CS5800 09F: Homework 04

Assigned: Wed 30 Sep 2009
Due: Wed 7 Oct 2009

Instructions

1. Please review the course syllabus and make sure that you understand the course policies for grading, late homework, and academic honesty.

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

   Problem	01	02	03	04	05	06	07	08	09	...
   Credit	RC	RC	RC	EC	RC	RC	NA	RC	RC	...

   where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not applicable" (not attempted). 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.

Problems

Required: Do four of the five problems below. Problem 4 is an exercise in the book but counts as a problem here.
Points: 20 points per problem.
Unless otherwise indicated, exercises and problems are from Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein. The edition (2^nd or 3^rd) will be indicated if the numbering differs.

1. Problem 8-2
2. Problem 8-6
3. Suppose that you are given the task of devising an algorithm for sorting n records and that the basic operation in your algorithm is an exchange: Exchange(i, j) swaps the data records in positions i and j. (Insertion Sort is an example of such an algorithm.) The width of an exchange is defined to be the distance across which data records are moved, i.e., j - i for Exchange(i, j). Prove that any algorithm for sorting which uses only constant width exchanges must run in Ω(n²) time in the worst case.
4. Exercise 9.3-7
5. Let the multi-selection problem be defined as follows. The input is a set S of n distinct keys drawn from some totally ordered universe, and a set of k + 1 integers r_i such that 1 = r₀ < r₁ . . . < r_k = n +1. The output is a partition of S into k sets S₀, . . ., S_k-1 such that S_i is the set of all keys with ranks greater than or equal to r_i and strictly less than r_i+1. Give an O(n lg k) time comparison-based algorithm for the multi-selection problem. You may assume for simplicity that k is a power of 2.

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