Programming Exercise 3

This exercise has two purposes: to strengthen your "data structures with pointers" skills and your debugging skills. You will implement a linked list version of the mathematical object called a "vector" and implement the inner-product (or dot-product). This is a trivial computation if vectors are implemented as arrays but becomes interesting when vectors are implemented as linked lists.

To start with, let's review vectors:

• A vector is an ordered collection of real numbers. A vector has a "size" (or dimension, in math terminology). Thus, (0.4, 1.55, 78.9, 0.33) is a vector of size 4 (or dimension 4). Similarly, (38.3, -14.4, 5.0, -6.0, 19.8, 445.2) is a vector of of size 6.
• In general, a vector v of size n is a collection of numbers (v₀, v₁, ..., v_n-1).
• The inner product of two vectors u = (u₀, u₁, ..., u_n-1) and v = (v₀, v₁, ..., v_n-1) of the same size is defined as: u . v = u₀v₀ + u₁v₁ + ... + u_n-1v_n-1. Thus, each element of one vector is multiplied with its corresponding element in the other vector, and all of these partial products are added up to give a single number.
• One way to represent vectors is to use arrays. Then, the following for-loop computes the inner-product:

      double innerProduct = 0;
      for (int i=0; i<v.length; i++) {
        innerProduct = innerProduct + (vector1[i] * vector2[i]);
      }
    

• The array representation works fine for small vectors. However, in many applications, we wish to store extremely large vectors with sizes in the range of 1 million. In these applications, it's frequently the case that most individual elements are zero, as in this 20-element vector:

      (0, 0, 0, 0, 0, 5.23, 0, 0, 0, 33.1, 0, 0, 0, 0, 0, 0, -0.9, 0, 0, 0)
      

• For such applications, it's preferable to use linked lists to store only the non-zero values. The zero values don't need to be stored because they don't contribute to the inner-product.
• So, the above vector can be stored as the following list:

        [ (5, 5,23)   (9, 33.1)    (16, -0.9) ]
      

  This says that the first non-zero value 5.23 occurs at position 5, then the next non-zero value 33.1 occurs at position 9 and so on.
• Your goal in this exercise is to implement a linked list version of vectors, in particular, the inner-product operation using lists. This is a good exercise in lists because you will have to simultaneously traverse two lists to compute the inner product. As part of testing your implementation, you will compute the average inner product from two randomly generated vectors.

Details:

• Start by downloading the following jar file. Upon unpacking, you will see the files RandomVector.class, LinkedVector.java and UniformRandom.class.
• Your code will be written in the file LinkedVector.java. This has one constructor and three methods:
  • The parameter to the constructor is the size of the vector.
  • The addValue method will be called to insert a particular element at a particular position. This position is a position in the vector, not in the list. You will need to implement this method.
  • The innerProduct method will be called with another LinkedVector as parameter. You are to compute the inner product by walking along the two lists in a single sweep.
  • Finally, you are to implement toString() that prints out the full vector, including the zeroes.
• To create a random vector of a given size, simply call the method RandomVector.makeRandomVector (int size) with the desired size, as in:

       // To create a vector of 8 elements:
       LinkedVector V = RandomVector.makeRandomVector (8);
     

  This method will call your addValue method to add elements to a LinkedVector instance.
• Create a main method in your LinkedVector class. Add some tests in your main to ensure that your list implementation of vectors is working correctly and submit evidence of the same to us. These tests should include one test in which you add at least 5 elements and in which you print out the exact memory locations of each element in the list. Include a drawing showing step-by-step insertions (with memory addresses) for this one test case.
• Suppose you create two vectors of size K randomly, and you compute the inner product. What is the average value of this inner product when K=10? When K=20?

Submission:

• Submit the source code as before, in an encrypted tar-file. This time, the name of subdirectory for this exercise should be: your username followed by 3.
• Follow the submission instructions carefully as usual.
• Submit a printout (hardcopy) of your code before the due date in my mailbox.