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Data Structures CSCI 132, Spring 2014 Lecture 16 Tail Recursion

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Recall Recursive Skating

void skateLaps( int n) {

if ( n < = 0 ) {

//do nothing

} else {

cout << "Skating 1 lap." << endl;

skateLaps( n - 1);

}

}

Base Case

General Case

Tail Recursion

- When the recursive function call is the last thing that occurs in the recursive function, this is called a Tail Recursion.
- In a tail recursion, there are no pending operations.
- Any tail recursion can be written as an iteration (a loop).

Iterating skateLaps

void skateLaps( int n) {

while (n > 0 ) {

cout << "Skating 1 lap." << endl;

n = n - 1;

}

}

Factorial Revisited

int Fact (int n) {

if (n <= 0) {

return 1;

} else {

return n * Fact (n - 1);

}

}

Is this a tail recursion?

No.

Can we write this as a tail recursion?

Yes!

Keep track of the answer in a parameter that gets passed on.

Tail Recursion Factorial

int factIter (int n) { //Helper function to initialize ans

return factTail(n, 1);

}

int factTail (int num, int ans) {

if (num <= 0) {

return ans;

} else {

return factTail(num - 1, num*ans);

}

}

Execution of FactTail

num ans num*ans Function Calls

4 1 4 Fact(4, 1):24

3 4 12 Fact(3, 4):24

2 12 24 Fact(2, 12):24

1 24 24 Fact(1, 24):24

0 24 (returned) Fact(0, 24):24

Writing factTail as an iteration

int factWhile (int n) {

int num = n;

int ans = 1;

while (num > 0) {

ans = ans * num;

num = num - 1;

}

return ans;

}

Fibonacci in C++ code

int Fib( int n ) {

if (n < 2) {

return n;

} else {

return Fib(n-1) + Fib(n -2);

}

}

Fibonacci as a Tail Recursion

(fib(i) + fib(i-1))

n i fib(i-1) fib(i) fib(i_next)

5 1 0 1 1

5 2 1 1 2

5 3 1 2 3

5 4 2 3 5

5 5 3 5 8

Pass values as parameters of the recursive function:

n, i, fibIminus1, fibI (we will work on this in lab).

The Tower of Hanoi

Setup: A stack of rings in order of largest to smallest is on one of three pegs (largest on the bottom, smallest on the top).

Object: Move stack from one peg to another peg following these rules:

1) Cannot put a ring on top of a smaller ring.

2) Can only move 1 ring at a time.

Strategy for Tower of Hanoi

1) Move top (n-1) rings from origin to spare

(following all the rules).

2) Move bottom ring from origin to destination

3) Move top (n-1) rings from spare to destination

(following all the rules).

Implementing Tower of Hanoi

void hanoi(int n, char start, char end, char spare) {

if (n < = 0) {

//do nothing

} else {

hanoi(n-1, start, spare, end);

cout << "Move disk " << n << "from "

<< start << " to " << end << endl;

hanoi(n-1, spare, end, start);

}

}

//Outputs sequence of moves to solve tower of hanoi!

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