Search code examples
algorithmdynamic-programmingnp-complete

how to find the least number of operations to compute x^n

here is the problem from

ACM International Collegiate Programming Contest Asia Regional Contest, Yokohama, 2006-11-05

Starting with x and repeatedly multiplying by x, we can compute x^31 with thirty multiplications:

x^2 = x * x, x^3 = x^2 * x, x^6 = x^3 * x^3, x^7 = x^6 *x, x^14 = x^7 * x^7 ,
x^15 = x^14 * x, x^30 = x^15 * x^15 , x^31 = x^30 * x

write a program to find the least number of operations to compute x^n by multiplication and division starting with x for the given positive integer n and n<=200

for n = 31 the least #operations is 6
for n = 50 the least #operations is 7

Any ideas are welcome.

Solution

  • See this: http://en.wikipedia.org/wiki/Addition-chain_exponentiation

    There is no efficient algorithm that will get you the minimum number of steps, and dynamic programming does not work.

    I am guessing that n is small enough to allow a brute force solution to pass, although it might need to be optimized. Do you know how to brute force it?