1 data structures and algorithms sorting. 2 sorting is the process of arranging a list of items...
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Data Structures and Algorithms
Sorting
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Sorting Sorting is the process of arranging a list of items
into a particular order
There must be some value on which the order is based
There are many algorithms for sorting a list of items
These algorithms vary in efficiency
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SORTING
We will examine several algorithms:
Selection Sort
Bubble Sort
Insertion Sort
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Selection SortThe approach of Selection Sort:
select one value and put it in its final place in the sort list
repeat for all other values
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In more detail:
find the smallest value in the list
switch it with the value in the first position
find the next smallest value in the list
switch it with the value in the second position
repeat until all values are placed
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Selection SortAn example:
original: 3 9 6 1 2 smallest is 1: 1 9 6 3 2 smallest is 2: 1 2 6 3 9 smallest is 3: 1 2 3 6 9 smallest is 6: 1 2 3 6 9
Pick any item and insert it into its proper place in a sorted sublist
repeat until all items have been inserted
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In more detail Selection Sort:
consider the first item to be sorted as a sub list (of one item)
insert the second item into the sorted sub list, shifting items as necessary to make room for the new addition
insert the third item into the sorted sublist (of two items), shifting as necessary
repeat until all values are inserted into their proper position
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Insertion Sort An example of an insertion sort occurs in everyday
life while playing cards.
To sort the cards in your hand you extract a card, shift the remaining cards, and then insert the extracted card in the correct place.
This process is repeated until all the cards are in the correct sequence. Both average and worst-case time is O(n2).
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Sorting Card players all know how to sort …
First card is already sortedWith all the rest,
Scan back from the end until you find the first card larger than the new one,
Move all the lower ones down one slot
insert it
A
K
10
J
Q
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Insertion Sort– Another n^2 sortAn example, :
original: 3 9 6 1 2
insert 9: 3 9 6 1 2
insert 6: 3 6 9 1 2
insert 1: 1 3 6 9 2
insert 2: 1 2 3 6 9
Each loop iteration produces a sorted sub list.
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Insertion Sort
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INSERTION SORTAssuming there are nn elements . elements .
For each entry, we may need to examine and shift up to nn - 1 other entries, resulting in a - 1 other entries, resulting in a
O(n2) algorithm.
The insertion sort is an in-place sort. That is, we sort the array in-place, no needed helper array.
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INSERTION SORTNo extra memory is required.
The insertion sort is also a stable sort.
Stable sorts retain the original ordering of keys when identical keys are present in the input data.
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Comparing Sorts Both Selection and Insertion sorts are similar in
efficiency
The both have outer loops that scan all elements,
and inner loops that compare the value of the outer loop with almost all values in the list
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Comparing Selection & Insertion Sort
Therefore approximately nn22 number of comparisons number of comparisons are made to sort a list of size nare made to sort a list of size n
We therefore say that these sorts are of order norder n22
Other sorts are more efficient: order n log2 n
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Sorting - Insertion sort
ComplexityFor each card
Scan for smaller card O(n)Shift cards down O(n)Insert O(1)Total O(n)
First card requires O(1), second O(2), …For n cards operations
O(n2) ii=1
n
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Sorting - Insertion sort
ComplexityFor each card
Scan O(n) O(O(loglog n) n)Shift up O(n)O(n)Insert O(1)Total O(n)O(n)
First card requires O(1), second O(2), …
For n cards operations O(n2) ii=1
n
Unchanged!Because the
shift up operationstill requires O(n)
time
Use binary search to find location!
