Recursion

Review:A recursive function calls itself, but with a smaller

problem – at least one of the parameters must

decrease.The function uses the results of the recursive call to

create its result.There must be a base case – when the problem is

sufficiently small to be solved directly, or by

another method.

Recursion - Example

Consider the following problem:

We want to print out the digits of a number (in

decimal) one by one. For example, the number

16425 would be printed out as:

1 6 4 2 5

How do we isolate a single digit of a number? Hint –

how do we get the one's place (5 in this example),

i.e., what arithmetic operation could we do to get 5

from 16425?

Example (cont'd)

Now that we can get the one's place (5), how do we

print out the rest of the number? What is the rest of

the number? Hint – pass the buck.

Exercise

How could we change the previous code to print out

the number in binary?

Recursion in General

We've seen examples where we code right a program

using either iteration or recursive – with the same

running times (finding the minimum element,

insertion sort, selection sort, mergesort, finding the

exponential), and two examples where it's easier to

write the program using recursion (quicksort and

the Tower of Hanoi).

Recursion (cont'd)

In theory, we can always use either to solve a

problem – but some problems are more naturally

solved using recursion, and some with iteration.

Bad Recursive Example

A classic problem:You have a pair of rabbits. Every

two years on the dot, a pair of rabbits produces a

new litter of exactly two rabbits. We assume that

our rabbits never die. How many pairs of rabbits

will we have in, say 6, years.

Year 1 2 3 4 5 6 7# Pairs 1 1 2 3 5 8 13

Fibonacci Sequence

This problem gives us the Fibonnaci sequence, f1 = 1,

f2 = 1, f

n = f

n-1 + f

n-2. Since the sequence is defined

recursively (the next value of f is defined in terms

of two previous values), it would seem natural to

write a recursive function to find the nth number in

the sequence.

Recursive Function

int fib(n) { if(n == 1 || n == 2) return 1; else return fib(n-1) + fib(n-2);}

What's wrong with this solution?

How the Computer Does Recursion

Basically, a recursive function is one that plays “Pass

the Buck” with itself as its own neighbor. For each

instance of the problem, the function keeps a record

of who called it (the calling function, and the point

in that function when it was called), and the

arguments – that is, the problem that it was given.

When it passes the buck to its neighbor (really

itself) it creates a new record with the new problem,

and places it on top of the old record.

How it Works (cont'd).

When it finishes a problem, it removes the top

record, and writes the answer in the record that is

now on top – the record with the previous problem

which has just been solved.

When it gets to the base case, it does not play Pass

the Buck, but rather just solves that problem

directly (runs the appropriate code) and returns the

answer to the calling function.

Example - Mergesort

Suppose that we use the mergesort function to sort

the list {4,2,1,7,6}, that is, a = {4,2,1,7,6}, low = 1,

and high = 5. A record is created with the calling

function (say main), the position in the calling

function, and the arguments. Since it's not the base

case, the mergesort function splits the list into two

pieces, and recurses, first posing the problem: a, 1,

3.

Example (cont'd)

A new record is created with the calling function

(mergesort), the position in that function (the first

recursive call to mergesort) and the new arguments:

a, 1, 3. That record is placed on top of the original

record (which does not reappear until the new

problem is solved). The mergesort function is called

again with the new arguments, a new record is

created (if not the base case) and so on...

Example (cont'd)

When the function finishes, it removes the top

record, and records the answer (the return value) in

the record that now is on top – the one with the

problem that the answer is the solution for.

Example (cont'd)

Calling function: main, line 3Arguments: a = {4, 2, 1, 7, 6}, 1, 6Result:

Calling function: mergesort, line 4Arguments: a = {4, 2, 1, 7, 6}, 1, 3Result:

Calling function: mergesort, line 4Arguments: a = {4, 2, 1, 7, 6}, 1, 2Result:

Calling Function: mergesort, line 4Arguments: a = {4, 2, 1, 7, 6}, 1, 1Result: {4}

Calling function: main, line 3Arguments: a = {4, 2, 1, 7, 6}, 1, 6Result:

Calling function: mergesort, line 4Arguments: a = {4, 2, 1, 7, 6}, 1, 3Result:

Calling function: mergesort, line 4Arguments: a = {4, 2, 1, 7, 6}, 1, 2Result:

Calling function: mergesort, line 5Arguments: a = {4, 2, 1, 7, 6}, 2, 2Result: {2}

Example (cont'd)

Calling function: main, line 3Arguments: a = {2, 4, 1, 7, 6}, 1, 6Result:

Calling function: mergesort, line 4Arguments: a = {2, 4, 1, 7, 6}, 1, 3Result:

Calling Function: mergesort, line 5Arguments: a = {2, 4, 1, 7, 6}, 3, 3Result: {1}

Example (cont'd)

Calling function: main, line 3Arguments: a = {1, 2, 4, 7, 6}, 1, 6Result:

Calling function: mergesort, line 4Arguments: a = {1, 2, 4, 7, 6}, 1, 3Result: {1, 2, 4}

Example (cont'd)

Calling function: main, line 3Arguments: a = {1, 2, 4, 7, 6}, 1, 6Result:

Calling function: mergesort, line 5Arguments: a = {1, 2, 4, 7, 6}, 4, 6Result: