Implement pow(x,y) in logarithm time

The problem would be straightforward if we are allowed to do it in O(n) time as we can simply multiple x by itself y times. But how to do it in log time?

The problem description itself is telling us how to implement it in log time. If we need to find power of x raised to y in logarithmic time then somehow we need to make the computation tree into half in each iteration. For example, a^5 = a^2*a^2*a. a^10 = a^5*a^5. So, basically if we can compute a^(y/2) once then we use this computation to find a^y by simply multiplying the result to itself if y is even. If y is odd then we need to multiply x once more with the even multiplication result. Below is the implementation of this idea in both recursive and iterative approach. Both code runs in O(lgn) time (why?).

//O(lgn) public static int powRec(int x, int y){ if(y == 0){ return 1; } if(y == 1){ return x; } int pow = powRec(x, y/2); if((y&1) != 0){ return pow*pow*x; } else{ return pow*pow; } } //O(lgn) public static int powIter(int x, int y){ int pow = 1; if(y == 0){ return 1; } if(y == 1){ return x; } while(y > 0){ //if y is odd if((y&1) != 0){ pow*=x; } //divide by 2 y >>= 1; //calclate x^2 x = x*x; } return pow; }