CSC 344
Homework 3

Assigned Mar. 8, due Mar. 28

Problems from the textbook

• Chap. 4, problem 3 (packing trucks)
• Chap. 4, problem 6 (scheduling triathlon competitors)
• Chap. 4, problem 9 (a variant on minimum-spanning-tree)
• Chap. 4, problem 15 (scheduling student workers)

Weighed in the Balance

A balance scale is a device with two pans, on each of which you can put heavy objects. It tells you whether the stuff in the left pan is lighter, heavier, or the same weight as the stuff in the right pan. The most common way to use one is to put an unknown object on the right pan, then add one or more pre-calibrated weights to the left pan until they balance, at which point you can tell what the unknown object weighs by adding up the calibrated weights.

Suppose you're given a bunch of objects that each weigh an integer number of ounces, and you need to find out how much each one weighs. Before you start, you are allowed to choose n calibrated weights: for example, if n were 2, you might choose to have a 1-ounce weight and a 3-ounce weight. What weights of unknown objects could you handle with these?

Given the number n, how would you choose calibrated weights so as to be able to handle every possible integer number of ounces 1, 2, 3, ... W where W is as large as possible?
Hint: Solve this problem first for the situation in which you can only put calibrated weights on the left pan; then solve it for the situation in which you can put calibrated weights on either or both pans.

Making change

Consider the problem of making change using pennies, nickels, dimes, and quarters (i.e. coin values 1, 5, 10, 25).

1. Give a greedy algorithm (in pseudocode) which, for any specified number of cents, makes that amount of change with as few coins as possible. (1 point)

2. The aforementioned greedy algorithm works fine in the U.S, but it doesn't necessarily work with other coin values. Give an example of a set of coin values for which the aforementioned greedy algorithm does not choose the fewest possible coins. (1 point)

3. Extra credit: What properties of a set of coin values would guarantee that the greedy algorithm works? (2 more points)

Programming:

Implement, test, and debug the coin-changing algorithm from above in a real programming language.

The part of the program that does the real work should be callable from other parts of the program, e.g. with known test cases or to actually use the results for some other purpose.