Lecture-02.md

Asymptotic notation

Smaller than or equal to

f(n) = O(g(n)) means there are constants C>0, n_0>0, such that 0 <= f(n) <= Cg(n) for all n>=n_0. Ex. 2n^2 = O(n^3).
Set definition: O(g(n)) is a set. n>=n_0 means n is sufficient large.

Greater than or equal to

f(n) = Ω(g(n)) means there are constants C>0, n_0>0, such that 0 <= Cg(n) <= f(n) for all n>=n_0. Ex. n^(1/2) = Ω(lgn)

Equal to

θ(g(n)) = O(g(n)) join Ω(g(n). Ex. n^2 = θ(n^2)

Solving recurrences

• Substitution method
  • Guess the form of the solution
  • Verify by induction
  • Solve for constants
  • Example: T(n) = 4T(n/2) + n
• Recursion-tree method
  • Example: T(n) = T(n/4) + T(n/2) + n^2
  • Summation at each level, find the pattern at each level, such as arithmetical series or geometrical series.
  • 1 + 1/2 + 1/4 + 1/8 + … = 2
• Master method
  • Very easy to use, can be viewed as an application of recursion tree.
  • Only applies to a particular family of recurrences.
  • T(n) = aT(n/b) + f(n): a sub problems and each size is n/b.
  • a >= 1, b > 1, f(n) is asymptotic positive (when n>n_0, f(n) is positive)
  • Three cases about f(n) and n^(logb_a)
  • n^(logb_a) is number of leaves in recursion tree