1. Counting sort

2. Outline In this lesson, we will: Describe the counting sort algorithm
Look at an implementation
Consider the run time of this algorithm

3. Sorting Suppose we have the following numbers:
How could you sort these numbers quickly?
How about just counting how often each number appears:
0 appears once
1 appears 6 times
2 appears 10 times
3 appears 10 times, as well
4 appears 7 times
Thus, the sorted array is:

4. Sorting algorithms
This nice algorithm only works because we not sorting that many different numbers
Suppose we know, a priori, that we are only sorting numbers from 0 to 9
What could we do in this case?
Create an array of capacity 10 and initialize all of the entries to zero
Walk through the array, incrementing the counter for each entry
Then, use this to copy the values sorted back to the array
This is called a counting sort, because we are essentially counting the number of values that equals the target

5. Sorting algorithms Here is the algorithm:
First, create 10 counters (an array) and ensure they are all zero
Go through the array, and for each entry in the array, increment the corresponding counter
Now, for each value between 0 to 9, copy into the array as many entries as we counted

6. Sorting algorithms Looking at these steps:
Go through the array, and for each entry in the array, increment the corresponding counter
This requires a for-loop going through each entry of the array
for ( std::size_t k{0}; k < capacity; ++k ) {
// Record what we found }
Now, for each value between 0 to 9, copy into the array as many entries as we counted
Inside a for-loop going from 0 to 9, we will have to copy to the array as many entries were found in the array
for ( unsigned int j{0}; j < 10; ++j ) {
for ( std::size_t i{0}; i < count[j]; ++i, ++k ) {
array[k] = j;

7. Counting sort Here is an implementation:
void counting_sort( unsigned int array[], std::size_t const capacity ) {
std::size_t count[10]{0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
// Count the number of objects being sorted
for ( std::size_t k{0}; k < capacity; ++k ) {
++count[array[k]]; }
// Copy the appropriate number of each back to the array
std::size_t posn{0};
for ( unsigned int i{0}; i < 10; ++i ) {
for ( std::size_t k{0}; k < count[k]; ++k, ++posn ) {
array[posn] = i;

8. Counting sort Questions:
What happens if you pass this function an array that has a value outside of 0 through 9?
Does it make sense to use counting sort on an array of type double?
Which types would this work for?

9. Counting sort
Now, in general, we will not be sorting numbers from 0 to 9, but more likely, in general, we may know that we are sorting numbers from some minimum value to some maximum value
We can deal with this either by:
Explicitly searching for both the minimum and maximum values in the array
Require the user explicitly pass the minimum and maximum values:
void counting_sort( unsigned int array[],
std::size_t const capacity,
unsigned int const minimum,
unsigned int const maximum );

10. Sorting algorithms
In your homework, you will be required to implement both of these algorithms
You will have to use dynamically allocated memory
You must use the minimum sized array possible
For example, the following array should only require an array of size 12 (why?) to sort:
int array[20]{
, , , , ,
, , , , ,
, , , , ,
, , , , };

11. Run times
We know that sorting this array using counting sort is the correct approach:
int array[100]{
8, 8, 6, 5, 9, 9, 6, 9, 5, 6, 5, 6, 7, 8, 9, 8, 7, 5, 7, 7,
5, 9, 9, 7, 5, 9, 9, 6, 6, 5, 8, 6, 5, 9, 8, 8, 8, 8, 6, 8,
7, 5, 6, 5, 9, 6, 5, 9, 6, 8, 8, 7, 6, 6, 9, 8, 8, 8, 5, 5,
6, 8, 9, 9, 7, 7, 6, 7, 5, 8, 5, 6, 9, 9, 5, 7, 7, 9, 6, 5,
8, 9, 6, 6, 6, 6, 5, 5, 8, 5, 8, 7, 9, 7, 8, 6, 7, 8, 6, 7 };
Question: Would counting sort be good for sorting this array?
int array[20]{
, , , , ,
, , , ,
Why or why not?

12. Summary Following this lesson, you now
Understand the counting sort algorithm
Know that this is a fast algorithm under specific circumstances
Know that this algorithm cannot be used in general

13. References
[1] Wikipedia:

14. Colophon
These slides were prepared using the Georgia typeface. Mathematical equations use Times New Roman, and source code is presented using Consolas. The photographs of lilacs in bloom appearing on the title slide and accenting the top of each other slide were taken at the Royal Botanical Gardens on May 27, 2018 by Douglas Wilhelm Harder. Please see for more information.

15. Disclaimer
These slides are provided for the ece 150 Fundamentals of Programming course taught at the University of Waterloo. The material in it reflects the authors’ best judgment in light of the information available to them at the time of preparation. Any reliance on these course slides by any party for any other purpose are the responsibility of such parties. The authors accept no responsibility for damages, if any, suffered by any party as a result of decisions made or actions based on these course slides for any other purpose than that for which it was intended.