Question

i need to prove that following algorithm works correctly,i know induction,but don’t know how to use it here?also i will be happy if would know complexity of algorithm,how optimal it is?
what is it’s run time?please help me

#include <cstdlib>
#include <iostream>
#define c 2
//we should take c   more ot equal  then 2
using namespace std;
int multiply(int y,int z){
      // product yz
    if(z==0) return 0;
    return (multiply(c*y,int(z/c))+y*(z %c));
}

int main(int argc, char *argv[])
{
    int y=5;
    int z=7;
    cout<<multiply(y,z)<<endl;
    system("PAUSE");
    return EXIT_SUCCESS;
}

thanks

Answer 1

1) for z=0 your function is obviously correct

2) suppose multiply(x, y) returns x*y for 0 <= y < y0. Then

  x*y
= x*((y/c)*c + y%c)    # by the definition of %
= x*c*(y/c) + x*(y%c)  # distributive, commutative laws
= multiply(x*c, y/c) + x*(y%c) # 0 <= y/c < y, induction hypothesis