We have discussed (in tail recursion) that a recursive function is tail recursive if the recursive call is the last thing executed by the function.
C++
// An example of tail recursive function
void print(int n)
{
if (n < 0)
return;
cout << " " << n;
// The last executed statement is recursive call
print(n-1);
}
// An example of tail recursive function
static void print(int n)
{
if (n < 0)
return;
System.out.print(" " + n);
// The last executed statement
// is recursive call
print(n - 1);
}
// This code is contributed by rutvik_56
# An example of tail recursive function
def display(n):
if (n < 0):
return
print(" ", n)
# The last executed statement is recursive call
display(n - 1)
# This code is contributed by sanjoy_62
// An example of tail recursive function
static void print(int n)
{
if (n < 0)
return;
Console.Write(" " + n);
// The last executed statement
// is recursive call
print(n - 1);
}
// This code is contributed by pratham76
<script>
// An example of tail recursive function
function print(n)
{
if (n < 0)
return;
document.write( " " + n);
// The last executed statement is recursive call
print(n-1);
}
</script>
We also discussed that a tail-recursive is better than a non-tail recursive as tail-recursion can be optimized by modern compilers. Modern compiler basically does tail call elimination to optimize the tail-recursive code.
If we take a closer look at the above function, we can remove the last call with goto. Below are examples of tail call elimination.
C++
// Above code after tail call elimination
void print(int n)
{
start:
if (n < 0)
return;
cout << " " << n;
// Update parameters of recursive call
// and replace recursive call with goto
n = n-1
goto start;
}
// Java code after tail call elimination
public static void print(int n)
{
start:
if (n < 0)
return;
System.out.print(" "+n);
// Update parameters of recursive call
// and replace recursive call with goto
n = n - 1
goto start;
}
// This code is contributed by ishankhandelwals.
# Python code after tail call elimination
def print(n):
while n >= 0:
print(" ", n)
# Update parameters of recursive call
# and replace recursive call with goto
n = n - 1
# This code is contributed by ishankhandelwals.
// Above code after tail call elimination
public static void print(int n)
{
while(n>=0)
{
if (n < 0)
return;
Console.Write(" " + n);
// Update parameters of recursive call
// and replace recursive call with goto
n = n-1;
}
}
// The code is contributed by Arushi Jindal.
// Above code after tail call elimination
function print( n)
{
while(n>=0)
{
if (n < 0)
return;
Console.Write(" " + n);
// Update parameters of recursive call
// and replace recursive call with goto
n = n-1;
}
}
QuickSort : One more example
QuickSort is also tail recursive (Note that MergeSort is not tail recursive, this is also one of the reasons why QuickSort performs better)
C++
/* Tail recursive function for QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
void quickSort(int arr[], int low, int high)
{
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
int pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
/* Java program to implement the above approach
Tail recursive function for quick sort
arr[] --> to get the elements
low --> starting index
high --> ending index
*/
static void quickSort(int[] arr, int low, int high)
{
if (low < high)
{
/* pi is partitioning index,after the call
to partition function the arr[p] will be
at its correct index*/
int pi = partition(arr, low, high);
// Sort elements before and after
// the partition separately
// thus whole array will get sorted eventually
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
// This code was contributed by Abhijeet Kumar
# Python program to implement
# the above approach
# Tail recursive function for QuickSort
# arr[] --> Array to be sorted,
# low --> Starting index,
# high --> Ending index */
def quickSort(arr,low,high):
if(low<high):
# pi is partitioning index, arr[p] is now
# at right place
pi=partition(arr,low,high)
# Separately sort elements before
# partition and after partition
quickSort(arr,low,pi-1)
quickSort(arr,pi+1,high)
# This code is contributed by Pushpesh Raj.
// C# program to implement
// the above approach
using System;
class GFG{
/* Tail recursive function for QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static void quickSort(int[] arr, int low, int high)
{
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
int pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
}
// This code is contributed by code_hunt.
/* Tail recursive function for QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
function quickSort( arr, low, high)
{
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
let pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
// This code was contributed by poojaagrawal2.
