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Sorting algorithms are one kind of algorithm whose performance may
depend upon the data. Choose one of the sorting algorithms or any
other algorithm and explain whether the there are any differences in
the best, average and worst cases. If there are no differences, explain
why not. If there are differences, describe the data in the different
cases and explain how the performance differs in each case.
CMSC 451 Homework 3
1. Shown below is the code for the insertion sort consisting of two recursive methods that
replace the two nested loops that would be used in its iterative counterpart:
void insertionSort(int array[])
{
insert(array, 1);
}
void insert(int[] array, int i)
{
if (i < array.length)
{
int value = array[i];
int j = shift(array, value, i);
array[j] = value;
insert(array, i + 1);
}
}
int shift(int[] array, int value, int i)
{
int insert = i;
if (i > 0 && array[i – 1] > value)
{
array[i] = array[i – 1];
insert = shift(array, value, i – 1);
}
return insert;
}
Draw the recursion tree for insertionSort when it is called for an array of length 5 with
data that represents the worst case. Show the activations of insertionSort, insert and
shift in the tree. Explain how the recursion tree would be different in the best case.
2. Refer back to the recursion tree you provided in the previous problem. Determine a
formula that counts the numbers of nodes in that tree. What is Big- for execution time?
Determine a formula that expresses the height of the tree. What is the Big- for memory?
3. Provide a generic Java class named SortedPriorityQueue that implements a priority
queue using a sorted list implemented with the Java ArrayList class. Make the
implementation as efficient as possible.
4. Consider the following sorting algorithm that uses the class you wrote in the previous
problem:
void sort(int[] array)
{
SortedPriorityQueue queue = new SortedPriorityQueue();
for (int i = 0; i < array.length; i++)
queue.add(array[i]);
for (int i = 0; i < array.length; i++)
array[i] = queue.remove();
}
Analyze its execution time efficiency in the worst case. In your analysis you may ignore
the possibility that the array list may overflow and need to be copied to a larger array.
Indicate whether this implementation is more or less efficient than the one that uses the
Java priority queue.
Grading Rubric
Problem
Problem 1
Meets
10 points
Does Not Meet
0 points
Recursion tree is drawn correctly (8)
Recursion tree is not drawn correctly
(0)
Best case tree is described correctly (2)
Best case tree is not described
correctly (0)
10 points
0 points
Provided correct formula for number of Did not provide correct formula for
nodes in tree (3)
number of nodes in tree (0)
Problem 2
Problem 3
Provided correct Big-Theta for
execution time (2)
Did not provide correct Big-Theta for
execution time (0)
Provided correct formula for tree
height (3)
Did not provide correct formula for
tree height (0)
Provided correct Big-Theta for memory
(2)
Did not provide correct Big-Theta for
memory (0)
10 points
0 points
Provided class correctly implements a
priority queue (4)
Provided class does not correctly
implement a priority queue (0)
Provided class is generic (1)
Provided class uses an array list (1)
Provided class is not generic (0)
Provided class does not use an array
list (0)
List in class is maintained in sorted
order (2)
List in class is not maintained in sorted
order (0)
Implementation is most efficient (2)
Implementation is not most efficient
(0)
10 points
Problem 4
0 points
Provided correct worst case analysis (8)
Did not provide correct worst case
analysis (0)
Provided correct efficiency comparison
to Java priority queue (2)
Did not provide correct efficiency
comparison to Java priority queue (0)
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Sorting Algorithms
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