This blog teaches you how to optimize quick sort in C by using tail recursion and iterative methods. You will learn the benefits and drawbacks of each approach and how to compare them.
1. Introduction
Quick sort is one of the most popular and efficient sorting algorithms. It works by dividing an array into two subarrays based on a pivot element, and then recursively sorting the subarrays. The average time complexity of quick sort is O(n log n), where n is the number of elements in the array.
However, quick sort has some drawbacks. One of them is that it uses a lot of stack space, which can cause stack overflow errors for large arrays or deep recursion levels. Another one is that it depends on the choice of the pivot element, which can affect the performance of the algorithm.
In this blog, you will learn how to optimize quick sort in C by using two methods: tail recursion and iterative. Tail recursion is a technique that reduces the stack space by eliminating the recursive calls that are not essential. Iterative is a technique that converts the recursive algorithm into a loop-based one, using a stack data structure to store the subarray boundaries.
By the end of this blog, you will be able to implement both versions of quick sort in C, and compare their advantages and disadvantages. You will also understand the concept of tail recursion and how it can improve the efficiency of recursive algorithms.
Are you ready to dive into the world of quick sort optimization? Let’s get started!
2. Quick Sort Algorithm
The quick sort algorithm is based on the idea of divide and conquer. It works by choosing a pivot element from the array, and partitioning the array into two subarrays: one with elements smaller than the pivot, and one with elements larger than the pivot. Then, it recursively sorts the subarrays until the whole array is sorted.
The main steps of the quick sort algorithm are:
- Select a pivot element from the array. There are different ways to choose the pivot, such as the first element, the last element, the median element, or a random element.
- Partition the array into two subarrays: one with elements less than or equal to the pivot, and one with elements greater than the pivot. This can be done by using two pointers, one from the left and one from the right, and swapping the elements that are out of place.
- Recursively sort the left subarray and the right subarray by repeating the above steps.
The following code snippet shows a simple implementation of the quick sort algorithm in C, using the last element as the pivot:
// A function to swap two elements
void swap(int* a, int* b)
{
int temp = *a;
*a = *b;
*b = temp;
}
// A function to partition the array
int partition(int arr[], int low, int high)
{
// Choose the last element as the pivot
int pivot = arr[high];
// Initialize the index of the smaller element
int i = low - 1;
// Loop through the array from low to high - 1
for (int j = low; j <= high - 1; j++)
{
// If the current element is smaller than or equal to the pivot
if (arr[j] <= pivot)
{
// Increment the index of the smaller element
i++;
// Swap the current element with the smaller element
swap(&arr[i], &arr[j]);
}
}
// Swap the pivot with the element at i + 1
swap(&arr[i + 1], &arr[high]);
// Return the index of the pivot
return i + 1;
}
// A function to sort the array using quick sort
void quickSort(int arr[], int low, int high)
{
// If the array has more than one element
if (low < high)
{
// Partition the array and get the index of the pivot
int pi = partition(arr, low, high);
// Recursively sort the left subarray and the right subarray
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
2.1. Partitioning Strategy
The partitioning strategy is a crucial part of the quick sort algorithm. It determines how the array is divided into two subarrays based on the pivot element. A good partitioning strategy can improve the performance and stability of the algorithm, while a bad one can degrade it.
There are different ways to choose the pivot element and partition the array. Some of the common ones are:
- Lomuto partition scheme: This is the simplest and most widely used partitioning scheme. It chooses the last element as the pivot, and partitions the array into two subarrays: one with elements less than or equal to the pivot, and one with elements greater than the pivot. It uses a single loop and a single index to keep track of the smaller elements.
- Hoare partition scheme: This is the original partitioning scheme proposed by C.A.R. Hoare, the inventor of quick sort. It chooses the first element as the pivot, and partitions the array into two subarrays: one with elements less than the pivot, and one with elements greater than or equal to the pivot. It uses two loops and two indices to move the elements from both ends of the array.
- Dutch national flag partition scheme: This is a variation of the Lomuto partition scheme that can handle arrays with duplicate elements more efficiently. It chooses the last element as the pivot, and partitions the array into three subarrays: one with elements less than the pivot, one with elements equal to the pivot, and one with elements greater than the pivot. It uses a single loop and three indices to keep track of the smaller, equal, and larger elements.
Each partitioning scheme has its own advantages and disadvantages, depending on the characteristics of the array and the choice of the pivot. In general, the Lomuto partition scheme is easier to implement and understand, but it can perform poorly when the array has many duplicate elements or when the pivot is not close to the median. The Hoare partition scheme is more efficient and stable, but it is more complex and prone to errors. The Dutch national flag partition scheme is optimal for arrays with many duplicate elements, but it requires more space and comparisons.
Which partitioning scheme do you think is best for quick sort? How can you choose the pivot element wisely? In the next section, you will learn how to implement the recursive version of quick sort using the Lomuto partition scheme.
2.2. Recursive Implementation
In this section, you will learn how to implement the recursive version of quick sort in C, using the Lomuto partition scheme. You will also see how the algorithm works on an example array.
The recursive implementation of quick sort consists of two functions: partition and quickSort. The partition function takes an array and two indices, low and high, as parameters. It chooses the last element of the array as the pivot, and partitions the array into two subarrays: one with elements less than or equal to the pivot, and one with elements greater than the pivot. It returns the index of the pivot after the partitioning. The quickSort function takes an array and two indices, low and high, as parameters. It calls the partition function to get the index of the pivot, and then recursively sorts the left subarray and the right subarray by calling itself.
The following code snippet shows the recursive implementation of quick sort in C, using the Lomuto partition scheme:
// A function to swap two elements
void swap(int* a, int* b)
{
int temp = *a;
*a = *b;
*b = temp;
}
// A function to partition the array
int partition(int arr[], int low, int high)
{
// Choose the last element as the pivot
int pivot = arr[high];
// Initialize the index of the smaller element
int i = low - 1;
// Loop through the array from low to high - 1
for (int j = low; j <= high - 1; j++)
{
// If the current element is smaller than or equal to the pivot
if (arr[j] <= pivot)
{
// Increment the index of the smaller element
i++;
// Swap the current element with the smaller element
swap(&arr[i], &arr[j]);
}
}
// Swap the pivot with the element at i + 1
swap(&arr[i + 1], &arr[high]);
// Return the index of the pivot
return i + 1;
}
// A function to sort the array using quick sort
void quickSort(int arr[], int low, int high)
{
// If the array has more than one element
if (low < high)
{
// Partition the array and get the index of the pivot
int pi = partition(arr, low, high);
// Recursively sort the left subarray and the right subarray
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
3. Tail Recursion Optimization
Tail recursion is a technique that can reduce the stack space used by recursive functions. It involves transforming the recursive calls into tail calls, which are the last operations performed by the function before returning. A tail call does not need to keep any information on the stack, as the caller does not need to do anything after the call. This allows the compiler or the interpreter to optimize the tail calls by reusing the same stack frame, instead of creating a new one for each call.
How can you apply tail recursion to quick sort? The idea is to eliminate the second recursive call to quickSort, which is not a tail call, by using a loop instead. This way, you only need to keep one subarray on the stack at a time, instead of two. The algorithm works as follows:
- Choose a subarray to sort, initially the whole array.
- Partition the subarray and get the index of the pivot.
- Recursively sort the smaller subarray, either the left or the right one, depending on the size.
- Update the subarray to be the larger subarray, either the left or the right one, depending on the size.
- Repeat the above steps until the subarray has only one element or is empty.
The following code snippet shows the tail recursive implementation of quick sort in C, using the Lomuto partition scheme:
// A function to swap two elements
void swap(int* a, int* b)
{
int temp = *a;
*a = *b;
*b = temp;
}
// A function to partition the array
int partition(int arr[], int low, int high)
{
// Choose the last element as the pivot
int pivot = arr[high];
// Initialize the index of the smaller element
int i = low - 1;
// Loop through the array from low to high - 1
for (int j = low; j <= high - 1; j++)
{
// If the current element is smaller than or equal to the pivot
if (arr[j] <= pivot)
{
// Increment the index of the smaller element
i++;
// Swap the current element with the smaller element
swap(&arr[i], &arr[j]);
}
}
// Swap the pivot with the element at i + 1
swap(&arr[i + 1], &arr[high]);
// Return the index of the pivot
return i + 1;
}
// A function to sort the array using quick sort
void quickSort(int arr[], int low, int high)
{
// While the array has more than one element
while (low < high)
{
// Partition the array and get the index of the pivot
int pi = partition(arr, low, high);
// If the left subarray is smaller than the right subarray
if (pi - low < high - pi)
{
// Recursively sort the left subarray
quickSort(arr, low, pi - 1);
// Update the subarray to be the right subarray
low = pi + 1;
}
else
{
// Recursively sort the right subarray
quickSort(arr, pi + 1, high);
// Update the subarray to be the left subarray
high = pi - 1;
}
}
}
3.1. Tail Call Elimination
Tail call elimination is a technique that can optimize the tail calls by reusing the same stack frame, instead of creating a new one for each call. This can reduce the stack space used by recursive functions, and avoid stack overflow errors. Some compilers or interpreters can perform tail call elimination automatically, while others require manual intervention.
