Due: Wednesday, February 8th
Your scheduled presentation time:
Grading Rubric

A: Solution meets the stated requirements and is completely correct. Presentation is clear, confident, and concise.

B: The main idea of the solution is correct, and the presentation was fairly clear. There may be a few small mistakes in the solution, or some faltering or missteps in the explanation.

C: The solution is acceptable, but there are significant flaws or differences from the stated requirements. Group members have difficulty explaining or analyzing their proposed solution.

D: Group members fail to present a solution that correctly solves the problem. However, there is clear evidence of significant work and progess towards a solution.

F: Little to no evidence of progress towards understanding the problem or producing a correct solution.

ProblemFinal assessment
1
2
3
4

Instructions: Review the course honor policy: you may not use any human sources outside your group, and must document anything you used that's not on the course webpage.

This cover sheet must be the front page of what you hand in. Use separate paper for the your written solutions outline and make sure they are neatly done and in order. Staple the entire packet together.

Comments or suggestions about this problem set:
Comments or suggestions about the course so far:
Citations (be specific about websites):

1 Sums of Squares

A "perfect square" is an integer multiplied by itself; the first few are 0, 1, 4, 9, 16, etc. Some integers can be written as the sum of two perfect squares. For example, $10 = 1^2 + 3^2$. But some cannot: for example, 7. And some can be written as the sum of 2 squares in more than one way: for example, $50 = 1^2 + 7^2 = 5^2 + 5^2$.
(Caution: mathematical enrichment irrelevant to the assignment.) Integers that can be written as the sum of two squares have interested mathematicians for a long time. For example, Fermat showed that a prime number $p$ is the sum of two perfect squares if and only if $p-1$ is divisible by 4.
The following algorithm takes a given integer $n$ and determines whether it can be written as the sum of two perfect squares. If so, it returns $a$ and $b$ such that $n = a^2 + b^2$. Otherwise, it returns "NO".
Algorithm: sumsq(n)
Input : n, a non-negative integer
Output: (a,b), where a and b are integers such that
        n = a*a + b*b, if such integers exist, 'NO' otherwise
        
a = 0, b = n
s = a*a + b*b

 while a <= b and s != n:
   if s < n
     a = a + 1
   else
     b = b - 1
   s = a*a + b*b

if s == n
  return (a, b)
else
  return 'NO'      
      
  1. Use a loop invariant to show that this algorithm is correct. State the invariant, then go through the three steps from class to show correctness.
  2. Determine the worst-case running time of the algorithm. Give a $\Theta$ bound on the number of primitive operations, in terms of the input integer $n$, and simplify as much as possible. Show your work.
  3. Develop an improved algorithm that solves the same problem. Present your new algorithm, briefly explain why it is correct (you do not have to do a formal proof with a loop invariant), and state the worst-case running time.

2 Meet and Greet

There are $n$ strangers in a room. Everyone wants to get acquainted and meet everyone else. Assume that:
  • It takes exactly one minute for a pair of people to meet each other.
  • There is plenty of space in the room, and people can move instantly from one stranger to another.
  • Each individual can only meet one other person at a time, but multiple meetings can be happening simultaneously in the room.

For example, if $n=4$, if we call the people A, B, C, D, then you could have everyone meet everyone else in the following algorithm:

  1. A and B meet, and C and D meet
  2. A and C meet, and B and D meet
  3. A and D meet, and B and C meet
At this point everyone has met, and the total time taken was 3 minutes.

Your task is to develop an efficient algorithm if there are any number $n$ of people, not just 4. It doesn't have to match with my algorithm for $n=4$; that's just an example to see how it could be done. Answer the following questions to get you on the right track:

  1. When $n=4$ above, there are a total of 6 meetings that take place. How many meetings are required for any $n$? Your answer should be a formula in terms of $n$.
  2. Only two people can meet at once, so for example when $n=4$ only 2 meetings can take place at once, and therefore the 3-minute solution above is the best possible for $n=4$.

    Using this same logic, what would be the least possible number of minutes to meet for any value of $n$? (Again, your answer should be a formula in terms of $n$.)

  3. Imagine you have split everyone up into two equal-sized groups $G_1$ and $G_2$. Everyone within $G_1$ has met everyone else in $G_1$, and the same goes for $G_2$, but no one from $G_1$ has met anyone from $G_2$ yet.

    Describe an algorithm for everyone in $G_1$ to meet everyone in $G_2$. If there are $n/2$ people in each group, your algorithm should take $n/2$ minutes. Hint: look up how "Speed Dating" works.

  4. Come up with an efficient algorithm for everyone in the whole group of $n$ people to meet each other. You might be able to use your algorithm from part (c) as a useful subroutine, but that's not required. (There are many efficient ways to solve this problem.)

