How to explain this algorithm for calculating the power of a number?

Question

I am currently reading Skiena\'s \"The Algorithm Design Manual\".

He describes an algorithm for calculating the power of a number i.e. calculate a^n.

Answer 1

First of all, let's fix your algorithm:

function power( a, n )
    if (n = 0) 
        return(1)

    x = power(a,n/2)

    if (n is even) 
        return(x*x)
    else 
        return(a*x*x)

Say you want to calculate power(2,8), that is, 2^8 (^ is not XOR here, of course).

• 1) (a=2, n=8). 2^8 = (2^4)^2, so we have to calculate x=2^4, and then x*x to yield the final result. We have to call power() again to get x. power(2,4).

• 2) (a=2, n=4). 2^4 = (2^2)^2, so we have to calculate x=2^2, and then x*x to yield the final result. We have to call power() again to get x. power(2,2).

• 3) (a=2, n=2). 2^2 = (2^1)^2, so we have to calculate x=2^1, and then x*x to yield the final result. We have to call power() again to get x. power(2,1).

• 4) (a=2, n=1). 2^1 = (2^0)^2, so we have to calculate x=2^0, and then a*x*x to yield the final result. We have to call power() again to get x. power(2,0).

• 5) (a=2, n=0). 2^0 = 1 because n is 0, so we have the value of x that is returned to step #4.

• 4) (a=2, n=1, x=1). The final result for this step is a*x*x = 2*1*1=2, which is the value to be assigned to x in step #3.

• 3) (a=2, n=2, x=2). The final result for this step is x*x = 2*2 = 4. This is the result to be assigned to x in step #2.

• 2) (a=2, n=4, x=4). The final result for this step is x*x = 4*4 = 16. This is the result to be assigned to x in step #1.

• 1) (a=2, n=8, x=16). The final result for this step is x*x = 16*16 = 256. This is the result to be returned by power(2,8).