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### Section 3.5

Recursive Algorithms

Recursive Algorithms

- Sometimes we can reduce solution of problem to solution of same problem with set of smaller input values
- When such reduction is possible, solution to original problem can be found with series of reductions, until problem is reduced to case for which solution is known
- Algorithms which take this approach are called recursive algorithms

Example 1: computing an

- Can base algorithm on recursive definition of an:
- for n=0, an = 1
- for n>0, an+1 = a(an)
- where a is a non-zero real number and n is a non-negative integer

procedure power(inputs: a, n)

if (n=0) then power(a,n) = 1

else power (a,n) = power(a, n-1)

Extending example 1

- The algorithm in example 1 works only for non-negative powers of non-zero a - we can extend the algorithm to work for all powers of any value a, as follows:

procedure power (inputs: a, n)

if (a = 0) power(a,n) = 0

else if (n = 0) power(a,n) = 1

else if (n > 0) power(a,n) = (a * power(a, n-1))

else power(a,n) = (1 / rpower(a, -n))

Example 1

int power ( int num, int p)

{

if (num == 0 )

return 0;

if (p ==0)

return 1;

if (p < 0)

return 1 / power(num, -p);

return num * power(num, p-1);

}

Example 2: computing gcd

- The algorithm on the following slide is based on the reduction gcb(a,b) = gcd(b mod a, a) and the condition gcd(0,b) = b where:
- a < b
- b > 0

Example 2: computing gcd

Procedure gcd (inputs: a, b with a < b)

if (a=0) then gcd(a,b) = b

else gcd(a,b) = gcd(b mod a, a)

Example 2

int gcd (unsigned int smaller, unsigned int larger)

{

if (larger < smaller)

{

int tmp = larger;

larger = smaller;

smaller = tmp;

}

if (larger == smaller || smaller == 0)

return larger;

return gcd (larger % smaller, smaller);

}

Linear search revisited

- The linear, or sequential search algorithm was introduced in section 2.1, as follows:
- Examine each item in succession to see if it matches target value
- If target found or no elements left to examine, stop search
- If target was found, location = found index; otherwise, location = 0

Linear search revisited

- In the search for x in the sequence a1 … an, x and ai are compared at the ith step
- If x = ai, the search is finished; otherwise the search problem is reduced by one element, since now the sequence to be searched consists of ai+1 … an
- Looking at the problem this way, a recursive procedure can be developed

Linear search revisited

- Let search(x,y,z) be the procedure that searches for z in the sequence ax … ay
- The procedure begins with the triple (x,y,z), terminating when the first term of the sequence is z or when no terms are left to be searched
- If z is not the first term and additional terms exist, same procedure is carried out but with input of (x+1, y, z), then with (x+2, y, z), etc.

Linear search revisited

Procedure lsearch (x,y,z)

if ax = z then location = x

else if x = y then location = 0 (not found)

else lsearch(x+1, y, z)

Linear search

int lsearch(int index, int len, int target, int array[])

{

if (index == len)

return len; // not found

if (array[index]==target)

return index; // found

return lsearch(index+1, len, target, array);

}

Binary search revisited

- The binary search algorithm was also introduced in section 2.1:
- Works by splitting list in half, then examining the half that might contain the target value
- if not found, split and examine again
- eventually, set is split down to one element
- If the one element is the target, set location to index of item; otherwise, location = 0

Binary search - recursive version

procedure bsearch (inputs: x,y,z)

mid = (x+y)/2

if z = amid then location = mid (found)

else if z < amid and x < mid then

bsearch(x, mid-1, z)

else if z > amid and y > mid then

bsearch (mid+1, y, z)

else location = 0 (not found)

Implementation of binary search

void BinarySearch(int array[], int first, int size,

int target, bool& found, int& location)

{

size_t middle;

if (size == 0)

found = false; // base case

else

{

middle = first + size / 2;

if (target == array[middle])

{

location = middle;

found = true;

}

Binary search code continued

// target not found at current midpoint -- search appropriate half

else if (target < array[middle])

BinarySearch (array, first, size/2, target, found, location);

// searches from start of array to index before midpoint

else

BinarySearch (array, middle+1, (size-1)/2, target, found, location);

// searches from index after midpoint to end of array

} // ends outer else

} // ends function

Recursion Vs. Iteration

- A recursive definition expresses the value of a function at a positive integer in terms of its value at smaller integers
- Another approach is to start with the value of the function at 1 and successively apply the recursive definition to find the value at successively larger integers - this method is called iteration

Example 3: finding n!

- Recursive algorithm:

int factorial (int n) {

if (n==1) return 1;

return n * factorial(n-1);}

- Iterative algorithm:

int factorial (int n) {

int x = 1;

for (int c = 1; c<= n; c++)

x = c * x ;

return x; }

Recursion Vs. Iteration

- Iterative approach often requires much less computation than recursive procedure
- Recursion is best suited to tasks for which there is no obvious iterative solution

Example 4: finding nth term of a sequence

- Devise a recursive algorithm to find the nth term of the sequence defined by:
- a0 = 1, a1 = 2
- an = an-1 * an-2 for n=2, 3, 4 …

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