    Try to get an exact formula in terms of $n$ for how many minutes your algorithm will take. Ideally, your formula should match your lower bound from part (b), or at least it should have the same big-oh.


3 X Marks the Spot

I'm sure you are all missing Turing Machines right now, so let's revisit them. Suppose you have a 2-way infinite tape Turing Machine. One cell on the tape has an X on it, the rest are blank. Your job is to design an algorithm to find the X using the following three Turing Machine operations:
  • lmove(), which moves one tape square to the left,
  • rmove(), which moves one tape square to the right, and
  • isblank(), which returns true if the current tape square is blank, and false otherwise.
In your algorithm, besides the Turing Machine operations and the usual pseudocode operations (variables, adding/subtracting/multiplying integers, loops, etc.), I'll let you write "lmove(k)" as a short-hand for a loop that calls lmove() k times, and "rmove(j)" as a short-hand for a loop that calls rmove() k times. Your algorithm should exit with the Turing Machine tape reader on the square that contains X.
  1. Present an algorithm for the FindX problem.
  2. Analyze the worst-case cost of your algorithm. The "measure of difficulty" will be $d$, the distance (in tape squares) of the square containing X from the square you start at. So if the X is on the square you start at, $d = 0$. If it's on one of the squares adjacent to the start square, the distance is 1. The "cost measure" will be the total number of moves, i.e. the total number of calls to lmove() or rmove(). [Remember: "lmove(k)" and "rmove(j)" are just short-hand for multiple calls to lmove()/rmove().] Give a big-O bound on the worst-case number of moves in terms of $d$. The best solutions will have worst-case cost $O(d)$.

    Note: this difficulty measure is not typical! Since there is no "input" to this algorithm, the measure of difficulty cannot be the input size. And the cost measure is also unusual - we're not counting primitive operations in the algorithm, rather the number of moves made.

  3. Asymptotics are great and all, but in this case the "hidden constants" in the big-O really matter — tape moves on my Turing Machine are really slow! Refine your analysis from (b) to give an exact upper bound on the number of miles sailed (i.e. not a big-O bound). There will be a tangible prize for the group who gets the smallest coefficient in front of $d$.

4 Swap counting

For this problem, you will study three sorting algorithms you should already be familiar with: InsertionSort, SelectionSort, and HeapSort. The first two of these are in the class notes, and here is the version of HeapSort that you can use for the purposes of this problem:
Algorithm: heapSort(A,n)
Input : array A of n elements
Output: leaves A is sorted order

for i from n-1 to 0 do     ← This loop does the heapify operation
  bubbleDown(A,i,n)

for i from n-1 to 0 do     ← This loop does the heap removals.
  swap(A,0,i)
  bubbleDown(A,0,i)
      
And here is a bubbleDown algorithm that you should use:
Algorithm: bubbleDown(A,start,end)
Input : A - an array, 
        start - the index of the heap root, 
        end - the index of the last/rightmost element of the heap
        Note: A[start],...,A[end] is a heap except that the
              element at the root might be out of place (hence the
              call to bubbleDown!)
Output: at exit, A[start],...,A[end] is a heap

more = true
while more and 2*start + 1 ≤ end do            ← as long as start is not a leaf, loop!
  child = 2*start + 1                          ← set child to start's left child
  if child + 1 ≤ end and A[child+1] > A[child] ← set child to start's right child if
                                                 that's larger
    child = child + 1
  if A[start] < A[child]                       ← if A[start] out of place, push it 
    swap(A, start, child)                        down to A[child]
    start = child
  else
    more = false                               ← start not out of place, we're done!
		
  1. Tell me the worst-case running time of InsertionSort, SelectionSort, and HeapSort. You should know this already.
  2. Analyze the number of *swaps* performed by a call to InsertionSort, in the worst case. To do this, you first have to describe a worst-case family of examples, i.e., what ordering of a list, for *any* size, makes InsertionSort do the maximum number of swaps.

    Second, you need to analyze the number of swaps that InsertionSort does on your worst case example. This should be an exact function of $n$, like $3n^3 - 6$ or something like that. Try to be as precise as possible.

    Finally, turn your formula into a $\Theta$-bound on the number of swaps.
  3. Repeat (b) for SelectionSort.
  4. Repeat (b) for HeapSort. It will be a little more difficult to get the exact analysis of the number of swaps, so for this one you just need to give a formula for an *upper bound* on the number of swaps. But still, try to make your formula as precise as possible.
  5. Show that *any* sorting algorithm that only uses swaps to move elements in the input array must always perform at least $\lceil n/2 \rceil$ swaps, in the worst case. Which of the three algorithms above comes closest to this lower bound?