How can you apply tail call elimination to quick sort in C? The answer is that you cannot, unless you use a compiler that supports it. C does not have a standard way to implement tail call elimination, and most C compilers do not support it. Therefore, you cannot rely on tail call elimination to optimize quick sort in C.
However, there is another way to achieve the same effect as tail call elimination, which is to use a loop instead of recursion. This is what we did in the previous section, when we implemented the tail recursive version of quick sort. By using a loop, we eliminated the second recursive call to quickSort, which was not a tail call, and reduced the stack space used by the algorithm.
Therefore, the tail recursive implementation of quick sort in C is equivalent to a tail call elimination implementation, as it uses a loop to optimize the tail calls. However, this is not the case for all recursive algorithms, as some of them cannot be easily converted into loops. For those algorithms, tail call elimination can be a useful technique to optimize the recursion.
In the next section, you will learn how to further optimize quick sort by using an iterative method, which eliminates the recursion altogether.
3.2. Tail Recursive Implementation
In this section, you will learn how to implement the tail recursive version of quick sort in C, using the Lomuto partition scheme. You will also see how the algorithm works on an example array.
The tail recursive implementation of quick sort consists of two functions: partition and quickSort. The partition function takes an array and two indices, low and high, as parameters. It chooses the last element of the array as the pivot, and partitions the array into two subarrays: one with elements less than or equal to the pivot, and one with elements greater than the pivot. It returns the index of the pivot after the partitioning. The quickSort function takes an array and two indices, low and high, as parameters. It calls the partition function to get the index of the pivot, and then recursively sorts the smaller subarray, either the left or the right one, depending on the size. It then updates the subarray to be the larger subarray, either the left or the right one, depending on the size, and repeats the process until the subarray has only one element or is empty.
The following code snippet shows the tail recursive implementation of quick sort in C, using the Lomuto partition scheme:
// A function to swap two elements
void swap(int* a, int* b)
{
int temp = *a;
*a = *b;
*b = temp;
}
// A function to partition the array
int partition(int arr[], int low, int high)
{
// Choose the last element as the pivot
int pivot = arr[high];
// Initialize the index of the smaller element
int i = low - 1;
// Loop through the array from low to high - 1
for (int j = low; j <= high - 1; j++)
{
// If the current element is smaller than or equal to the pivot
if (arr[j] <= pivot)
{
// Increment the index of the smaller element
i++;
// Swap the current element with the smaller element
swap(&arr[i], &arr[j]);
}
}
// Swap the pivot with the element at i + 1
swap(&arr[i + 1], &arr[high]);
// Return the index of the pivot
return i + 1;
}
// A function to sort the array using quick sort
void quickSort(int arr[], int low, int high)
{
// While the array has more than one element
while (low < high)
{
// Partition the array and get the index of the pivot
int pi = partition(arr, low, high);
// If the left subarray is smaller than the right subarray
if (pi - low < high - pi)
{
// Recursively sort the left subarray
quickSort(arr, low, pi - 1);
// Update the subarray to be the right subarray
low = pi + 1;
}
else
{
// Recursively sort the right subarray
quickSort(arr, pi + 1, high);
// Update the subarray to be the left subarray
high = pi - 1;
}
}
}
4. Iterative Optimization
Iterative optimization is a technique that can eliminate the recursion from the quick sort algorithm, and use a loop instead. This can reduce the stack space used by the algorithm, and avoid stack overflow errors. It also makes the algorithm independent of the choice of the pivot element, as it can sort any subarray regardless of its size.
How can you apply iterative optimization to quick sort in C? The idea is to use a stack data structure to store the subarray boundaries, and loop through the stack until it is empty. The algorithm works as follows:
- Create an empty stack.
- Push the low and high indices of the whole array onto the stack.
- While the stack is not empty, pop the top two elements from the stack, and assign them to low and high variables.
- Call the partition function to get the index of the pivot, and partition the subarray.
- If the left subarray has more than one element, push the low and pivot – 1 indices onto the stack.
- If the right subarray has more than one element, push the pivot + 1 and high indices onto the stack.
