> (mod-expt-mults signing-exponent)
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Q: If e is the exponent, then how is the number of multiplications expressed in big-Theta notation?

• A: Θ(log e) instead of Θ(e)

  The third version of mod-expt is an example of logarithmic recursion.

  Q: How is the amount of memory required by mod-expt expressed in big-Theta notation?

• A: Θ(log e)

• First pass: All n cards are handled one or two times
• Second pass: n-1 cards are handled one or two times
• Third pass: n-2 cards are handled one or two times
• ...
• n + (n-1) + (n-2) + ... + 1

  is at most n² and at least ¼n² (n/2 of the numbers are n/2 or larger). So:

• ¼n² ≤ H(n) ≤ 2n²

  (Factor of 2 on the upper bound because each card is handled one or two times.)

• First pass: All n cards are handled once to create n/2 stacks
• Second pass: All n cards are handled once to create n/4 stacks
• Third pass: All n cards are handled once to create n/8 stacks
• ...
• Each pass takes a certain number of stacks and cuts it in half, doubling the size of each stack.

  Q: How many passes are required?

• n = 2^p-1
• log₂n = log₂2^p-1
• log₂n = p-1
• p = log₂n + 1
• So the number of passes required by a merge sort of n numbers is log₂n + 1.

• H(n)	=	n(log2n + 1)
  	=	nlog2n + n

• Since n ≤ nlog₂n, then

  H(n)	≤	nlog2n + nlog2n
  	≤	2nlog2n

• But H(n) ≥ nlog₂n, so

  nlog2n

  ≤

  H(n)

  ≤

  2nlog2n

• H(n) is bounded by constant multiples of nlog₂n. (The constant for the lower bound is 1; the constant for the upper bound is 2.) Thus the time it takes to merge sort n cards is:
  Θ(nlog₂n)
• This can also be expressed without a logarithmic base as:
  Θ(nlogn)

  since a number's logarithms of different bases differ only by a constant factor (log_ca = log_ba × log_cb).

Comparing n, nlogn, and n²

Here are the values of the functions for n = 10:

The chart to the right shows the functions n (red), nlogn (purple), and n² (green) with the horizontal axis extending to n = 100, with n² = 10,000.

Note the difference between the general shapes (that is, the "orders of growth") of this graph and the preceding one:

As n increases by an order of magnitude, the difference between nlogn and n² becomes more dramatic.