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Sorting - BubbleFrom the first element
Exchange pairs if they’re out of orderExchange pairs if they’re out of order
Last one must now be the largest
Repeat from the first to n-1
Stop when you have only one element to check
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Bubble Sort
SWAP(a,b) { int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) { int i, j;
for( i=0; i < n; i++) { /* n passes thru the array */
/*From start to the end of unsorted part */ for( j=1; j < (n-i) ; j++) {
/* If adjacent items out of order, swap */ if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]); } }
}
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Bubble Sort - Analysisvoid SWAP(a,b) { int t; t=a; a=b; b=t; // see below}void SWAP(a,b) { int t; t=a; a=b; b=t; // see below}
void bubble( int a[], int n ) { int i, j; for( i=0; i < n; i++) { /* n passes thru the array */ /* From start to the end of unsorted part */ for(j=1; j < (n-i); j++) { /* if adjacent items out of order, swap */ if( a[j-1] > a[j] ) SWAP ( a[j-1], a[j]);} }
O(1) statement
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Bubble Sort - Analysisvoid bubble( int a[], int n ) { int i, j; for(i=0;i<n;i++) for(i=0;i<n;i++) { /* n passes thru the arrayn passes thru the array */
/* From start to the end of unsorted part */ for( j=1;j < (n-i); j++) {for( j=1;j < (n-i); j++) { /* If adjacent items out of order, swap */ if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]); } } }
Inner loopn-1, n-2, n-3, … , i iterations
O(1) statement
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Bubble Sort - Analysis
Overall
ii=n-1
1=
n(n+1)2
= O(n2)
n outer loop iterations inner loop iteration count
BubbleSort
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Complexity of the Bubble Sort
Thus:Thus:
((N-1) + (N-2) + N-1) + (N-2) + ...... + 2 + 1 = N*(N-2)/2 = O(N^2) + 2 + 1 = N*(N-2)/2 = O(N^2)
Thus Bubble Sort is an O(N^2) algorithmThus Bubble Sort is an O(N^2) algorithm. .
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Sorting - SimpleBubble sort
O(n2)Very simple!!! code – slow, should be retired.
Insertion sortSlightly better than bubble sort
Fewer comparisons
Also O(n2) Selection sort is similar
But HeapSort is O(n log n) Where would you use selection or
insertion sort?
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Simple Sorts Selection Sort or Insertion Sort
Use when n is small
Simple code compensates for low efficiency!
n^2 and n log n
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60
n
Tim
e n log n
n 2̂
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QuicksortEfficient sorting algorithm
Discovered by C.A.R. Hoare
Example of Divide and Conquer algorithm
Two phasesPartition phase
Divides the work into half
Sort phaseConquers the halves!
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QuicksortPartition
Choose a pivotFind the position for the pivot so that
all elements to the left are lessall elements to the right are greater
< pivot > pivotpivot
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Quicksort
ConquerApply the same algorithm to each half
< pivot > pivot
pivot< p’ p’ > p’ < p” p” > p”
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QuickSortquick_sort(left, right) { // Boundary Condition: when left >= right -- Boundary Condition: when left >= right --
do nothing do nothing // Place first item of current sublist in its
correct position, C, in the sublist itself, such that
// // List[i] <= List[C], for all i, First <= i List[i] <= List[C], for all i, First <= i
< C < C
// List[J] > List[C], for all J, C < J <= // List[J] > List[C], for all J, C < J <= Last Last
quick_sort (left, C-1); quick_sort (C+1, right);
}
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QuickSort codeint partition(int array [], int left, int right)int partition(int array [], int left, int right)
{{ int i = left, j = right;int i = left, j = right;
int tmp;int tmp;
int pivot = array[(left + right) / 2];int pivot = array[(left + right) / 2]; while (i <= j)while (i <= j)
{{
while (array[i] < pivot)while (array[i] < pivot)
i++;i++;
while (array[j] > pivot)while (array[j] > pivot)
j--;j--;
if (i <= j)if (i <= j)
{{
tmp = array[i];tmp = array[i];
array[i] = arrar[j];array[i] = arrar[j];
array[j] = tmp;array[j] = tmp;
i++;i++;
j--;j--;
}}
};};
return i; return i; }
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Quicksort
Implementationquicksort( int [] a, int low, int high ) { int pivot; /* Termination condition! */ if ( high > low ) { pivot = partition( a, low, high ); quicksort( a, low, pivot-1 ); quicksort( a, pivot+1, high ); } }
Divide
Conquer
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QuickSortvoid quickSort(int []array, int left, int right)
{
int index = partition(array, left, right);
// sorts the left side of the array
if (left < index - 1)
quickSort(array, left, index - 1);
// sorts the right side of the array
if (index < right)
quickSort(array, index, right);
}
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right; int pivot_item; pivot_item = a[low]; // pivot is first element in the array pivot = left = low; right = high;
while ( left < right ) { /* Move left while item < pivot */ while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */ while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right); } /* right is final position for the pivot */ a[low] = a[right]; a[right] = pivot_item; return right; }