- Repeat the above steps until the stack is empty.
The following code snippet shows the iterative implementation of quick sort in C, using the Lomuto partition scheme and a stack data structure:
// A function to swap two elements
void swap(int* a, int* b)
{
int temp = *a;
*a = *b;
*b = temp;
}
// A function to partition the array
int partition(int arr[], int low, int high)
{
// Choose the last element as the pivot
int pivot = arr[high];
// Initialize the index of the smaller element
int i = low - 1;
// Loop through the array from low to high - 1
for (int j = low; j <= high - 1; j++)
{
// If the current element is smaller than or equal to the pivot
if (arr[j] <= pivot)
{
// Increment the index of the smaller element
i++;
// Swap the current element with the smaller element
swap(&arr[i], &arr[j]);
}
}
// Swap the pivot with the element at i + 1
swap(&arr[i + 1], &arr[high]);
// Return the index of the pivot
return i + 1;
}
// A function to sort the array using quick sort
void quickSort(int arr[], int low, int high)
{
// Create an empty stack
int stack[high - low + 1];
// Initialize the top of the stack
int top = -1;
// Push the low and high indices of the whole array onto the stack
stack[++top] = low;
stack[++top] = high;
// While the stack is not empty
while (top >= 0)
{
// Pop the top two elements from the stack
high = stack[top--];
low = stack[top--];
// Call the partition function to get the index of the pivot
int pi = partition(arr, low, high);
// If the left subarray has more than one element
if (pi - 1 > low)
{
// Push the low and pivot - 1 indices onto the stack
stack[++top] = low;
stack[++top] = pi - 1;
}
// If the right subarray has more than one element
if (pi + 1 < high)
{
// Push the pivot + 1 and high indices onto the stack
stack[++top] = pi + 1;
stack[++top] = high;
}
}
}
4.1. Stack Data Structure
A stack is a linear data structure that follows the last in, first out (LIFO) principle. It means that the last element that is added to the stack is the first one that is removed from it. A stack has two main operations: push and pop. Push adds an element to the top of the stack, and pop removes the top element from the stack.
A stack can be implemented using an array or a linked list. In this blog, we will use an array to implement a stack. To do so, we need to keep track of the top of the stack, which is the index of the last element that is pushed onto the stack. Initially, the top is -1, indicating that the stack is empty. When we push an element onto the stack, we increment the top by 1, and store the element at that index. When we pop an element from the stack, we return the element at the top index, and decrement the top by 1.
The following code snippet shows the implementation of a stack data structure in C, using an array:
// Define the maximum size of the stack
#define MAX_SIZE 100
// Declare an array to store the stack elements
int stack[MAX_SIZE];
// Initialize the top of the stack
int top = -1;
// A function to check if the stack is empty
int isEmpty()
{
// Return 1 if the top is -1, 0 otherwise
return top == -1;
}
// A function to check if the stack is full
int isFull()
{
// Return 1 if the top is MAX_SIZE - 1, 0 otherwise
return top == MAX_SIZE - 1;
}
// A function to push an element onto the stack
void push(int x)
{
// If the stack is full, print an error message and return
if (isFull())
{
printf("Error: Stack overflow\n");
return;
}
// Increment the top by 1
top++;
// Store the element at the top index
stack[top] = x;
}
// A function to pop an element from the stack
int pop()
{
// If the stack is empty, print an error message and return -1
if (isEmpty())
{
printf("Error: Stack underflow\n");
return -1;
}
// Store the element at the top index
int x = stack[top];
// Decrement the top by 1
top--;
// Return the element
return x;
}
A stack is a useful data structure for many applications, such as reversing a string, evaluating a postfix expression, or implementing a backtracking algorithm. In this blog, we will use a stack to store the subarray boundaries for the iterative version of quick sort.
In the next section, you will learn how to implement the iterative version of quick sort in C, using the Lomuto partition scheme and a stack data structure.
4.2. Iterative Implementation
An alternative way to optimize quick sort is to use an iterative method, instead of a recursive one. This means that you will use a loop to sort the array, instead of calling the function repeatedly. To do this, you will need a stack data structure to store the subarray boundaries, and pop them when you need to sort them.
The main steps of the iterative quick sort algorithm are:
- Create an empty stack and push the initial low and high indices of the array.
- Pop the top two elements from the stack and assign them to low and high variables.
- Partition the subarray from low to high using the same function as before, and get the index of the pivot.
