CSG713 04F: Homework 07

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General Instructions

1. Please review the 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

   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 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.

Problems

1. Exercise 24.3-4 and Problem 24-3 (a)

2. In each of the following problems, you should create a single graph and appropriate edge weights in such a way that the output of a "shortest paths" algorithm run on your graph can be used to solve the original problem.
   • Solve the "rental car" problem by using a graph algorithm for determining shortest paths. Argue that your solution is correct, and analyze its running time and space requirements.

   • Solve the "chess board" problem by using a graph algorithm for determining shortest paths. Argue that your solution is correct, and analyze its running time and space requirements. (Note: Your solution must involve creating a single graph and running a shortest path algorithm only once.)

3. Problem 25-2, parts (a, b, c).

   Hint: A d-ary heap is an array indexed starting at 0 where the parent of the element in position i, p(i), and the k-th child of the element in position i, child_k(i), are defined as follows:

   • p(i) = floor((i-1)/d)
   • child_k(i) = d i + k

4. Exercise 26.2-8

5. Exercise 26.2-9

6. Problem 26-5

7. In class, we proved that the expected height of a randomly built binary search tree on n distinct elements, E[X_n], is at most 3 lg n - Omega(1). (Since the constant is relevant here, it is important to note that "lg" is log base 2.)

   The purpose of this problem is to derive the optimal constant in front of the log by using the optimal base (instead of 2) in the definition of exponential height.

   • Let Y_n = c^X_n, for any constant c > 1. Show that

   • Show that
     

     for all k >= 1.

     Hint: Use an integral approximation to the summation; see pg. 1067 in CLRS.

   • Now let Z_n = E[Y_n]. Show that a solution to the recurrence inequality

     is

     

     Hint: Use (strong) induction on n and the fact you proved above.

   • Show that Hint: Use Jensen's inequality as we did in class.

     Finally, we note that the quantity

     

     is minimized when c is approximately 2.155, yielding a constant factor of 2.9882 in front of the lg n. This constant is provably as small as possible.

8. Exercise 35.1-3

9. Problem 35-1

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