Read [Gray], Chapters 2 and 3. Consider the "Checkpoint" questions as part of your reading assignment: answer them but don't turn in your answers. This assignment is very unusual because it has a lot of "busy work," but these exercises are designed to give you the practice you need at working with sets, functions, logs, and asymptotic complexity. No whining.

Submit answers to the following problems, in a nicely typeset pdf file:

1. Write 2^{{a, b, c}} in a form in which all elements are listed.
2. How many partitions are there of {x | x ∈ N ∧ 1 ≤ x ∧ x ≤ 10}?
3. Let A be a set and m and n be positive integers. Would you say that A^m+n = A^m × Aⁿ? Give arguments for or against accepting this equality as fact. Hint: you may want to consider the sets A × A² and A² × A.
4. Express, in lambda notation, the function which when passed two functions f and g, returns the composition of f and g. (Don't use the predefined "o" symbol for composition. Actually define the meaning of composition in your answer.
5. Consider the function λx. 4x-3x². Call this function f.
   1. What is f⁰(0.01)?
   2. What is f¹(0.01)?
   3. What is f²(0.01)?
   4. What is f⁵(0.01)?
   5. What is f³⁰(0.01)?
   6. What is f⁵⁰(0.01), according to Java?
   7. What is f⁵⁰(0.01), according to your handheld calculator?
   8. Explain why we can never write out the value of f⁷⁵(0.01) in 10 point Times New Roman on printer paper that is produced on Earth.
   9. Do you think it will possible in our lifetime whether we will ever know the first digit of the decimal expansion of f¹⁰⁰(0.01). Why or why not?
6. What is the time complexity of this code fragment? (Use Θ-notation)

   int x = 1;
   for (int i = 0; i < n; i++) {
       for (int j = 1; j <= x; j++) {
           System.out.println("*");
       }
       x = x + x;
   }
   

7. What is the time complexity of this code fragment? (Use Θ-notation)

   for (int i = 0; i < n; i++) {
       for (int j = 0; j < i; i *= 2) {
           System.out.println("*");
       }
   }
   

8. What is the time complexity of this code fragment? (Use Θ-notation)

   for (int i = 1; i * i <= n; i++) {
       for (int j = 1; j <= n; j += j) {
           System.out.println("*");
       }
   }
   

9. What is the time complexity of this code fragment? (Use Θ-notation)

   for (int i = 1; i <= n; i *= 2)
       for (int j = 1; j <= i; j++)
           System.out.println("*");
       }
   }
   

10. What is the time complexity of this code fragment? (Use Θ-notation)

    for (int i = 1; i <= n * n; i++) {
        for (int j = 1; j <= n; j *= 2) {
            System.out.println("*");
        }
    }
    

11. What is the time complexity of this code fragment? (Use Θ-notation)

    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= i; j *= 2) {
            System.out.println("*");
        }
    }
    

12. What is the time complexity of this code fragment? (Use Θ-notation)

    for (int i = n; n >= 1; n /= 4) {
        for (int j = 0; j < i; j++) {
            System.out.println("*");
        }
    }
    

13. Give both the best-case and worst-case time complexities of this code fragment. (Use Θ-notation)

    // Assume some global variable t is defined out here
    for (int i = n; n >= 1; n /= 2) {
        if (new Date().getTime() > t) {
            for (int j = 0; j < i; j++) {
                System.out.print("*");
            }
        }
    }
    

14. Give both the best-case and worst-case time complexities of this function. (Use Θ-notation)

        public static long power(int x, int n) {
            return n == 0 ? 1
                : n % 2 == 0 ? power(x * x, n / 2)
                : x * power(x * x, n / 2)
        }
    

15. What is the time complexity (in Θ-notation) of a procedure to print out the exact value of 2ⁿ, where n is a nonnegative integer? The procedure described is supposed to print a result no matter how large the result may be.
16. An algorithm with time complexity T(n) = n³ can process a 100-element list on our PC in 10 seconds.
    1. How long would it take to process a 200-element list?
    2. If we ran the algorithm on a machine that was 10 times faster than our PC, how large of a list could we process in 30 seconds?
    3. How much faster than our PC would a computer have to be in order to process a 1000000000-element list in a time span of 1 hour?
17. An algorithm with complexity function T(n) = n log n processes a 64-element list in three minutes and 12 seconds on our PC.
    1. How long does it take to process a 128-element list?
    2. How large of a list could a computer that is 8 times faster than our PC process in 10 seconds?
    3. How much faster would a computer have to be than our PC to process a list of size 10 in a second?
18. An algorithm of with time complexity function T, where T(n) = 2n log n, can process a 32 element list in 2 minutes and 40 seconds on our PC.
    1. How long would it take to process a 64 element list?
    2. If we ran the algorithm on a machine that was 4 times faster than our PC, how large of a list could we process in 16 seconds?
19. We have seen that there is little hope of solving problems of size 100 or so with algorithms of complexity Θ(2ⁿ) even when billions of operations can be carried out per second. But what about algorithms of complexity Θ(1.1ⁿ)? How do these algorithms compete with quadratic algorithms? In particular, if a billion operations can be performed in one second, up to what problem size will the Θ(1.1ⁿ) be faster than a Θ(n²) algorithm?
20. Rank each of the following functions by growth rate: λn. n, λn. n², λn. n^1.5, λn. sqrt(n), λn. n log n, λn. n log log n, λn. n² log n, λn. n log n², λn. n (log n)², λn. 2, λn. n³, λn. 2/n, λn. 2 log n, λn. 2ⁿ, λn. 2^log n, λn. 2^n*n, λn. 2^n/2, λn. n!, λn. nⁿ, λn. (log n)ⁿ, λn. log nⁿ, λn. log(sqrt(n)),

Compose the solutions to the project using the LaTeX document preparation system, whether you want to or not. Create a pdf from your document source, print it out, and hand it in by the due date. It is not necessary to email your solutions to this assignment.

Keep the LaTex source of your work around. If you have not yet set up a CVS or SVN repository of your work don't worry, because it will be required for next assignment. At that point you can backfill your repository with your work from this and the previous assignment.

If you are already building up a repository, add to it as follows:

homework
  cmsi281
    src
      main
        docs
          hw3.tex
          <diagram source files>, if any
          <images>, if any