Introduction - What is the World’s Best Sorting Algorithm?
There is no single best sorting algorithm. Quicksort is the fastest known comparison-based sorting algorithm when applied to large, unordered, sequences. It also has the advantage of being an in-place (or nearly in-place) sort. Unfortunately, quicksort has some weaknesses: it’s worst-case performance is $O(n^2)$, and it is not stable. Merge sort shares neither of these disadvantages: it is stable and it requires $O(n \log n)$ steps in the worst case. Unfortunately, merge sort requires $O(n)$ additional space and it runs more slowly than quick sort on most inputs.
What we really want is a sorting algorithm that is as fast as quicksort, stable, in-place, with $O(n \log n)$ worst-case performance. Sadly, no such algorithm has yet been discovered. In practice, most code libraries follow one of two paths: either they provide a modified version of quicksort that is able to avoid worst-case behavior on typical inputs, or they provide a modified version of merge sort that is able to close some of the performance gap through careful optimizations*.
The objective of this assignment is to write modified versions of merge and quicksort that exhibit better performance than the standard versions described in our textbook. You will implement two improvements (one for merge sort and one for quicksort) that should improve some aspect of the algorithm’s performance without requiring a great deal of additional code. You will also experimentally evaluate the impact of your improvements.
The following zip file contains starter code, including the sort implementations provided by our textbook.
This zip file has the following contents:
Classes for generating testing data:
Generator.java- Abstract base class.
EvilGenerator.java(UNFINISHED. See Part 4 below.)
Sorting classes. Normally we would write sorting algorithms as static utility methods. For this project we are creating sorting classes so that we can take advantage of polymorphism in the development of the profiling tool described in Part 2.
Sorter.java- Abstract base class for all sorting classes.
TimSorter.java- The default Java sorting algorithm.
MergeSorter1.java(UNFINISHED. See Part 3 below.)
IntroSorter.java(UNFINISHED. See Part 5 below.)
CliDemo.java- Demo application illustrating the use of the JOpt Simple library. (See Part 2 below.)
DemoDriver.java- Demo driver illustrating how to use the sort and generator classes.
Part 1 - Unit Testing
Develop unit tests for each of the provided sorting classes. These tests don’t need to evaluate the running time of the sorts, they only need to ensure that each implementation produces the correct output. Your tests must test the stability of the sorts that should be stable.
Note that there is at least one defect in the provided sorts. For full credit your testing must uncover all defects and you must correct them before submission.
Part 2 - Sort Profiling Tool
So far in this course we have focused on asymptotic analysis. This usually gives us the information we want. When deciding between two algorithms we generally prefer the one with better asymptotic performance.
Our goals for this project are somewhat different. The focus now is on fine-tuning existing algorithms to improve their performance. This is a situation where we really care about the constant factors. (We would be very happy if we could write a version of merge sort that is twice as fast, even if the asymptotic performance were still $O(n \log n)$.)
The goal for this part of the project is to develop a command-line tool that can be used to systematically evaluate the run-time performance of different sorting algorithms.
Your tool should conform to the following help message:
Option (* = required) Description --------------------- ----------- * -s <Integer: NUMBER> Starting (smallest) input size * -i <Integer: NUMBER> Input size increment * -m <Integer: NUMBER> Maximum input size to test * -t <Integer: NUMBER> Number of trials for each input size -w [String: SORT1,SORT2,...] Comma separated list of sorts. Options include InsertionSort, SelectionSort, MergeSort, MergeSort1, QuickSort, IntroSort and TimSort. Default is to execute all sorts. -g [String: GENERATOR] Sequence generator. Options include random, ordered or evil. The default is random
For example, we could compare the performance of quicksort and merge sort on small inputs by executing the tool as follows:
$ java SortProfiler -s 1000 -i 1000 -m 10000 -t 10 -w QuickSort,IntroSort
The resulting output would look something like the following:
N, Quicksort, IntroSort 1000, 0.00008631, 0.00000076 2000, 0.00018698, 0.00000068 3000, 0.00028221, 0.00000066 4000, 0.00041534, 0.00000069 5000, 0.00056695, 0.00000062 6000, 0.00063308, 0.00000063 7000, 0.00073361, 0.00000064 8000, 0.00084264, 0.00000063 9000, 0.00096724, 0.00000063 10000, 0.00108252, 0.00000064
Note that IntroSort is very fast in these results because it hasn’t been implemented yet. Your exact timing results will vary depending on the speed of your computer.
Specific functional requirements for the tool include the following:
- Columns must be separated by a comma character followed by a tab character.
- Each row (including the last) must be terminated with a newline character.
- Averaged times must be reported in seconds with 8 decimal places of precision.
- The order of the columns must match the order that the sorts appear on the command line.
- There should be error checking for the command line arguments. In the case of negative integers or other incorrect arguments a help message should be printed and the application should exit.
- The command-line arguments should be interpreted correctly regardless of their ordering.
Processing Command-Line Arguments
It’s lot of work to write code that properly handles command line
arguments. Rather than tackling this problem from scratch, you are
encouraged to make use of the
JOpt Simple library. You’ll
need to place this jar file on your build path to compile your
CliDemo.java provides an example of using this library.
Your implementation should follow the pseudocode below:
Print column headers For each input size n: For t trials: Generate a sequence of size n For each sort: Time the execution of that algorithm on a copy of the sequence For each sort Calculate and store the average running time across all t trials Output the data for the current row
Note that this pseudocode is organized so that each algorithm is tested against the same sequences. This results in a more “fair” comparison between the different algorithms, particularly when the number of trials is small.
