CSOR 4231 05F: Homework 1

Assigned: Monday, September 12, 2005
Due: Wednesday, September 21, 2005

Required for Undergraduates: 5 of the following 8 problems
Required for Graduate Students: 6 of the following 8 problems
Points: 10 pts per problem

General Instructions

1. Please review the homework and grading policy outlined in the course information page.

2. On the first page of your solution write-up, you must make explicit which problems are to be graded for "regular credit", which problems are to be graded for "extra credit", and which problems you did not attempt. Please use a table something like the following

Problem01020304 0506070809...
CreditRCRCRCECRC RCNARCRC...

where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not attempted" (not applicable). Failure to do so will result in an arbitrary set of problems being graded for regular credit, no problems being graded for extra credit, and a five percent penalty assessment.

3. You must also write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.
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, 2nd edition.

Problems

1. Problem 2-3. Horner's Rule.
2. Problem 2-4. Inversions
3. Problem 3-4 d,e,f,h. Big-O notation.
4. Problem 4-1 a, d, h. Problem 4-4 b,g,j. Recurrences.
5. Problem 4-6. VLSI Chip Testing.
6. 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 the second, by giving a big-O proof.
• n2
• n.01
• 2n
• 2n/2
• log8 n
• 8log n
• n
• (log n) log n
7. Exercise 6.1-4, 6.1-5. Heaps.
8. Exercise 6.4-2. Correctness of Heapsort.

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cliff@ieor.columbia.edu