The above function can be replaced by following after tail call elimination.
C++
/* QuickSort after tail call elimination
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
void quickSort(int arr[], int low, int high)
{
start:
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
int pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
// Update parameters of recursive call
// and replace recursive call with goto
low = pi+1;
high = high;
goto start;
}
}
// QuickSort after tail call elimination
// arr[] --> Array to be sorted,
// low --> Starting index,
// high --> Ending index
void quickSort(int arr[], int low, int high) {
start:
if (low < high) {
// pi is partitioning index, arr[p] is now
// at right place
int pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
// Update parameters of recursive call
// and replace recursive call with goto
low = pi + 1;
high = high;
goto start;
}
}
// This code is contributed Aman Kumar.
def quickSort(arr, low, high):
def start():
nonlocal low, high
if low < high:
# pi is the partitioning index, arr[p] is now at the right place
pi = partition(arr, low, high)
# Separately sort elements before partition and after partition
quickSort(arr, low, pi - 1)
# Update parameters of the recursive call
# and replace recursive call with a function call
low = pi + 1
start()
start()
using System;
class QuickSort {
// Function to perform the partitioning of the array
int Partition(int[] arr, int low, int high) {
int pivot = arr[high];
int i = low - 1;
for (int j = low; j < high; j++) {
if (arr[j] < pivot) {
i++;
// Swap arr[i] and arr[j]
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
// Swap arr[i+1] and arr[high] (or pivot)
int temp1 = arr[i + 1];
arr[i + 1] = arr[high];
arr[high] = temp1;
return i + 1;
}
// Main sorting function with a different name
void Sort(int[] arr, int low, int high) {
Action start = null;
start = () => {
if (low < high) {
int pi = Partition(arr, low, high);
// Separately sort elements before partition
Sort(arr, low, pi - 1);
// Replace the recursive call with the start function call
low = pi + 1;
start();
}
};
start();
}
// Function to print the sorted array
void PrintArray(int[] arr) {
int n = arr.Length;
for (int i = 0; i < n; i++) {
Console.Write(arr[i] + " ");
}
Console.WriteLine();
}
// Driver code
static void Main() {
QuickSort quickSort = new QuickSort();
int[] arr = { 64, 25, 12, 22, 11 };
int n = arr.Length;
Console.WriteLine("Unsorted array:");
quickSort.PrintArray(arr);
quickSort.Sort(arr, 0, n - 1);
Console.WriteLine("Sorted array:");
quickSort.PrintArray(arr);
}
}
// QuickSort after tail call elimination
// arr[] --> Array to be sorted,
// low --> Starting index,
// high --> Ending index
function quickSort(arr, low, high) {
let start = () => {
if (low < high) {
// pi is partitioning index, arr[p] is now at right place
let pi = partition(arr, low, high);
// Separately sort elements before partition and after partition
quickSort(arr, low, pi - 1);
// Update parameters of recursive call
// and replace recursive call with goto
low = pi + 1;
high = high;
start();
}
};
start();
}
// The code is contributed by Arushi Goel.
Therefore job for compilers is to identify tail recursion, add a label at the beginning and update parameter(s) at the end followed by adding the last goto statement.
Function stack frame management in Tail Call Elimination :
Recursion uses a stack to keep track of function calls. With every function call, a new frame is pushed onto the stack which contains local variables and data of that call. Let's say one stack frame requires O(1) i.e, constant memory space, then for N recursive call memory required would be O(N).
Tail call elimination reduces the space complexity of recursion from O(N) to O(1). As function call is eliminated, no new stack frames are created and the function is executed in constant memory space.
It is possible for the function to execute in constant memory space because, in tail recursive function, there are no statements after call statement so preserving state and frame of parent function is not required. Child function is called and finishes immediately, it doesn't have to return control back to the parent function.
As no computation is performed on the returned value and no statements are left for execution, the current frame can be modified as per the requirements of the current function call. So there is no need to preserve stack frames of previous function calls and function executes in constant memory space. This makes tail recursion faster and memory-friendly.
Next Article:
QuickSort Tail Call Optimization (Reducing worst case space to Log n )
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