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CSC 263H               Tutorial Outline for Week  2             Winter 2004
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This week's tutorial is a review of asymptotic notation and basic
complexity analysis.  It's all material that you should know from CSC148H,
CSC165H, and CSC236H.

Asymptotic notation:

 1. Show that (n+1)^5 is O(n^5).

 2. Show that 2^{n+1} is O(2^n).

 3. Show that 2^{2n} is not O(2^n).

 4. Show that n is o(n log n).

 5. Show that n^2 is omega(n).

 6. Show that O(f+g) = O(max{f,g}).

 7. Show that for any k >= 0, 1^k + 2^k + ... + n^k is O(n^{k+1}).

 8. Give an example of a positive function f(n) such that
    f(n) is not O(n) and f(n) is not Omega(n).

Complexity analysis:

 9. Find the worst-case complexity of the following code fragment:

        p := 1
        for i := 1 to n:
            for j := 1 to i:
                p := p * j

10. Find the worst-case complexity of the following code fragment:

        p := 1
        for i := 1 to k^2:
            for j := 1 to i:
                p := p * j

11. Find the worst-case complexity of the following code fragment:

        while s > 1:
            s := s / 2     // integer division, rounding down

12. Find the worst-case complexity of the following recursive algorithm

        // Sort L[b..e]: the portion of list L between indices b and e
        // (inclusive).  Precondition: 1 <= b <= e <= length[L].
        MergeSort(L, b, e):
            if b < e:
                m := (b + e) / 2     // integer division
                MergeSort(L, b, m)
                MergeSort(L, m+1, e)
                Merge(L, b, m, e)
            // else L[b..e] contains only 1 element and is already sorted

    Assume that a call to Merge(L, b, m, e) takes time Theta(e-b+1)
    (i.e., makes Theta(e-b+1) comparisons).

13. Consider computing the value of a polynomial

        p(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n

    for a given value of x.

    (a) Describe a simple Theta(n^2) algorithm for doing this.

    (b) What is the complexity (number of additions and multiplications)
        of the following method, known as "Horner's method":

        p(x) = a_0 + x(a_1 + x(a_2 + ... + x(a_{n-1} + x a_n)...))