Fund. CS II (CS152 2005S)

Exam 2: ADTs, Data Structures, Algorithms

Distributed: Friday, April 8, 2005
Due: 11:00 a.m., Friday, April 15, 2005
No extensions.

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There are four problems on the exam. Some problems have subproblems. Each problem is worth 25 points. The point value associated with a problem does not necessarily correspond to the complexity of the problem or the time required to solve the problem.

This examination is open book, open notes, open mind, open computer, open Web. However, it is closed person. That means you should not talk to other people about the exam. Other than that limitation, you should feel free to use all reasonable resources available to you. As always, you are expected to turn in your own work. If you find ideas in a book or on the Web, be sure to cite them appropriately.

Although you may use the Web for this exam, you may not post your answers to this examination on the Web (at least not until after I return exams to you). And, in case it's not clear, you may not ask others (in person, via email, via IM, by posting a please help message, or in any other way) to put answers on the Web.

This is a take-home examination. You may use any time or times you deem appropriate to complete the exam, provided you return it to me by the due date.

This exam is likely to take you about four to six hours, depending on how well you've learned topics and how fast you work. You should not work more than eight hours on this exam. Stop at eight hours and write There's more to life than CS and you will earn at least 80 points on this exam. I would appreciate it if you would write down the amount of time each problem takes. Each person who does so will earn two points of extra credit. I expect that someone who has mastered the material and works at a moderate rate should have little trouble completing the exam in a reasonable amount of time. Since I worry about the amount of time my exams take, I will give two points of extra credit to the first two people who honestly report that they've spent at least five hours on the exam or completed the exam. (At that point, I may then change the exam.)

You must include both of the following statements on the cover sheet of the examination. Please sign and date each statement. Note that the statements must be true; if you are unable to sign either statement, please talk to me at your earliest convenience. You need not reveal the particulars of the dishonesty, simply that it happened. Note also that inappropriate assistance is assistance from (or to) anyone other than Professor Rebelsky (that's me).

1. I have neither received nor given inappropriate assistance on this examination.
2. I am not aware of any other students who have given or received inappropriate assistance on this examination.

Because different students may be taking the exam at different times, you are not permitted to discuss the exam with anyone until after I have returned it. If you must say something about the exam, you are allowed to say This is among the hardest exams I have ever taken. If you don't start it early, you will have no chance of finishing the exam. You may also summarize these policies. You may not tell other students which problems you've finished. You may not tell other students how long you've spent on the exam.

You must both answer all of your questions electronically and turn in a printed version of your exam. That is, you must write all of your answers on the computer, print them out, number the pages, put your name on every page, and hand me the printed copy. You must also email me a copy of your exam by copying your exam and pasting it into an email message. Put your answers in the same order as the problems. Please write your name at the top of each sheet of the printed copy. Doing so will earn you two points of extra credit.

In many problems, I ask you to write code. Unless I specify otherwise in a problem, you should write working code and include examples that show that you've tested the code.

Just as you should be careful and precise when you write code and documentation, so should you be careful and precise when you write prose. Please check your spelling and grammar. Since I should be equally careful, the whole class will receive one point of extra credit for each error in spelling or grammar you identify on this exam. I will limit that form of extra credit.

I will give partial credit for partially correct answers. You ensure the best possible grade for yourself by emphasizing your answer and including a clear set of work that you used to derive the answer.

I may not be available at the time you take the exam. If you feel that a question is badly worded or impossible to answer, note the problem you have observed and attempt to reword the question in such a way that it is answerable. If it's a reasonable hour (before 10 p.m. and after 8 a.m.), feel free to try to call me in the office (269-4410) or at home (236-7445).

I will also reserve time at the start of classes next week to discuss any general questions you have on the exam.


Problem 1: Techniques for Building Data Structures

Topics: Data structure design, Linked structures, Arrays

At this point, you've seen three basic techniques for building data structures:

Suppose you are presented with a new data structure implementation technique. You will need to select one of these techniques. What are some advantages and disadvantages of each? (You should list at least two advantages and two disadvantages for each technique.)

Problem 2: Minimizing Change

Topics: Algorithm design, experimental algorithm analysis, arrays

As you may recall, in the first exam you wrote a method that figured out the fewest coins that make a particular price. That method was a particular instance of a famous computer science problem known as the coin minimization problem. Given a set of coin values (e.g., 1, 5, 10, 25, 50 or 2, 3, 8, 25, 40), and a particular price, find the fewest coins that make that particular value. For example, given coin values of 2, 3, 8, 25, 40, the fewest coins to make a value of 19 is 3 (8, 8, and 3) and the fewest coins to make a value of 50 is 2 (25 and 25).

How do you figure out the fewest coins? One possible solution is the greedy solution: To find the fewest coins, assume you use as many as possible of the most expensive coin, and then as many as possible of the next most expensive coin, and so on and so forth. You used that solution on the previous exam. Unfortunately, that solution won't work for all combinations of coins. For example, if the coins have values 2, 3, 8, 25, and 40, the greedy solution will select 40, 8, and 2 to make 50 cents, when the best solution is to use 25 and 25.

