insert One Element
Copy the new element to a separate location.
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How to Insert One ElementHow to Insert One Element
Shift elements in the sorted side, creating an open space for the new element.
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How to Insert One ElementHow to Insert One Element
Shift elements in the sorted side, creating an open space for the new element.
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How to Insert One ElementHow to Insert One Element
Continue shifting elements...
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How to Insert One ElementHow to Insert One Element
Continue shifting elements...
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How to Insert One ElementHow to Insert One Element
...until you reach the location for the new element.
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How to Insert One ElementHow to Insert One Element
Copy the new element back into the array, at the correct location.
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How to Insert One ElementHow to Insert One Element
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• The last element must also be inserted. Start by copying it...
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Sorted ResultSorted Result
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Insertion Sort
• animation
template <class Item>void insertion_sort(Item data[ ], size_t n){ size_t i, j; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 1; i < n; ++i) {
// take next item at front of unsorted part of array // and insert it in appropriate location in sorted part of arraytemp = data[i];for(j = i; data[j-1] > temp and j > 0; --j)
data[j] = data[j-1]; // shift element forward
data[j] = temp; }}
Insertion Sort Time Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 1 to n-1
take next key from unsorted part of array
insert in appropriate location in sorted part of array:
for j = i down to 0,
shift sorted elements to the right if key > key[i]
increase size of sorted array by 1
Insertion Sort Time Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 1 to n-1
take next key from unsorted part of array
insert in appropriate location in sorted part of array:
for j = i down to 0,
shift sorted elements to the right if key > key[i]
increase size of sorted array by 1
Outer loop:O(n)
Insertion Sort Time Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 1 to n-1
take next key from unsorted part of array
insert in appropriate location in sorted part of array:
for j = i down to 0,
shift sorted elements to the right if key > key[i]
increase size of sorted array by 1
Outer loop:O(n)
Inner loop:O(n)
template <class Item>void insertion_sort(Item data[ ], size_t n){ size_t i, j; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 1; i < n; ++i) {
// take next item at front of unsorted part of array // and insert it in appropriate location in sorted part of arraytemp = data[i];for(j = i; data[j-1] > temp and j > 0; --j)
data[j] = data[j-1]; // shift element forward
data[j] = temp; }}
O(n)
O(n)
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Bubble sort
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed.
[0] [1] [2] [3] [4] [5] Swap?
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Yes!
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Swap?
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Swap?
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Swap?
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Yes!
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Swap?
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• The Bubble Sort algorithm looks at pairs of entries in the array, and swaps their order if needed. [0] [1] [2] [3] [4] [5]
Yes!
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Repeat.
[0] [1] [2] [3] [4] [5] Swap? No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? No.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? Yes.
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[1] [2] [3] [4] [5] [6]
The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? Yes.
0
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40
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70
[1] [2] [3] [4] [5] [6]
The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? No.
0
10
20
30
40
50
60
70
[1] [2] [3] [4] [5] [6]
The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Loop over array n-1 times, swapping pairs of entries as needed.
[0] [1] [2] [3] [4] [5] Swap? No.
0
10
20
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[1] [2] [3] [4] [5] [6]
The Bubble Sort AlgorithmThe Bubble Sort Algorithm
• Continue looping, until done.
[0] [1] [2] [3] [4] [5] Swap? Yes.
Bubble sort
• animation
template <class Item>void bubble_sort(Item data[ ], size_t n){ size_t i, j; Item temp;
if(n < 2) return; // nothing to sort!!
for(i = 0; i < n-1; ++i) {
for(j = 0; j < n-1;++j) if(data[j] > data[j+1]) // if out of order, swap!
{ temp = data[j]; data[j] = data[j+1]; data[j+1] = temp; }
}}
template <class Item>void bubble_sort(Item data[ ], size_t n){ size_t i, j; Item temp; bool swapped = true;
if(n < 2) return; // nothing to sort!! for(i = 0; swapped and i < n-1; ++i) { // if no elements swapped in an iteration,
// then elements are in order: done!for(swapped = false, j = 0; j < n-1;++j)
if(data[j] > data[j+1]) // if out of order, swap! { temp = data[j]; data[j] = data[j+1]; data[j+1] = temp;
swapped = true; }
}}
Bubble Sort Time Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 0 to n-1
for j =0 to n-2
if key[j] > key[j+1] then swap
if no elements swapped in this pass through array, done.
otherwise, continue
Bubble Sort Time Analysis
• In O-notation, what is:– Worst case running time for n items?– Average case running time for n items?
