Lab: Fast Sorting Algorithms

• Exercises
  • Exercise 0: Preparation
  • Exercise 1: Merging
  • Exercise 2: Splitting
  • Exercise 3: Splitting, Revisited
  • Exercise 4: Merge Sorting
  • Exercise 5: Special Cases
  • Exercise 6: Comparing Sorts
  • Exercise 7: Quicksorting Lists
  • Exercise 8: Observing Quicksort
  • Exercise 9: Improving Quicksort
  • Exercise 10: Verifying Sorts
  • Exercise 11: Experimentally Comparing Sorts

Exercises

Exercise 0: Preparation

a. Make a copy of mergesort.ss, my implementation of merge sort. Scan through the code and make sure that you understand all the procedures.

b. Make a copy of quicksort.ss, my implementation of Quicksort. Scan through the code and make sure that you understand the procedures.

c. Start DrScheme

Exercise 1: Merging

a. Write an expression to merge two lists of strings. (You may choose the words yourself. Each list should have at least three elements.)

b. What will happen if you call merge with unsorted lists as the first two parameters?

Exercise 2: Splitting

Use split to split:

a. A list of numbers of length 6

b. A list of strings of length 5

c. A length-4 list of lists, where each element list is of the form (last-name first-name major)

Exercise 3: Splitting, Revisited

One of my colleagues prefers to define split something like the following

(define split
  (lambda (ls)
    (let kernel ((rest ls)
                 (left null)
                 (right null))
      (if (null? rest)
          (list left right)
          (kernel (cdr rest) (cons (car rest) right) left)))))

a. How does this procedure split the list?

b. Why might you prefer one version of split over the other?

Exercise 4: Merge Sorting

a. Run merge-sort on a list you design of fifteen integers.

b. Run new-merge-sort on a list you design of twenty strings.

c. Uncomment the lines in new-merge-sort that print out the current list of lists. Rerun new-merge-sort on a list you design of twenty strings. Is the output what you expect?

Exercise 5: Special Cases

a. Run both versions of merge sort on the empty list.

b. Run both versions of merge sort on a one-element list.

c. Run both versions of merge sort on a list with duplicate elements.

d. Consider a list of lists, with each element list of the form (last-name first-name major). What's the difference between using each of the following as a comparison algorithm in new-merge-sort?

• (lambda (val1 val2) (string<=? (car val1) (car val2)))
• (lambda (val1 val2) (string<? (car val1) (car val2)))

e. Verify your answer experimentally.

Exercise 6: Comparing Sorts

Which version of the merge sort algorithm do you prefer, merge-sort or new-merge-sort? Why?

Exercise 7: Quicksorting Lists

a. Quicksort a list of ten integers in increasing order.

b. In sorting a list of ten integers, does it make a difference whether you use < or <= as the comes-before? parameter? Why or why not?

c. Verify your results experimentally.

Exercise 8: Observing Quicksort

a. Use random-list to create a list, values of fifty values.

b. Uncomment the lines that display the segmentation in quicksort.

c. Run quicksort four times on values. Do you see the same steps every time? Why or why not?

Exercise 9: Improving Quicksort

Rewrite quicksort so that it no longer needs same? as a parameter.

Exercise 10: Verifying Sorts

a. Write a procedure, verify-sort, that verifies the reasonably verifiable postconditions of merge-sort. That is, (verify-sort unsorted sorted may-precede?) should return true (#t) if sorted in sorted order and sorted is a permutation of unsorted. It should return false (#f) otherwise.

Note that (verify-sort '(1 1 2) '(1 2 2) <=) should return false (#f).

b. Use verify-sort to verify that merge-sort correctly sorts lists of 1000 random numbers.

c. Use verify-sort to verify that new-merge-sort correctly sorts lists of 1000 random numbers.

d. Use verify-sort to verify that quicksort correctly sorts lists of 1000 random numbers.

Exercise 11: Experimentally Comparing Sorts

a. Using DrScheme's built-in timing mechanism (you may have to look through the online help to find information about that mechanism), make a table of the running time of insertion sort, merge-sort, new-merge-sort, and quicksort on inputs of size 0, 1, 100, 500, 1000, 5000, 10000, 50000, and 100000.

b. Graph your data.

c. Based on your data, what can you say about the relative speeds of the three sorting methods?