A more correct solution is the exhaustive search solution. For each denomination, we assume that we use one of that denomination, reduce the amount appropriately, and recurse. We then take the smallest of those solutions. Here is an implementation of that solution:

package rebelsky.exam2;

 * Something that helps us count coins.
 * @author Samuel A. Rebelsky
 * @author YOUR NAME HERE
public class CoinCounter
// +--------+--------------------------------------------------
// | Fields |
// +--------+

   * The valid coin values.
  int[] denominations;

// +--------------+--------------------------------------------
// | Constructors |
// +--------------+

   * Build a new coin counter for a particular set of denominations.
   * @pre
   *   The values in _denominations must be stored in 
   *   decreasing order.
   * @pre
   *   All values in _denominations must be positive.
  public CoinCounter(int[] _denominations)
    this.denominations = _denominations;
  } // CoinCounter(int[])

// +----------------+------------------------------------------
// | Public Methods |
// +----------------+

   * Determine the fewest number of coins necessary to
   * make a particular price.
   * @exception Exception
   *   If it is not possible to make that price.
  public int fewestCoins(int price)
    throws Exception
    // Simple solution: No coins needed for no price.
    if (price == 0)
      return 0;
    // Fewest represents the fewest coins we've done so far
    int fewest = Integer.MAX_VALUE;
    for (int i = 0; i < this.denominations.length; i++) {
      int coin = this.denominations[i];
      if (coin <= price) {
        try {
          int sub = this.fewestCoins(price-coin);
          if (sub+1 < fewest)
            fewest = sub+1;
        } // try
        catch (Exception e) {
          // It's okay we failed, we'll just
          // try a different denomination.
        } // catch
      } // if (coin <= price)
    } // for
    // Make sure that we found a solution
    if (fewest == Integer.MAX_VALUE)
      throw new Exception("Can't make change for " + price);
    return fewest;
  } // fewestCoins(int)

} // class CoinCounter

You may find it useful to test the program with

a. Extend CoinCounter so that it lets you count the number of recursive calls that the fewestCoins does. Make a chart to show the growth in the number of recursive calls as the price increases.

As you may have noted, that implementation is not very efficient because we repeatedly compute the same value again and again and again. For example, in computing the number of coins for 40 cents using denominations of 2, 3, and 5, we check the number of coins for 35 cents at least three times (40-5, 40-2-3, 40-3-2). As importantly, each of those three computations requires at least three computations of the number of coins to make 30 cents (35-5, 35-2-3, 35-3-2). Hence, we do at least nine calls to fewestCoins(30). Similarly, we do at least twenty-seven calls to fewestCoins(25), at least 81 calls to fewestCoins(20), and so on and so forth.

We can make this computation significantly more efficient by remembering the number of coins for a particular price the first time we compute it and then checking it the next time. In pseudocode:

To compute the number of coins to make N cents
  if we have previously computed that result
    return the previously computed result
    BEST = N+1 // A very large number
    for each coin value, V
      compute C, the number of coins to make N-V
      if (C+1 < BEST)
        BEST = C+1
    remember that BEST coins are needed to make N cents
    return BEST

How do we remember the number of coins? We use an array indexed by the price. We initialize each cell to some value (say 0 or -1) to indicate not yet computed. How do we know the array is big enough? We allocate it at the first (outermost) call to fewestCoins. Approximately,

public int fewestCoins(int price)
  throws Exception
  // If the array of previously computed values is not big
  // enough, build one.
  if ( (this.previouslyComputed == null)
       || (this.previouslyComputed.length <= price) ) {
    this.previouslyComputed = new int[price+1];
    for (int i = 0; i < price; i++) {
      this.previouslyComputed[i] = this.NOT_COMPUTED;
  } // if
  // If we've previously determined that no such value exists,
  // throw an exception.
  if (this.previouslyComputed[price] == this.NO_SUCH_VALUE)
    throw new Exception("No combination of coins works.");
  // If we've previously found a number of coins, return it
  if (this.previouslyComputed[price] != this.NOT_COMPUTED)
    return this.previouslyComputed[price];
  // ...
} // fewestCoins(int)

b. Rewrite fewestCoins to use this improvement.

c. Make a chart to show the growth in the number of recursive calls as the price increases. What does the new growth curve look like?

Problem 3: Randomized Linear Structures

Topics: Linear structures, Randomness, Implementing ADTs, Asymptotic analysis

An interesting (and perhaps even useful) kind of linear structure is the Random List. A random list is one in which the value returned by get is difficult to predict. (In particular, it should be equally likely that any member of the structure be returned.)

a. Write a RandomList interface. For this part of the problem, you should focus on writing good documentation.

b. Implement random lists, either using linked nodes or an array.

c. Indicate the running time of put, get, peek, and isEmpty for your data structure.

Problem 4: Generalized Heaps

Topics: Heaps, Trees, Comparing objects

In a class and laboratory, you have learned about and contributed to the implementation of a heap of integers.

Generalize that implementation to permit a heap of any kind of comparable value.

Some Questions and Answers

These are some of the questions students have asked about the exam and my answers to those questions.


Here you will find errors of spelling, grammar, and design that students have noted. Remember, each error found corresponds to a point of extra credit for everyone. I limit such extra credit to five points.



March and April 2005 [Samuel A. Rebelsky]

Thursday, 7 April 2005 [Samuel A. Rebelsky]

Friday, 8 April 2005 [Samuel A. Rebelsky]

Monday, 11 April 2005 [Samuel A. Rebelsky]

Wednesday, 13 April 2005 [Samuel A. Rebelsky]


Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes.

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The source to the document was last modified on Wed Apr 13 11:07:49 2005.
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Samuel A. Rebelsky,