Homework 1

  • Assigned: Thursday, September 10, 2015
  • Due: Friday, September 25, 2015
  • General Instructions: 
  1. Please review the homework and grading policy outlined in the course information and homework page.
  2. Please use a cover sheet, as described in the homework page.
  3. You must also write down with whom you worked on the assignment, and any sources used.
  4. Start early, be concise but rigorous, and enjoy!
  • Specific Instructions:
  1. Be sure to review any parts of Chapters 1-4 with which you are not familiar.
  2. All logs are base 2 unless otherwise specified.
  3. Unless otherwise specified, all problems and exercises are from CLRS, Introduction to Algorithms, 3rd edition.
  • Problems to do but not hand in:
    1. Problem 2-3.
    2. Problem 2-4.
  • Problems To Hand In:
    1. Show that for any positive real constants a, b, and d, (an + b)d = Θ(nd).
    2. Is 34n = O(24n)? Is 34n = 2O(4n)
  2. Exercise 3.2-4 and 3.2-5. Bounds on Functions.
  3. Problem 3-4 a, b, g, h . Big-O and other notation.
  4. Place the following functions by order of growth, from smallest to largest. For each adjacent pair in the list, give a proof that the earlier one is assymptotically smaller than (or equal to ) the second, by giving a big-O proof.
      • n2
      • n2 - n/ lg n
      • n1.01
      • 2n
      • 3n/2
      • 4lg lg n
      • lg3 n
      • 4lg n
      • 16 n + 10000
      • (lg n)lg n / lg lg n
  5. Exercise 4.4-3. 4.4-8. and Problem 4-3. b, c, h, j. For each recurrence, provide an upper bound. If the recurrence cannot be solved by the master method, you must justify the solution via a substitution proof or by drawing a recursion tree.
  6. Problem 4-5. Chip Testing.