- If the left subarray has more than one element, push its low and high indices to the stack.
- If the right subarray has more than one element, push its low and high indices to the stack.
- Repeat steps 2 to 5 until the stack is empty.
The following code snippet shows an implementation of the iterative quick sort algorithm in C, using the same partition function as before:
// A function to sort the array using iterative quick sort
void iterativeQuickSort(int arr[], int low, int high)
{
// Create an empty stack
int stack[high - low + 1];
// Initialize the top of the stack
int top = -1;
// Push the initial low and high indices to the stack
stack[++top] = low;
stack[++top] = high;
// Loop until the stack is empty
while (top >= 0)
{
// Pop the high and low indices from the stack
high = stack[top--];
low = stack[top--];
// Partition the subarray from low to high
int pi = partition(arr, low, high);
// If the left subarray has more than one element, push its low and high indices to the stack
if (pi - 1 > low)
{
stack[++top] = low;
stack[++top] = pi - 1;
}
// If the right subarray has more than one element, push its low and high indices to the stack
if (pi + 1 < high)
{
stack[++top] = pi + 1;
stack[++top] = high;
}
}
}
5. Comparison and Analysis
In this section, you will compare and analyze the three versions of quick sort that you have learned: recursive, tail recursive, and iterative. You will evaluate them based on their time complexity, space complexity, and performance. You will also discuss the pros and cons of each version, and when to use them.
The time complexity of quick sort depends on the choice of the pivot element and the partitioning strategy. In the best case, when the pivot is the median of the array, the time complexity is O(n log n), where n is the number of elements in the array. In the worst case, when the pivot is the smallest or the largest element of the array, the time complexity is O(n2). In the average case, the time complexity is also O(n log n).
The space complexity of quick sort depends on the implementation of the algorithm. In the recursive version, the space complexity is O(log n), where n is the number of elements in the array. This is because the recursive calls are stored in the function call stack, which has a depth of log n. In the tail recursive version, the space complexity is O(1), because the recursive calls are eliminated by using a loop and updating the low and high indices. In the iterative version, the space complexity is also O(log n), because the subarray boundaries are stored in the stack data structure, which has a size of log n.
The performance of quick sort depends on the size and distribution of the array, and the compiler optimization. In general, the tail recursive and iterative versions are faster and more efficient than the recursive version, because they avoid the overhead of recursive calls and reduce the stack space. However, the difference may not be significant for small or nearly sorted arrays, or when the compiler can optimize the recursive calls. The tail recursive version may have an edge over the iterative version, because it does not need to push and pop the stack elements, which can save some time and memory.
The pros and cons of each version are summarized in the following table:
| Version | Pros | Cons |
|---|---|---|
| Recursive | Simple and elegant | Uses a lot of stack space Possible stack overflow errors Depends on compiler optimization |
| Tail recursive | Reduces the stack space Eliminates the recursive calls Faster and more efficient | Requires a loop and extra variables Less intuitive and readable |
| Iterative | Reduces the stack space Eliminates the recursive calls Faster and more efficient | Requires a stack data structure Less intuitive and readable |
As you can see, there is no definitive answer to which version of quick sort is the best. It depends on the situation and the trade-offs that you are willing to make. You should consider the following factors when choosing a version:
- The size and distribution of the array
- The available stack space and memory
- The compiler optimization and support
- The readability and maintainability of the code
Quick sort is a powerful and versatile sorting algorithm that can be optimized in different ways. You have learned how to use tail recursion and iterative methods to improve the efficiency and performance of quick sort in C. You have also compared and analyzed the three versions of quick sort, and discussed when to use them. You have gained a deeper understanding of the concept of tail recursion and how it can benefit recursive algorithms.
We hope you enjoyed this blog and learned something new and useful. If you have any questions or feedback, please leave a comment below. Thank you for reading!
6. Conclusion
In this blog, you have learned how to implement and optimize quick sort in C using three different methods: recursive, tail recursive, and iterative. You have also compared and analyzed the time and space complexity, performance, and pros and cons of each method. You have gained a deeper understanding of the concept of tail recursion and how it can improve the efficiency of recursive algorithms.
Quick sort is a powerful and versatile sorting algorithm that can be adapted to different situations and trade-offs. You should consider the following factors when choosing a method:
- The size and distribution of the array
- The available stack space and memory
- The compiler optimization and support
- The readability and maintainability of the code
We hope you enjoyed this blog and learned something new and useful. If you have any questions or feedback, please leave a comment below. Thank you for reading!
