Lab: Performance

In Sorting Algorithms we took advantage of a few ideas to show how to do basic benchmarking to compare the various approaches.

using randomly-generated data
making sure each algorithm is working with the same data
making sure that we try a range of sizes to observe the effects of scaling
using a timer with sufficiently high resolution (the Stopwatch gives us measurements in milliseconds).

In this lab, you get your chance to learn a bit more about performance by comparing searches. The art of benchmarking is something that is easy to learn but takes a lifetime to master (to borrow a phrase from the famous Othello board game).

Most of the algorithms we cover in introductory courses tend to be polynomial in nature. That is, the execution time can be bounded by a polynomial function of the data size \(n\). A more accurate measure may also include a logarithm. Examples include but are not limited to:

\(O(1)\) is constant time, characterized by a calculation with a limited number of steps.
\(O(n)\) is linear time; often characterized by a single loop
\(O(n^2)\) is the time squared; often characterized by a nested loop
\(O(log\ n)\) is logarithmic (base 2) time; often characterized by a loop that repeatedly divides its work in half. The binary search is a well-known example.
\(O(n\ log\ n)\) is an example of a hybrid. Perhaps there is an outer loop that is linear and an inner loop that is logarithmic.

And there are way more than these shown here. As you progress in computing, you’ll come to know and appreciate these in greater detail.

In this lab, we’re going to look at a few different data structures and methods that perform searches on them and do empirical analysis to get an idea of how well each combination works. Contrasted with other labs where you had to write a lot of code, we’re going to give you some code to do all of the needed work but ask you to write the code to do the actual analysis and produce a basic table.

The Experiments

We’re going to measure the performance of data structures we have been learning about (and will learn about, for lists and sets). For this lab, we’ll focus on:

Integer arrays using Linear Searching and Binary Searching
Lists of integers with linear searching
Sets of integers; checking if an item is contained in the set

In the interest of fairness, we are only going to look at the time it takes to perform the various search operations. We’re not going to count the time to randomly-generate the data and actually build the data structure. The reasoning is straightforward. We’re interested in the search time, which is completely independent of other aspects that may be at play. We’re not at all saying that the other aspects are unimportant but want to keep the assignment focus on search.

The experimental apparatus that we are constructing will do the following for each of the cases:

create the data structure (e.g. new array, new list, new set)
use a random seed seed, initialize a random generator that will generate n values.
insert the random values into the data structure . For the case of sets, which eliminate duplicates, it is entirely possible you will end up with a tiny fraction of a percent fewer than n values.
to measure the performance of any given search method, we need to perform a significant number of lookups (based on numbers in the random sequence) to ensure that we get an accurate idea of the average lookup time in practice. We’ll call this parameter, rep. We will spread out the values looked for by checking data elements that have indices at a regular interval throughout the array. The separation is m = n/rep when rep < n. The separation is 1, and we wrap around at the end of the array if rep > n.
We’ll start a Stopwatch just before entering the loop to perform the lookups.

Starter Project

To make your life easier, we have put together a project that refers to all the code for all of the experiments you need to run. (That’s right, we’re giving you the code for the experiments, but you’re going to write some code to run the various experiments and then run for varying sizes of n.) The stub file is performance_lab_stub/performance_lab.cs.

Recreate example project performance_lab_stub in your solution as performance_lab, so you have your own copy to modify. You can either

copy into the lab project the files sorting/sorting.cs, searching/searching.cs, and binary_searching/binary_searching.cs. If you copy them into the lab project, rename the unused Main method from binary_searching.cs to something else (since Xamarin Studio allows only one Main method in a project).
An alternative is to recreate their whole projects, and reference them from the lab project.

