CSE 331 Homework 2 Part I - Winter 2012

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Due: Wednesday, Jan. 18, at 11 pm. Sunday, Jan. 22, at 11 pm; no late days allowed and no assignments accepted after that time. You may turn in part 1 on paper during office hours Friday Jan. 20 from 3:30-4:30, or in the lab on Sunday Jan. 22 from noon to 1:30. Part two should be turned in online by the deadline, and you can also turn in an electronic copy of your solutions to part 1 online by that time instead of handing in a paper copy.

Part I . Answer the questions later on this page.

Part II. Go to this web page for part II of the assignment, which involves setting up Eclipse and other Java tools.

Part I. Loops, assertions, and invariants.

Answer the following questions on paper. In questions that ask for proofs of algorithms or code you only need to show that the code is correct; you do not need to prove termination. You do not need to show all of your intermediate work, but your answers and proofs should be clear and easy to follow.

1. (Warmup) In class we developed an algorithm to find the largest value in a non-empty list of integers items[0..size-1]. In our code, the loop had the following invariant:

   max = largest value in items[0..k-1]

   Suppose we had used the following slightly different invariant instead:

   max = largest value in items[0..k]

   (The difference is that the upper bound of the array section is k instead of k-1.)
   
   Rework the code in the example to use this new invariant instead of the original one and show that the modified code is correct. Insert or modify assertions and code as needed.
   
   After solving this problem, give a brief description of how this change to the invariant affected the algorithm. What were the major changes? Did this change make the code easier or harder to write or prove compared to the original version? Why? (You should be able to keep your answers brief and to the point.)
   
   
2. Given two non-negative integers x and y, we can calculate the exponential function x^y (x raised to the power y) using repeated multiplications by x a total of y times. The following code is alleged to compute x^y much more quickly (it actually takes time proportional to lg y).

   int expt(int x, int y) {
     int z = 1;
     while (y > 0) {
       if (y % 2 == 0) {
         y = y/2;
         x = x*x;
       } else {
         z = z*x;
         y = y-1;
       }
     }
     return z;
   }

   Give a proof that this code is correct. A suitable precondition for the function would be x==X && y==Y && X>=0 && Y>=0, and an appropriate postcondition would be that the returned value z==X^Y. (We define 0^0 to be 1.) You will need to develop a loop invariant and pre- and post-conditions for the statements inside the loop, and verify that the code has the necessary properties to ensure that these assertions hold. You are not required to give pre- and post-conditions for each individual assignment statement inside the if if these are not needed to clearly show the code is correct, but you should include whatever is needed so that the grader can follow your proof and see that it is valid.
   
   
3. We are given an integer array a containing n elements and would like to develop an algorithm to rearrange the array elements as follows and prove that the algorithm is correct. If the original elements of the array are A[0], A[1], ..., A[n-1], we would like to rearrange the array so that all of the elements less than or equal to the original value in A[0] are at the beginning of the array, followed by the value originally stored in A[0], followed by all the elements in the original array that are larger than A[0]. In pictures, the postcondition we want is the following:

       0                                  n
      +-------------+----+---------------+
     a|  <= A[0]    |A[0]|      >A[0]    |
      +-------------+----+---------------+
    

   The operation swap(a[i],a[j]) can be used to interchange any pair of elements in the array, and this is the only operation that can be used to modify the contents of the array. (As a result of this restriction, the array will be a permutation of its original contents, so you do not need to prove this.) Your code should run in linear (O(n)) time.
   
   You should develop a suitable loop invariant, then write code to partition the array using that invariant and prove that when the code terminates the postcondition has been established. You do not need to provide assertions for every trivial statement in the code, but there should be sufficient annotations so that the correctness of your code is obvious. You do not need to write a complete procedure, function, or method, just the necessary loop and supporting statements, including any necessary initialization and final statements before and after the loop.
   
   Hints: As you are partitioning the values in the array, you might find it simpler if you are not constantly moving the original value from A[0] into new locations. Try different invariants and see which one makes things simplest. Also remember that you can include additional statements before and after the main loop if useful to establish the loop invariant or the postcondition.
   
   
4. Give an implementation of the algorithm selection sort and a proof of its correctness. The algorithm should sort an array a containing n integer values. The precondition is that the array contains n integer values in some unknown order. The postcondition is that the array holds a permutation of the original values such that a[0] <= a[1] <= ... <= a[n-1]. As in the previous problem you should use the operation swap(a[i],a[j]) to interchange array elements when needed.
   
   Selection sort proceeds as follows. First we find the smallest element in the original array and swap it with the original value in a[0]. (If a[0] is the smallest element in the original array it is fine to swap it with itself to avoid having additional special cases in the code.) Next we find the smallest element in the remaining part of the array beginning at a[1] and swap it with a[1]. Then we find the smallest element in the remaining part of the array starting at a[2] and move it to the front of that part of the array. This search-and-swap operation continues on the successively smaller remaining parts of the array until all elements are sorted.
   
   As with the previous problem, you should develop suitable invariants for the nested loops needed to perform the sort, then write the code and annotate it with appropriate assertions so that it is clear that your code is correct and that when it terminates the array is sorted.

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