UMBC CMSC441, Design & Analysis of Algorithms, Fall 1999, Section 0101

Project: Select versus Quicksort

Due Date

The report for this project is due November 11, 1999. You will also be asked to submit the source code for your programs online (details will be given later).


After reading the description of the linear-time Select algorithm, Professor X made the following statement:
This algorithm is so complicated that if you were to implement it in real life it would be really slow. Even though the theoretical analysis gives it a O(n) running time, it will probably be faster to just use Quicksort.
The purpose of your project is to either support or refute Professor X's statement.

Implementation Issues

For this project you will implement the following algorithms to find the median element in an array.

  1. The deterministic linear time Select algorithm given in Chapter 10 of the textbook.

  2. The randomized selection algorithm given in Chapter 10 of the textbook.

  3. Using randomized Quicksort to first sort the set of numbers, then picking the middle element of the sorted array.
We fully expect that the randomized selection algorithm will the fastest of the three --- it will be used to check the other two algorithms.

In order for this to be a fair comparison, you must make each algorithm as fast as you can. The following is a minimal list of issues that you should address to make your implementation run as fast as possible:

  1. The Select algorithm in the textbook groups the items into segments of 5 items each. The number 5 was chosen purely for analysis purposes. Segments of 7, 9, 11, ... would also work. You should run experiments to determine the optimal segment size.

  2. The textbook does not describe exactly what happens when you recursively call Select to find the median of the medians (Step 3, page 190). You should not copy the medians of the segments into a new array. This is unnecessary and wastes a lot of time. Your implementation of Select should be flexible enough that you can use it to find the median of an array from one index to another considering only every k-th item.

  3. For small arrays it will be faster to use Insertion Sort to sort the items. You should run some experiments to find the optimal size to switch over to insertion sort and use this in your running time trials.
Other implementation issues will come up as you write your programs. You must document in your report the steps you have taken to make the implementations as fast as you can.


For this project you must use "real" data. Both Quicksort and the randomized selection algorithm will use a pseudo-random number generator on your system. It is not a fair experiment to also use this generator to produce your data. Furthermore, your arrays must be large enough to be meaningful. Your arrays should contain at least tens of thousands of items. Hundreds of thousands is preferable. Your experiments should include several trials for each size.

The internet is a useful source of ``random'' data. For example, you can download text or images and treat every four bytes as a 32-bit integer. Alternatively you can treat the binary code of an application (e.g., Microsoft Word) as a sequence of integers. Your report should fully document how you obtained the data for testing and give some indication of why you think this is a fair way to test your implementations.

Reporting Requirements

You should report your results as a technical report of roughly 5 to 10 pages. The report counts as 20% of the project grade. The report will be graded on its quality, not its length. A good report should present your case clearly and convincingly. The English will be judged according to the standards of a term paper submitted in a liberal arts course. Grammar counts.

The report should contain the following parts:


Your project will be graded on 5 parts weighted equally.

Note that coding is only one part of this project. You will not do well if you simply turned in 3 working programs. You must leave sufficient time after you code to test the code, run your experiments and write up the report.

Last Modified: 1 Nov 1999 22:23:36 EST by Richard Chang
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