Here is the code for the first experiment, to test the performance of linear searching on integer arrays:

public static long ExpIntArrayLinearSearch (int n, int rep, int seed)
{
   Stopwatch watch = new Stopwatch ();
   int[] data = new int[n];
   Sorting.IntArrayGenerate (data, seed);
   int m = Math.Max(1, n/rep);
   watch.Reset ();
   watch.Start ();
   // perform the rep lookups
   for (int k=0, i=0; k < rep; k++, i=(i+m)%n) {
      Searching.IntArrayLinearSearch (data, data [i]);
   }
   watch.Stop ();
   return watch.ElapsedMilliseconds;
}

Let’s take a quick look at how this experiment is constructed. We’ll also take a look at the other experiments but these will likely be presented in a bit less detail, except to highlight the differences:

On line 3, we create a Stopwatch instance. We’ll be using this to do the timing.
On lines 4-5, we are creating the data to be searched. Because we have already written this code in our sorting algorithms examples, we can refer to the Sorting class code in sorting.cs, as long as you made the lab project able to reference it. We use the Sorting class name to access the method IntArrayGenerate() within this class. We also take advantage of this in the other experiments.
Line 6 converts the number of repetitions into the increment in index values for each time.
Line 7 resets the stopwatch. It is not technically required; however, we tend to be in the habit of doing it, because we sometimes reuse the same stopwatch and want to make sure it is completely zeroed out. A call to Reset() ensures it is zero.
Line 8 actually starts the stopwatch. We are starting here as opposed to before line 4, because the random data generation has nothing to do with the actual searching of the array data structure.
Lines 10 through 12 are searching rep times for an item already known to be in the array.
Line 13 stops the stopwatch.
Line 14 returns the elapsed time in milliseconds between the Start() and Stop() method calls, which reflects the actual time of the experiment.

Each of the other experiments is constructed similarly. For linear search and binary search we use the methods created earlier. For the lists and the set we use the built-in Contains method to search. The list and set are directly initialized in their constructors from the array data. (More on that in later chapters.)

You need to fill in the Main method. The stub already has code to generate a random value for the seed for any run of the program. Read through to the end of the lab before starting to code. A step-by-step sequence is suggested at the end.

Your code must parse command line args for the parameters rep and any number of values for n. For instance:

mono PerformanceLab.exe 50000 1000 10000 100000

would generate the table shown below for 50000 repetitions for each of the values of n: 1000, 10000, and 100000.
In the end you will want to run each experiment for rep repetitions and iterate through each different value of n.
Present the result data in a nice printed right-justified table for all values of n, with a title including the number of repetitions. Print something like the following, with the number of seconds calculated.
```
Times in seconds with 50000 repetitions
       n    linear      list  binary     set
    1000  ????.???  ????.???  ??.???  ??.???
   10000  ????.???  ????.???  ??.???  ??.???
  100000  ????.???  ????.???  ??.???  ??.???
```
The table would be longer if more values of n were entered on the command line. Note that the experiments return times in milliseconds, (1/1000 of a second) while the table should print times in seconds.
Your final aim is to show your TA or instructor the results of a run with a table with at least three lines of data and with n being successive powers of 10, and non-zero entries everywhere. Read on for the major catch!

You will need to experiment and adjust the repetitions and n choices. In order to get all perceptible values (nonzero), you will need a very large number of repetitions to work for the fastest searches. Our choice of 50000 in the example is not appropriate with these n values. The catch is that without further tweaking, you will only get nonzero values for all the fastest searches if the slower ones take ridiculously long.

Because the range of speeds is so enormous, make an accommodation with the slow linear versions: If rep >= 100 and (long)n*rep >= 100000000, then, for the linear and list columns only, time with rep2 = rep/100 instead of rep, and then compensate by multiplying the resulting time by (double)rep/rep2 to produce the final table value. (This multiplier is not necessarily just 100, since the integer division creating rep2 may not be exact.)

Before making the modification for large numbers, be sure to test with small enough values (though some results will be 0). Once again, you are encouraged to develop this is steps, for example:

Make sure you can parse the command line parameters. In a testing version write code to print out rep, and separate code to print out all n values, for any number of n values.
Print out one linear test for rep and one value of n.
Print out the results for all tests for rep and one value of n. Keep rep*n small enough so the linear searches do not take too much time.
Do all values of n.
Make the printing be formatted as in the sample table.
Add the modification for large rep*n.
Experiment and get a table to show off!