Class 35: Algorithm Analysis (3)

Held: Tuesday, 1 November 2005

Summary: Today we continue our initial explorations of algorithm analysis with some further explorations of the notation and some experiments.

Related Pages:

• EBoard.
• Lab.
• Sample Code.

Notes:

• Eryn has noted that few of you are attending mentor sessions. Should we reschedule them?
• Homework to be determined at the end of class.

Overview:

• Review.
• Detour: Deletion, Revisited.
• About the Lab.

Review

• f(n) is in O(g(n)) iff
  • there exists a number n₀
  • there exists a number d > 0
  • for all but finitely many n > n₀, abs(f(n)) <= abs(d*g(n))
• This notation provides a convenient mechanism for talking about running time without worrying about too many potentially useless details.

decreasing

• At times when I've taught this course, students have worried about the analysis of the 'delete from left' problem.
• You may recall that the analysis goes something like this

  int len = stuff.length();       // 1 step
  for (int i = 0; i < len; i++) { // N repetitions
    stuff.delete(0);              // N steps to delete
  } // for
  

• The complaint is normally But after you've deleted a lot, it takes significantly fewer steps to delete something.
• It turns out that this decreasing number of steps has no real effect in the big picture.
• Here's another way to analyze this problem. We count only the steps in the deletion. That's N + (N-1) + (N-2) + ... + 2 + 1
• That sum is N(N+1)/2 (proof in class)
• N(N+1)/2 = (N^2)/2 + N/2, and is therefore also in O(N^2).

Lab

• Your homework for today was to do the lab on algorithm analysis.
• There were some problems with the lab (problems that were partially my fault), so we'll go over it in class.
• What did you find out about the more efficient exponentiation algorithm?