Question

From http://discuss.joelonsoftware.com/default.asp?interview.11.794054.1

The sequence A is defined as follows:

Start with the natural numbers
1,2,3,…

Initialize count = 1;

    while(there are uncrossed numbers)
    {
        pick the first uncrossed number say n.
      set A[count] = n.
      Cross out the number count+n.
      Cross out the number n
      Increment count.
    }

Give a fast algorithm to determine A[n], given n.

Try getting an algorithm which is polynomial in log n.

Answer 1

Sorry for posting this question.

Apparently this is a famous sequence called Wythoff’s sequence and there is a neat formula for A[n] given by A[n] = [n*phi] where [x] = integral part of x and phi is the golden ratio.

To calculate [nphi], we can approximate phi as a ratio of consecutive fibonacci numbers, giving an O(lognloglogn) algorithm. (O(logn) time to do arithmetic on O(log n) bit numbers).