Recursion

Exponentiation Problem

Input

Positive integers: b and n

Output

Puesdocode

function Exp(b,n)
    result = 1
    for i = i to n
        result = result * b
    return result

Multiplications: n

Goal: Less multiplications

1. compute
2. square it
   
   1. compute
   2. square it

Total: 252 multiplications

Puesdocode: Refined Algorithm

function Exp(b, n)
    if n = 0
        return 1
    if n is even
        return Exp(b, n/2)^2
    else
        return Exp(b, (n-1)/2)^2 * b

How many recursive calls are made if n = 2^k?

k+1 or k+2?

Logarithms

number of multiplications (worst case)

Binary Search Problem

Input

• sorted array A of numbers
• number X

Output

• i such that A[i] = x
• -1 if such i does not exist

Puesdocode

T(n) = max number of recursive calls for an array of length n