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right; int pivot_item; pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high; while ( left < right ) {while ( left < right ) { /* Move left while item < pivot *//* Move left while item < pivot */ while( a[left] <= pivot_item ) left++;while( a[left] <= pivot_item ) left++; /* Move right while item > pivot *//* Move right while item > pivot */ while( a[right] >= pivot_item ) right--;while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right);if ( left < right ) SWAP(a,left,right); }} /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }}
This exampleuses int’s
to keep thingssimple!
23 12 15 38 42 18 36 29 27
low high
Any item will do as the pivot,choose the leftmost one!
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right; int pivot_item; pivot_item = a[low];
pivot = left = low; right = high; while ( left < right ) {while ( left < right ) { /* Move left while item < pivot *//* Move left while item < pivot */ while( a[left] <= pivot_item ) left++;while( a[left] <= pivot_item ) left++; /* Move right while item > pivot *//* Move right while item > pivot */ while( a[right] >= pivot_item ) right--;while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right);if ( left < right ) SWAP(a,left,right); }} /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }}
Set left and right markers
23 12 15 38 42 18 36 29 27
low highpivot: 23
left right
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Quicksort - Partition
int partition( int a[], int low, int high ) { int left, right ,pivot_item; pivot_item = a[low]; pivot = left = low; right = high;
while ( left < right ) { /* Move left while item < pivot */ while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */ while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right);if ( left < right ) SWAP(a,left,right); }} /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }}
Move the markers until they cross over
23 12 15 38 42 18 36 29 27
low highpivot: 23
left right
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Quicksort - Partitionint partition( int []a, int low, int high ) { int left, right;int left, right; int pivot_item;int pivot_item; pivot_item = a[low];pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high;
while ( left < right ) { /* Move left while item < pivot */
while( a[left] <= pivot_item ) left++; while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right);if ( left < right ) SWAP(a,left,right); }} /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }}
Move the left pointer while items <= pivot
23 12 15 38 42 18 36 29 27
low highpivot: 23
left right Move right Items >=pivot
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right;int left, right; int pivot_item;int pivot_item; pivot_item = a[low];pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high;
while ( left < right ) {while ( left < right ) { /* Move left while item < pivot *//* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a, left, right); } /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }
Swap the two itemson the wrong side of the value of pivot
23 12 15 38 42 18 36 29 27
low highpivot: 23
left right
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right;int left, right; int pivot_item;int pivot_item; pivot_item = a[low];pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high;
while ( left < right ) { /* Move left while item < pivot */
while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right);if ( left < right ) SWAP(a,left,right); }} /* right is final position for the pivot *//* right is final position for the pivot */ a[low] = a[right];a[low] = a[right]; a[right] = pivot_item;a[right] = pivot_item; return right;return right; }}
left and right have swapped over,
so stop
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low highpivot: 23
leftright
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right;int left, right; int pivot_item;int pivot_item; pivot_item = a[low];pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high;
while ( left < right ) { /* Move left while item < pivot */
while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */
while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right); } /* right is final position for the pivot */
a[low] = a[right]; a[right] = pivot_item; return right; }
Finally, swap the pivotand a[right}
23 12 15 18 42 38 36 29 27
low highpivot: 23
leftright
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Quicksort - Partition
int partition( int []a, int low, int high ) { int left, right;int left, right; int pivot_item;int pivot_item; pivot_item = a[low];pivot_item = a[low]; pivot = left = low;pivot = left = low; right = high;right = high;
while ( left < right ) { /* Move left while item < pivot */
while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */
while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right); } /* right is final position for the pivot */ a[low] = a[right]; a[right] = pivot_item;
return right; }
Return the positionof the pivot
18 12 15 23 42 38 36 29 27
low high
pivot: 23right
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Quicksort - Conquer
pivot
18 12 15 23 42 38 36 29 27pivot: 23
Recursivelysort left half
Recursivelysort right half
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Quicksort - Analysis
PartitionCheck every item once O(n)
ConquerDivide data in half O(log2n)
TotalProduct O(n log n)
But there’s a catch …………….
