Understanding Recursive Algorithms: Concepts, Examples, and Implementations

1. Section 3.5 Recursive Algorithms

2. 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

3. 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)

4. 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))

5. 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); }

6. 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

7. 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)

8. 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); }

9. 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

10. 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

11. 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.

12. 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)

13. 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); }

14. 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

15. 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)

16. 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; }

17. 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

18. 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

19. 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; }

20. 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

21. 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 …

22. Example 4 int sequence (int n) { if (n < 2) return n+1; return sequence(n-1)*sequence(n-2); }

23. Section 3.5 Recursive Algorithms - ends -