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CSC 301.01, Class 07: The Master Theorem

Overview

  • Preliminaries
    • Notes and news
    • Upcoming work
    • Extra credit
    • Friday PSA
    • Questions
  • Detour - Recursion vs. iteration
  • Recursion trees, revisited
  • Proving asymptotic bounds of recursive functions
  • The master theorem

News / Etc.

  • We are now 1/7 of the way through CSC 301. After today’s class, we’ll be 1/6 of the way through CSC 301.
  • I hope to get homework back to you on Monday. (Probably only one of the two, but we’ll see.)

Upcoming work

  • Reading for Monday: Review the Master theorem in CLRS 4.
  • Assignment 3, due 10:30 pm next Wednesday

Extra credit (Academic)

  • CS Table, Tuesday, Machine Ethics

Extra credit (Peer)

  • Sign up for CS picnic (but only if you plan to attend). (You can reflect on the sign-up process or on the picnic itself.)

Extra Credit (Misc)

  • Host a prospective student [ohc]

Other good things

  • Women’s Tennis vs. Coe, TODAY at 4:30 p.m., High School
  • Women’s Tennis vs. Lake Forest, Saturday at 9 am High School
  • Women’s Tennis vs. Beloit, Saturday at 3 pm High School
  • Les Duke Cross Country Meet, Saturday at 9 a.m., Country Club
  • Men’s Soccer vs. North Central College, Saturday at 1:00 p.m., Springer Field
  • Women’s Soccer vs. University of Wisconsin-Oshkosh, Sunday at 1:00 p.m., Springer Field

Friday PSA

  • You’re great people.
  • Please stay that way.

Questions

On assignment 3, we have code and written.
Email Sam tarball of code.
Put written under Sam’s door (or in the envelope, if he ever puts one out)

Recursion trees, revisited

  • Drawing trees can help us estimate the running time!
  • We show the size of the problem at each level.
  • We can sum across each level to see the work at that level.
  • We then sum those sums to get an overall estimate of the running time.
  • This analysis requires that we identify the sum of the values at each level and the number of levels.
  • On Wednesday, we built the recursion tree for T(n) = 2*T(n/2) + cn.

Exercises

  • Build the recursion tree for T(n) = 2*T(n/2) + c.
  • Build the recursion tree for T(n) = 2*T(n/2) + n^2.
  • Build the recursion tree for T(n) = 3*T(n/2) + c.

Detour: Recursion vs. iteration

Solving recurrence relations with the Master Theorem

The master theorem works for recurrences of the form T(n) = aT(n/b)+f(n) or T(n) <= aT(n/b)+f(n)

  • What does a represent?
  • What does b represent?
  • What does f(n) represent?

There’s a simpler version (which I’m taking from Weiss). This is for recurrences of the form T(n) = aT(n/b) + O(n^k)

  1. If a > b^k, then T(n) is in O(n^(log_b(a)))

  2. If a = b^k, then T(n) is in O((n^k)*log_2(n)))

  3. If a < b^k, then T(n) is in O(n^k)

I may need to recheck those.

We’ll try some basic examples.

The more general computation depends on the relationship between f(n) and how quickly the n/b drops to 0.

  1. If f(n) is in O(n^(log_b(a)-e)) for some e > 0, then T(n) is in Theta(n^(log_b(a)))

  2. If f(n) is in Theta(n^log_b(a)), then T(n) is in Theta(n^(log_b(a)*log(n)))

  3. If f(n) is in Omega(n^(log_b(a)+e))_ for some e > 0, and af(n/b) <= cf(n) for some c < 1 and large enough n, then T(n) is in Theta(f(n)).