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Quicksort - The truth!
What happens if we use quicksorton data that’s already sorted(or nearly sorted)
We’d certainly expect it to perform well!
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Quicksort - The truth!
Sorted data
1 2 3 4 5 6 7 8 9
pivot
< pivot
?
> pivot
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Quicksort - The truth!
Sorted dataEach partition
produces
a problem of size 0and one of size n-1!
Number of partitions?
1 2 3 4 5 6 7 8 9
> pivot
2 3 4 5 6 7 8 9
> pivot
pivot
pivot
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Quicksort - The truth!
Sorted data
Each partitionproducesa problem of size 0and one of size n-1!
Number of partitions?n each needing time O(n)Total nO(n)
or O(n2)
? Quicksort is as bad as bubble or insertion sort
1 2 3 4 5 6 7 8 9
> pivot
2 3 4 5 6 7 8 9
> pivot
pivot
pivot
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Quicksort - The truth!
Quicksort’s O(n log n) behaviour
Depends on the partitions being nearly equalthere are O( log n ) of them
On average, this will nearly be the case
and quicksort is generally O(n log n)
Can we do anything to ensure O(n log n) time? In general, no
But we can improve our chances!!
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Quicksort - Choice of the pivot
Any pivot will work …Choose a different pivot …
so that the partitions are equal
then we will see O(n log n) time
1 2 3 4 5 6 7 8 9
pivot
< pivot > pivot
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Quicksort - Median-of-3 pivot Take 3 positions and choose the median
say … First, middle, last
median is 5perfect division of sorted data every time!O(n log n) time
Since sorted (or nearly sorted) data is common,median-of-3 is a good strategy
• especially if you think your data may be sorted!
1 2 3 4 5 6 7 8 9
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Quicksort - Random pivot Choose a pivot randomly
Different position for every partition
On average, sorted data is divided evenly
O(n log n) time
• Key requirement
• Pivot choice must take O(1) time
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Quicksort - Guaranteed O(n log n)? Never!!
Any pivot selection strategy could lead to O(n2) time
• Here median-of-3 chooses 2One partition of 1 and
• One partition of 7
• Next it chooses 4One of 1 and
• One of 5
1 4 9 6 2 5 7 8 3
1 2 4 9 6 5 7 8 3
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Sorting- Key PointsSorting
Bubble, Insertion, selection sortO(n2) sortsSimple codeMay run faster for small n,
n ~10 (system dependent)
Quick SortDivide and conquerO(n log n)
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Sorting - Key PointsQuick Sort
O(n log n) but ….Can be O(n2)
Depends on pivot selectionMedian-of-3Random pivot Better but not guaranteed
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Quicksort - Why bother? Use Heapsort instead?
• Quicksort is generally faster• Fewer comparisons and exchanges
• Some empirical data
n Quick Heap InsertComp Exch Comp Exch Comp Exch
100 712 148 2842 581 2595 899200 1682 328 9736 9736 10307 3503500 5102 919 53113 4042 62746 21083
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