Homework 3 Solution

1. Purpose: Apply various algorithm design strategies to solve a problem, practice formulating and analyzing algorithms, implement an algorithm. In the US, coins are minted with denominations of 50, 25, 10, 5, and 1 cent. An algorithm for making change using the smallest possible number of coins repeatedly returns the biggest coin smaller than the amount to be changed until it is zero. For example, 17 cents will result in the series 10 cents, 5 cents, 1 cent, and 1 cent. 
a) (4 points) Give a recursive algorithm that generates a similar series of coins for changing n cents. Don’t use dynamic programming for this problem. 

b) (4 points) Write an O(1) (non-recursive!) algorithm to compute the number of returned coins.

c) (1 point) Show that the above greedy algorithm does not always give the minimum number of coins in a country whose denominations are 1, 6, and 10 cents.

d) (6 points) Given a set of arbitrary denominations C =(c1,...,cd), describe an algorithm that uses dynamic programming to compute the minimum number of coins required for making change. You may assume that C contains 1 cent, that all denominations are different, and that the denominations occur in in increasing order.

e) (6 points) Implement the algorithm described in d). The code framework are given in the zip file: framework.zip. To avoid loss of marks please make sure that all provided test cases pass on remote-linux server by using the test file. Instructions for  setting up remote-linux server and testing are given in the document HW3_Programming_Assignment_Setup.pdf.


Problem 2. (10 points) In class we showed that multiplying two matrices
C            A  *  B
m☓p         m☓n  n☓p 
requires mnp scalar multiplications. You are given the following matrix chain:

A1    *    A2    *   A3     *   A4  
             20☓25   25☓5   5☓10    10☓30  
            d0☓d1    d1☓d2     d2_☓d3    d3☓d4

Denote m[i,j] the minimum number of scalar multiplications  to compute Ai * … *Aj. In class we showed the following recurrence:

    0 ;  if i=j                        
m[i,j]   :=
             min i-1<k<j m(i, k)+m(k +1, j)+di−1dkd j   ; otherwise

a) (6 points) Fill the Table below with the missing values for m[i,j]. Also, for each m[i,j] put the corresponding value k, where the recurrence obtains its minimum value, next to it.
i\j
1
2
3
4
1




2
Undefined



3
Undefined
Undefined


4
Undefined
Undefined
Undefined


Use the table to answer the following questions:
b) (1 point) What is the minimum number of scalar multiplications required to compute A1* A2 *A3 *A4 ? 

c) (2 points) Give the optimal order of computing the matrix chain by fully parenthesizing the matrix chain below.

A1         *      A2     *        A3        *        A4 
d) (1 points) How many scalar multiplications are used to compute (((A1*A2) * 􀀀A3 ) *A4)? Keep the order of matrix multiplications indicated by the brackets. Justify your solution.


Problem 3. Purpose: practice designing greedy algorithms. (10 points) Suppose you have a long straight country road with houses scattered at various points far away from each other. The residents all want cell phone service to reach their homes and we want to accomplish this by building as few cell phone towers as possible.
More formally, think of points x1, …, xn, representing the houses, on the real line, and let d be the maximum distance from a cell phone tower that will still allow reasonable reception. The goal is to find a minimum number of points y1,…,yk so that, for each i, there is at least one j with | yj - xi | ≤ d.
Describe a greedy algorithm for this problem. If the points are assumed to be sorted in increasing order your algorithm should run in time O(n). Be sure to describe the greedy choice and how it reduces your problem to a smaller instance. Prove that your algorithm is correct.


Problem 4. Purpose: reinforce your understanding of data structures for disjoint sets. For background on binomial trees and binomial heaps please read 19-2 on page 527. (6 points) Please solve Problem 21.3-3 on page 572 of our textbook.