Part 3 - Merge Sort Improvement
The merge algorithm described in our textbook consists of the following two stages:
1. Copy all values from the two merge-sorted halves int a temporary array. 2. Merge the values from the temporary array back into the original array.
If $n$ is the combined size of the two sub-arrays being merged, this algorithm requires a temporary array of size $n$, and requires $n$ assignment operations in stage 1. The following alternative approach cuts both of those values from $n$ to around $n/2$:
1. Copy the values from the first merge-sorted half to a temporary array. 2. Merge the values from the temporary array and the second merge-sorted half into the original array.
Here are some ASCII graphics illustrating the process:
Original sorted sub-arrays:
________________________________________ ...| 1 | 3 | 5 | 7 | 9 | 2 | 4 | 6 | 8 | 10 |... ---------------------------------------- start-^ mid-^ end-^
___________________ | | | | | |... ------------------- 0-^ ________________________________________ ...| 1 | 3 | 5 | 7 | 9 | 2 | 4 | 6 | 8 | 10 |... ---------------------------------------- start-^ mid-^ end-^
Copy the sorted first half into the temporary array:
___________________ | 1 | 3 | 5 | 7 | 9 |... ------------------- ________________________________________ ...| | | | | | 2 | 4 | 6 | 8 | 10 |... ----------------------------------------
i1 to be the index of the first position (0) in the
i2 to be the index of the first position of the
sorted right half, and
curr to be the next available position for
___________________ | 1 | 3 | 5 | 7 | 9 |... ------------------- i1-^ ________________________________________ ...| | | | | | 2 | 4 | 6 | 8 | 10 |... ---------------------------------------- curr-^ i2-^
temp[i1] < items[i2],
temp[i1] is copied to
curr are incremented.
___________________ | 1 | 3 | 5 | 7 | 9 |... ------------------- i1-^ ________________________________________ ...| 1 | | | | | 2 | 4 | 6 | 8 | 10 |... ---------------------------------------- curr-^ i2-^
items[i2] < new[i1],
items[i2] is copied to
next are incremented.
___________________ | 1 | 3 | 5 | 7 | 9 |... ------------------- i1-^ ________________________________________ ...| 1 | 2 | | | | 2 | 4 | 6 | 8 | 10 |... ---------------------------------------- curr-^ i2-^
Continue Merging until complete:
___________________ | 1 | 3 | 5 | 7 | 9 |... ------------------- ________________________________________ ...| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |... ----------------------------------------
Use your sort profiling tool to test your finished implementation. You
should not expect your improved merge sort to be dramatically faster
than the provided implementation. Since array assignments are very
fast relative to calls to
compareTo, the speed impact will be
minimal. The main advantage here is the savings in space.
Part 4 - Black Hat Sequence Generation
Quicksort is usually very fast. However for any quicksort implementation that selects a pivot point deterministically it is possible to construct an input sequence that will cause the running time to be $O(n^2)$.
For this part of the project you will channel your inner evil algorithmicist to write a method that maliciously generates hard-to-sort sequences. You will need to:
- Analyze the provided quicksort algorithm to determine how it selects the pivot point.
- Use this information to determine the characteristics of the worst possible input sequence.
- Complete the
GenerateEvilso that it produces these malicious inputs.
You must updated your profiling tool to use this generator when the appropriate command line argument is provided. It should be possible to use the tool to test the success of your evil method.
Part 5 - Introspective Sort
As mentioned above, quicksort is usually very fast. There are relatively few pathological inputs that result in $O(n^2)$ performance. The idea behind introspective sort is to perform a standard quick sort until there is evidence that the current input is pathological. If it looks like the current input will cause $O(n^2)$ performance, the algorithm switches strategies to a sorting algorithm with guaranteed $O(n \log n)$ performance. Typically, heap sort is used as the alternative, since it is in-place and takes $O(n \log n)$ time in the worst case. Since we haven’t yet studied heap sort, your version of introspective sort should use the improved merge sort from Part 3 as the alternate sorting algorithm. The resulting algorithm is almost as fast as quick sort for most inputs, but has a worst case performance of $O(n log n)$.
It is possible to detect pathological inputs by tracking the recursion depth of the quicksort algorithm. When quicksort is working well, the partition operation typically moves the pivot item somewhere near the center of the current sub-sequence. When this is the case, the maximum recursion depth will be $O(\log n)$. Therefore, introspective sort switches strategies when the recursion depth exceeds $2 \lfloor \log_2 n\rfloor$.
Note that the provided version of merge sort sorts a full
array. Introspective sort needs to be able to call a version of merge
sort that can sort a portion of a provided array. You’ll need to add a
public method with the following signature to
public void sort(T items, int left, int right)
You should confirm that introspective sort performs well even in the case of pathological inputs.
Submission A - Sorts and Tests
Submit your unit tests along with all of the sorting classes (including those that you didn’t modify) through Web-CAT.
Submission B - Sorts and Tools
Submit all of your completed files through Web-CAT. You don’t need to upload the jar file for the JOpt Simple library.
|Web-CAT Funcionality and Testing for Submission A||30%|
|Web-CAT Functionality and Testing for Submission B||30%|
|Web-CAT Style Checks||20%|
|Web-CAT Instructor Style Points||20%|
*The second approach has recently become popular. Both Python (starting in version 2.3) and Java (starting in version 7) use Timsort as their default sorting algorithm. Timsort is a modified version of merge sort that includes several enhancements: it uses a more space-efficient merge operation, it takes advantage of partially sorted arrays by finding and merging existing sorted regions, and it uses a non-recursive binary insertion sort to handle short unsorted regions.