• Steps of algorithm:for i = 0 to n-1
for j =0 to n-2
if key[j] > key[j+1] then swap
if no elements swapped in this pass through array, done.
otherwise, continue
O(n)O(n)
• Selection Sort, Insertion Sort, and Bubble Sort all have a worst-case time of O(n2), making them impractical for large arrays.
• But they are easy to understand, easy to program, easy to debug.
• Insertion Sort also has good performance when the array is nearly sorted to begin with.
Timing and Other Issues Timing and Other Issues
Next class
• More sophisticated sorting algorithms are needed when good performance is needed in all cases for large arrays.
• Next class: Merge Sort, Heap Sort. – O(N log N)
• Review the “recursion algorithm” and “heap structure” we discussed last week.
Merge sort
Merge sort
1. Partitioning: Divide the unsorted list into n sublists, each containing 1 element (a list of 1 element is considered sorted).
2. Merging: Recursively merge sublists to produce new sorted sublists until there is only 1 sublist remaining. This will be the sorted list.
Recursion
86
5 2 4 7 1 3 2 6
5 2 4 7 1 3 2 6
5 2 4 7 1 3 2 6
87
0
0
0
Main IndexMain Index ContentsContents88
The Merge Algorithm Example
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
s u b lis t Bs u b lis t A
firs t las tm id
The merge algorithm takes a sequence of elements in a vector v having index range [first, last). The sequence consists of two ordered sublists separated by an intermediate index, called mid.
Main IndexMain Index ContentsContents8989 Main IndexMain Index ContentsContents
The Merge Algorithm… (Cont…)
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
7
s u b lis t Bs u b lis t A
in d exA in d exB
t em p Vect o r
7 1 0
t em p Vect o r
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
s u b lis t Bs u b lis t A
in d exA in d exrB
Main IndexMain Index ContentsContents9090 Main IndexMain Index ContentsContents
The Merge Algorithm… (Cont…)
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
7 1 0 1 2
s u b lis t Bs u b lis t A
in d exA in d exB
t em p Vect o r
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
7 1 0 1 2 1 7 1 9 2 1 2 5
s u b lis t Bs u b lis t A
in d exA in d exB
t em p Vect o r
Main IndexMain Index ContentsContents91
The Merge Algorithm… (Cont…)
7 1 0 1 9 2 5 1 2 1 7 2 1 3 0 4 8
7 1 0 1 2 1 7 1 9 2 1 2 5 3 0 4 8
s u b lis t Bs u b lis t A
in d exA in d exB
t em p Vect o r
las t
7 10 12 17 19 21 25 30 48
t e m p Ve c t o r
7 10 12 17 19 21 25 30 48
firs t la s t
Merge sort
• animation
mergeSort()
Main IndexMain Index ContentsContents9494 Main IndexMain Index ContentsContents
Partitioning and Merging of Sublists in mergeSort()
(25 10 7 19 3 48 12 17 56 30 21)
(25 10 7 19 3 )
(7 19 3 )(25 10 )
[3 7 10 19 25]
(48 12 17 56 30 21 )
(56 30 21 )(48 12 17 )
[12 17 21 30 48 56]
[3 7 10 12 17 19 21 25 30 48 56]
(10 )
[10 25]
(25 ) (19 3 )(7 )
[3 7 19]
(56 )
[21 30 56]
(12 17 )(48 )
[12 17 48]
(21 )(3 )
[3 19]
(19 )
(30 21 )
[21 30]
(30 )(17 )
[12 17]
(12 )
Complexity of merge sort
Complexity of merge sort
Worst, best, average caseComplexity: O(n log n)
Heap sort
Heaps• For a maximum heap, the value of a parent is greater than or
equal to the value of each of its children. • For a minimum heap, the value of the parent is less than or
equal to the value of each of its children.• We assume that a heap is a maximum heap.
Heap sort
1. Build a heap of N elements – the maximum element is at the top of the heap
2. Perform N Delete operations– the elements are extracted in sorted order
Heap sort
• animation
Heap sort – Running Time Analysis
1. Build a binary heap of N elements O(N), see slide 21 of lecture ‘heap’
2. Perform N Delete operations– Each Delete operation takes O(log N), See slide 23 of lecture ‘heap’
Total: O(N log N)
Total time complexity: O(N) + O(N log N) O(N log N)
Reading
• Chapter 7
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