Tail Call Elimination

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. 

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.

// Non-tail recursive function
void print(int n) {
    if (n < 0)
        return;

    // Recursive call happens first (not last)
    print(n - 1);

    // Now process the current node
    cout << " " << n;
}

 static void print(int n) {
        if (n < 0)
            return;

        // Recursive call happens first
        print(n - 1);

        // Process after recursion
        System.out.print(" " + n);
    }

def print_n(n):
    if n < 0:
        return

    # Recursive call first
    print_n(n - 1)

    # Process after recursion
    print(n, end=" ")

static void Print(int n) {
        if (n < 0) return;

        // Recursive call first
        Print(n - 1);

        // Process after recursion
        Console.Write(n + " ");
    }

function printN(n) {
    if (n < 0) return;

    // Recursive call first
    printN(n - 1);

    // Process after recursion
    process.stdout.write(n + " ");
}

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) 

/* 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);
    }
}

# 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)
        

// 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);
    }
}
}

/* 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);
    }
}

The above function can be replaced by following after tail call elimination. 

// QuickSort after tail call elimination
void quickSort(int arr[], int low, int high) {
    while (low < high) {
        int pi = partition(arr, low, high);

        // Recursively sort left part
        quickSort(arr, low, pi - 1);

        // Tail recursion elimination for right part
        low = pi + 1;
    }
}

  // QuickSort after tail call elimination
    static void quickSort(int arr[], int low, int high) {
        while (low < high) {
            int pi = partition(arr, low, high);

            // Sort left part
            quickSort(arr, low, pi - 1);

            // Tail recursion elimination for right part
            low = pi + 1;
        }
    }

# QuickSort after tail call elimination
def quick_sort(arr, low, high):
    while low < high:
        pi = partition(arr, low, high)

        # Sort left part
        quick_sort(arr, low, pi - 1)

        # Tail recursion elimination for right part
        low = pi + 1

  // QuickSort after tail call elimination
 static void QuickSort(int[] arr, int low, int high) {
        while (low < high) {
            int pi = Partition(arr, low, high);

            // Sort left part
            QuickSort(arr, low, pi - 1);

            // Tail recursion elimination for right part
            low = pi + 1;
        }
    }

// QuickSort after tail call elimination
function quickSort(arr, low, high) {
    while (low < high) {
        let pi = partition(arr, low, high);

        // Sort left part
        quickSort(arr, low, pi - 1);

        // Tail recursion elimination for right part
        low = pi + 1;
    }
}

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 )