CS107 - Lab 4

Overview:

Some of the most important algorithms used in computing are in the category "search and sort." These are useful in a variety of settings, such as web search, alphabetic sorting of a list of names, etc. They are also related, since the most efficient search algorithms occur on lists that are sorted. This lab exercises algorithms for searching and sorting, and their efficiency. The (suggested) reading for this lab is Chapter 3 in the SG textbook.

Designing and analyzing algorithms

  1. Take a look at the following binary search algorithm discussed in class: 
    BinarySearch
get n, a1, a2, ...an, t
start = 1
end = n
found = "false"
repeat until found="true" or ...
     m = ceiling( (start + end)/2) //the middle between start and end
     if (t = am) then 
          print "Target found at position " m 
	  found = "true"
     if (t < am) then
          end = m-1
     if (t > am) then 
          start = m+1
if (found="false") then 
     print "Target not in the list"
    Using this algorithm, search for target t=3 in the list 2,4,5,7,9. At some point the values of the variables start and end will be equal. What happens if we let the loop execute when start=end? What will the values of start and end be after this last interation? With this insight, how would you write the condition to stop the repeat loop (so that it executes one last time start=end)?

  2. Assume you use the binary search algorithm to decide whether 35 is in the following list: 
    3, 6, 7, 9, 12, 14, 18, 21, 22, 31, 43
    What numbers are compared to 35?

  3. Consider we perform binary search on the following list of names: 
    Arturo, Elsa, JoAnn John, Jose, Lee, Snyder, Tracy
    1. When searching to see if Elsa is in the list, what names will be compared to Elsa?
    2. When searching to see if Tracy is in the list, what names will be compared to Tracy?
    3. When searching to see if Emille is in the list, what names will be compared to Emille?

  4. Draw the tree data structure that describes binary search on the 8-element list in the previous problem. What is the number of comparisons in the worst case? Give an example of a name to search that requires that many comparisons.

  5. Draw the tree structure that describes binary search on a list with 16 elements. What is the number of comparisons in the worst case?

  6. Assume we are using binary search to search for a target that is smaller than all elements in the list. What happens to the variable start throughout the algorithm? What values does it take? What about the variable end?

  7. In this problem you will explore the tradeoff between using sequential search on an unsorted list, as opposed to sorting the list and then using binary search. Assume we use a sorting algorithm that takes order of n2 in the worst case. If the list size is n=100,000, about how many worst-case searches must be done before the second before the second alternative is better in terms of number of comparisons? (Hint: Let p represent the number of searches done.)

  8. Write an algorithm that evaluates a polynomial at a point. The algorithm should read in from the user a list of n+1 coefficients, a0, a1, a2,...,an and a value x. The algorithm should compute and print the value a0 + a1x+a2x2 ...+anxn.

    As part of your algorithm you will need to compute the product xn (you cannot assume that you can use xn in pseudocode).

    Analyze the worst-case and best-case running time of your algorithm (the order of magnitude, or theta-notation is enough).

  9. Recall the SelectionSort algorithm discussed in class. The algorithm, at a high level of abstraction, is given bellow: 
    SelectionSort
get n, a1,...an
set unsortedEnd = n
repeat until (unsortedEnd = 1)
      find the position of the largest element among a1, a2,..,aunsortedEnd
      assign this position to pl
      swap apl with aunsortedEnd
      unsortedEnd = unsortedEnd - 1
print "The list in sorted order is" a1, a2,..,an
    1. Show how the algorithm works if the input list is 3,1,2,5,4
    2. Describe in words what happens in the first iteration of the repeat loop (one sentence).
    3. Describe in words what happens in the second iteration of the repeat loop (one sentence).
    4. How many times is the repeat loop executed for a list of size n?
    5. For each of the 4 steps within the repeat loop, write down how many instructions they take in the first iteration of the loop.
    6. Same, for the second iteration of the loop.
    7. Same, for the last iteration of the loop.
    8. By adding up the numbers that you got above, find out the total running time (number of instructions) of SelectionSort.

  10. Write an algorithm that reads a number n and a list of n-1 distinct integers from the user in the range 1,2,...,n and figures out what is the missing number. For instance, if the user types in 
    6
1 5 3 2 6
    your algorithm should print 
    The missing number is 4

    Analyze the worst-case and best-case running time of your algorithm (the order of magnitude, or theta-notation is enough).

  11. Write an algorithm that reads a number n and a list of n numbers from a user and computes and prints the median element(s) in the list (the median is an element which has an equal number of elements smaller and larger than it). For instance for the list 
     4 3 1 2 5 7  6
    the median is 4. For the list 
     5 4 3 8 6 7 1 2
    the medians are 4 and 5.

    You can use any of the algorithms studied in class as a black box, without writing the code for them.

    Analyze the worst-case and best-case running time of your algorithm (the order of magnitude, or theta-notation is enough).

What to turn in:

Be sure to justify your answers! You may do this work either with a word processor or by hand. If you type your solutions, bring a hard copy of your assignment and hand it in class. If you write by hand, do your best to write legibly and leave space between problems.

You may choose to do this assignment either by yourself or in a group. However, solutions should be written up individually and handed in on the due date, at the beginning of class.

Once you are finished in the lab, if you want to save any documents, you'll have to drag them to your network folder (mounted to your desktop). Files left on the desktop and in your local folder (on the local machine) will be erased!