©2010 Felleisen, Proulx, et. al.
Practice problems help you get started, if some of the lab and lecture
material is not clear. You are not required to do these problems, but
make sure you understand how you would solve them. Solving them on
paper is a great preparation for the exams.
Read through the documentation for Java Collections Framework library. Find how you can run the sorting algorithms defined there. Write a simple program that will test these algorithms and measure their timing in a manner similar to the last lab.
Your job is now to be an algorithm detective. The program we give you
allows you to run any of the six different sorting algorithms on data sets of
five different sizes using three different Comparators
to define the ordering of the data. When you run the program, the time
that each of these algorithms took to complete the task is shown in
the console.
To run the program you need to do the following:
Create a new Java Project in Eclipse (e.g. SortingTests).
Go to Preferences and choose to add a library then choose Add External jars to add the file sorting.jar to the project. Also add the library jpt.jar and, of course, tester.jar.
Create a new Package in Eclipse and give it a name student. Import into this package the two files: SortingHeapSort.java and Heapsort.java.
Download (or find on your computer) the file citydb.txt and save a copy in the Eclipse directory that has the src and bin directories for the SortingTests project.
Go to the Run menu, choose Run Configurations, select to make a new configuration. Name it SortingTests then click on the button to Select main. One of the choices should be sorting.Interactions. Choose that one. You can now run the program. It will come up with a GUI with several buttons.
To set up the timing tests you need to go through three steps:
You need to read in the data for the 29470 cities from the file citydb.txt. The button FileInput opens a file chooser dialog. Select the citydb.txt file.
Now hit the TimerInput button. It lets you select which algorithms to test, which Comparators to use, and what size data should be used in the tests.
Start with just a few small tests, to see how the program behaves, before you decide to run all tests.
The last choice if heapsort. The two files in the student package provide only hooks to the stress test program — the method heapsort in the class Heapsort just returns the original unsorted ArrayList.
Now you can run the actual tests by hitting the RunTests
button.
You can repeat the last two steps as many times as you want to.
Exploration:
Run the program a few times with small data sizes, to get familiar with what it can do. Then run experiments and try to answer the following questions:
Which algorithms run mostly in quadratic time, i.e. O(n2)?
Which algorithms run mostly in O(n.logn) time?
Which algorithms use the functional style, using Cons lists?
Which algorithm is the selection sort?
Why is there a difference when the algorithms use a different Comparator?
Copy the results into a spreadsheet. You may save the result portion in a text editor with a .csv suffix and open it in Excel (or some other spreadsheet of your choice). You can now study the data and represent the results as charts. Do so for at least three algorithms, where there is one of each — a quadratic algorithm and a linear-logarithmic algorithm.
Produce a report with a paragraph that explains what you learned, using the Excel charts to illustrate this.
Your report should have a professional look – typed, computer generated charts, reasonably organized. It does not have to be more than 2 pages long, one page is OK.
Add your implementation of the heapsort to
the sorting algorithms that will be measured (see Section
9.2). The skeleton for this is
already in place. Run additional tests to evaluate the performance of
the heapsort. Add to your report a sentence or two describing
your findings.
Your goal now is to design a priority queue using the heap algorithm. It will then be used to implement the heapsort algorithm that we can add to our collection of algorithms to be evaluated.
We start by reviewing the lecture on heap-based priority queue and the heapsort algorithm.
Recall the properties of the Heap from the lecture:
A heap is a complete binary tree. It means that every level is filled, except for the last level. The last level is filled from the left.
So a heap with five nodes will have the following shape:
node 1
/ \
node 2 node 3
/ \
node 4 node 5
The nodes are labeled by levels from left to right, starting at 1.
The value in each node is greater than (or equal) to the value in either of its children. So the item with the highest value (highest priority) is at the root of the tree.
The label of the parent of node i is i/2
The label of the left child of node i is 2i
The label of the right child of node i is 2i + 1
If it helps you, draw the picture of the tree and manipulate stickies or pieces of paper to see how the algorithms for insertion and deletion of nodes work.
Typically, we represent this heap as an ArrayList with the first item (at index 0) unused (maybe null).
Add ExamplesHeaps class to the package student (remember to include package student; as the first line.
Make three examples of the following heaps — by
defining
new ArrayList<Integer>() for each case and using
an initHeaps method to add the values to them in the
appropriate order.
So, to build the following heap
node 1
70
/ \
node 2 node 3
60 40
/ \
node 4 node 5
35 50
we would proceed as follows:
ArrayList<Integer> myheap = new ArrayList<Integer>();
void initMyheap(){
this.myheap.add(null); // the unused first item
this.myheap.add(70);
this.myheap.add(60);
this.myheap.add(40);
this.myheap.add(35);
this.myheap.add(50);
}
The three heaps to define:
node 1
80
/ \
node 2 node 3
50 40
/ \ /
node 4 node 5 node 6
45 20 30
node 1
50
/ \
node 2 node 3
45 40
/ \
node 4 node 5
30 20
node 1
70
/ \
node 2 node 3
45 50
/ \ /
node 4 node 5 node 6
30 20 40
Define the class PriorityQueue<T> that will represent the heap-based priority queue. It has two fields: an ArrayList<T> and a Comparator<T> that determines the ordering of the priorities.
Design the method isLeaf that consumes a node label (an int) and returns true if the node has no children.
Design the method higherPriorityChild that consumes the label of a node that is not a leaf and produces the index of its child with the higher priority.
Note: If the node has only one child, then this will be the one with the higher priority, of course.
Design the method insert that inserts a new node into the heap.
Here is what we did in class yesterday:
insert the new item at the next position in the bottom level (or the first position on the next level, if the previous level is filled). Say, this is position k.
Upheap from k:
While k > 1 and heap(k) > heap(k/2) {
swap heap(k) and heap(k/2)
set k to k/2 }
Design the method remove that removes a node with the highest priority from the heap.
Here is what we did in class yesterday:
save the top item as a temporary
move the last item into the top position
Downheap from k = 1:
While k is not a leaf {
find ck the node with the larger child of the node k
if heap(k) < heap(ck) {
swap heap(k) and heap(ck)
set k to ck }
else stop
}
Implement a simple variant of heapstort as follows:
In the first step insert the given data into your PriorityQueue, one item at a time.
In the second step, remove the data from your PriorityQueue and insert them into the resulting ArrayList, one at a time.
If you just add each item at the end, you will end up with a list ordered in descending order. If you wish to get the correct ordering, insert each item at the index 0.