Lab 8: Skiing

Ideally, each skier should obtain a pair of skis whose heigth matches his or her own height exactly. Unfortunately, this is generally not possible. We define the disparity between a skier and her pairs of skis as the absolute value of the difference between the height of the skier and the height of the skis.

The objective is to find an assignment of skis to skiers that minimizes the sum of the disparities.

1. First, let's assume that there are m skiers and n pairs of skis (that is, same number of pairs of skis and skiers). One day, while waiting for the lift, you make an interesting discovery: if we have a short person and a tall person, it would never be better to give to the shorter person a taller pair of skis than were given to the taller person. Prove that this is always true.

   Hint: Let P1, P2 be the length of two skiers, and S1, S2 the lengths of the skis. We assume P1 < P2 and S1 < S2. Prove that pairing P1 with S1 and P2 with S2 is better than pairing P1 with S2 and P2 with S1. Basically we'd like to show that |P1 - S1| + |P2 - S2| <= |P1 - S2| + |P2 - S1|. To prove this consider all possible cases:

   1. P1 < S1 < P2 < S2:
   2. P1 < S1 < S2 < P2:
   3. P1 < P2 < S1 < S2:
   4. S1 < P1 < P2 < S2:
   5. S1 < P1 < S2 < P2:
   6. S1 < S2 < P1 < P2:

2. Using this observation, describe a greedy algorithm to assign the (pairs of) skis to skiers, and argue why this algorithm is correct. What is the time complexity of your algorithm?

3. Your next task is to design an efficient algorithm for the more general case where there are m skiers and n pairs of skis and m < n (i.e. more skis than skiers). The greedy solution above does not work in this case (can you come up with a simple example?), so you'll come up with a dynamic programming solution.

   Here is some notation that may help you.

   • Sort the skiers and skis by increasing height.
   • Let h[i] denote the height of the i-th skier in sorted order, and s[j] denote the height of the j-th pair of skis in sorted order.
   • Let OPT(i,j) be the optimal cost (disparity) for matching the first i skiers with skis from the set 1, 2, ..., j.
   • The solution we seek is simply OPT(m, n).

   Define OPT(i,j) recursively.

4. Illustrate your algorithm by explicitly filling out the table OPT(i,j) for the following sample data: ski heights {1, 2, 5, 7, 13, 21}. Skier heights {3,4, 7, 11, 18}.

   Note: This requires a lot of patience but it makes you really understand what's going on. You may consider skipping this part and going to the programming part below.

The skis and skiers in practice

• One that corresponds to solving the problem recursively without dynamic programming. You could name it OPT-recursive(i,j)
• One that corresponds to the dynamic programming solution. You could name it OPT-dynprogr(i,j).

1. Each function should take i,j as parameters, and should return the optimal cost of the assignment (not the actual assignment of skis to skiers; as usual we claim that it can be added easily). In this assignment the goal is to get an understanding of the running time with and without dynamic programming.

2. The arrays that hold the heights of the skis and the skiers can be global variables.

3. The number of skiers and skis, m and n, need to be read from the command line using argv and argc (if you use c/c++); use something similar if you use python, but the idea is that the user does not need to edit your code and recompile to change n and m.

4. Initialize the skis and skiers height arrays with random values that are reasonable for skis and skiers heights (assume that heights are integers, in inches, so come up with a reasonable range).

5. We will run your program like this:

   g++ marty.cpp -o marty
   ./marty 100 200
   you entered m=100 n=200
   running opt_dynprogr  with m=100, n=200: total time = ...
   running opt_recursive with m=100, n=200: total time = 
   

6. Test both functions on increasing values of m and n, until you notice a significant difference in running time between the two methods.

7. Run your code on the heights in part (4) above, and include this table in your report.

8. Drive to Sugarloaf, take a picture of yourself in front of the lift, and post it on piazza.

9. Collaboration policy: This problem is individual, you cannot have a partner. But you can discuss it with anyone in the class, as long as you follow the honor code.

10. What to turn in:
    1. The solution to the version of this problem where the number of skis and skiers is tha same, together with a proof of correctness.
    2. email me the .cpp file so that we can test it.
    3. A hard-copy of your code
    4. A report including:
       1. The solution table for part (4) above, that is, for ski heights {1, 2, 5, 7, 13, 21} and skier heights {3,4, 7, 11, 18}.
       2. a summary of how you tested your algorithms and on what platform (what machine, what speed, what compiler, what IDE, if any)
       3. the running times on increasing values of n and m, until the difference in speed is significant (from seconds to minutes or more)
       4. conclusions
       The report can be very brief, but needs to be structured as above.

Enjoy!