An Extensive Examination of Data Structures Using C# 2.0

Welcome to the first in a six-part series on using data structures in .NET 2.0. This article series originally appeared on MSDN Online in October 2003, focusing on the .NET Framework version 1.x, and can be accessed at http://msdn.microsoft.com/vcsharp/default.aspx?pull=/library/en-us/dv_vstechart/html/datastructures_guide.asp. Version 2.0 of the .NET Framework adds new data structures to the Base Class Library, along with new features, such as Generics, that make creating type-safe data structures much easier than with version 1.x. This revised article series introduces these new .NET Framework data structures and examines using these new language features.

Throughout this article series, we will examine a variety of data structures, some of which are included in the .NET Framework's Base Class Library and others that we'll build ourselves. If you're unfamiliar with the term, data structures are classes that are used to organize data and provide various operations upon their data. Probably the most common and well-known data structure is the array, which contains a contiguous collection of data items that can be accessed by an ordinal index.

Before jumping into the content for this article, let's first take a quick peek at the roadmap for this six-part article series, so that you can see what lies ahead.

In this first part of the six-part series, we'll look at why data structures are important, and their effect on the performance of an algorithm. To determine a data structure's effect on performance, we'll need to examine how the various operations performed by a data structure can be rigorously analyzed. Finally, we'll turn our attention to two similar data structures present in the .NET Framework: the Array and the List. Chances are you've used these data structures in past projects. In this article, we'll examine what operations they provide and the efficiency of these operations.

In the Part 2, we'll explore the List's "cousins," the Queue and Stack. Like the List, both the Queue and Stack store a collection of data and are data structures available in the .NET Framework Base Class Library. Unlike a List, from which you can retrieve its elements in any order, Queues and Stacks only allow data to be accessed in a predetermined order. We'll examine some applications of Queues and Stacks, and see how these classes are implemented in the .NET Framework. After examining Queues and Stacks, we'll look at hashtables, which allow for direct access like an ArrayList, but store data indexed by a string key.

While arrays and Lists are ideal for directly accessing and storing contents, when working with large amounts of data, these data structures are often sub-optimal candidates when the data needs to be searched. In Part 3, we'll examine the binary search tree data structure, which is designed to improve the time needed to search a collection of items. Despite the improvement in search time with the binary tree, there are some shortcomings. In Part 4, we'll look at SkipLists, which are a mix between binary trees and linked lists, and address some of the issues inherent in binary trees.

In Part 5, we'll turn our attention to data structures that can be used to represent graphs. A graph is a collection of nodes, with a set of edges connecting the various nodes. For example, a map can be visualized as a graph, with cities as nodes and the highways between them as edged between the nodes. Many real-world problems can be abstractly defined in terms of graphs, thereby making graphs an often-used data structure.

Finally, in Part 6 we'll look at data structures to represent sets and disjoint sets. A set is an unordered collection of items. Disjoint sets are a collection of sets that have no elements in common with one another. Both sets and disjoint sets have many uses in everyday programs, which we'll examine in detail in this final part.

Analyzing the Performance of Data Structures

When thinking about a particular application or programming problem, many developers (myself included) find themselves most interested about writing the algorithm to tackle the problem at hand or adding cool features to the application to enhance the user's experience. Rarely, if ever, will you hear someone excited about what type of data structure they are using. However, the data structures used for a particular algorithm can greatly impact its performance. A very common example is finding an element in a data structure. With an unsorted array, this process takes time proportional to the number of elements in the array. With binary search trees or SkipLists, the time required is logarithmically proportional to the number of elements. When searching sufficiently large amounts of data, the data structure chosen can make a difference in the application's performance that can be visibly measured in seconds or even minutes.

Since the data structure used by an algorithm can greatly affect the algorithm's performance, it is important that there exists a rigorous method by which to compare the efficiency of various data structures. What we, as developers utilizing a data structure, are primarily interested in is how the data structures performance changes as the amount of data stored increases. That is, for each new element stored by the data structure, how are the running times of the data structure's operations effected?

Consider the following scenario: imagine that you are tasked with writing a program that will receive as input an array of strings that contain filenames. Your program's job is to determine whether that array of strings contains any filenames with a specific file extension. One approach to do this would be to scan through the array and set some flag once an XML file was encountered. The code might look like so:

public bool DoesExtensionExist(string [] fileNames, string extension)
{
      int i = 0;
      for (i = 0; i < fileNames.Length; i++)
         if (String.Compare(Path.GetExtension(fileNames[i]), extension, true) == 0)
            return true;

      return false;   // If we reach here, we didn't find the extension
   }
}

Here we see that, in the worst-case—when there is no file with a specified extension, or when there is such a file but it is the last file in the list—we have to search through each element of the array exactly once. To analyze the array's efficiency at sorting, we must ask ourselves the following: "Assume that I have an array with n elements. If I add another element, so the array has n + 1 elements, what is the new running time?" (The term "running time," despite its name, does not measure the absolute time it takes the program to run, but rather refers to the number of steps the program must perform to complete the given task at hand. When working with arrays, typically the steps considered are how many array accesses one needs to perform.) Since to search for a value in an array we need to visit, potentially, every array value, if we have n + 1 array elements, we might have to perform n + 1 checks. That is, the time it takes to search an array is linearly proportional to the number of elements in the array.

This sort of analysis described here is called asymptotic analysis, as it examines how the efficiency of a data structure changes as the data structure's size approaches infinity. The notation commonly used in asymptotic analysis is called big-Oh notation. The big-Oh notation to describe the performance of searching an unsorted array would be denoted as O(n). The large script O is where the terminology big-Oh notation comes from, and the n indicates that the number of steps required to search an array grows linearly as the size of the array grows.

A more methodical way of computing the asymptotic running time of a block of code is to follow these simple steps:

1. Determine the steps that constitute the algorithm's running time. As aforementioned, with arrays, typically the steps considered are the read and write accesses to the array. For other data structures, the steps might differ. Typically, you want to concern yourself with steps that involve the data structure itself, and not simple, atomic operations performed by the computer. That is, with the block of code above, I analyzed its running time by only bothering to count how many times the array needs to be accessed, and did not bother worrying about the time for creating and initializing variables or the check to see if the two strings were equal.
2. Find the line(s) of code that perform the steps you are interested in counting. Put a 1 next to each of those lines.
3. For each line with a 1 next to it, see if it is in a loop. If so, change the 1 to 1 times the maximum number of repetitions the loop may perform. If you have two or more nested loops, continue the multiplication for each loop.
4. Find the largest single term you have written down. This is the running time.

Let's apply these steps to the block of code above. We've already identified that the steps we're interested in are the number of array accesses. Moving onto step 2 note that there is only one line on which the array, fileNames, is being accessed: as a parameter in the String.Compare() method, so mark a 1 next to that line. Now, applying step 3 notice that the access to fileNames in the String.Compare() method occurs within a loop that runs at most n times (where n is the size of the array). So, scratch out the 1 in the loop and replace it with n. This is the largest value of n, so the running time is denoted as O(n).

O(n), or linear-time, represents just one of a myriad of possible asymptotic running times. Others include O(log₂ n), O(n log₂ n), O(n²), O(2ⁿ), and so on. Without getting into the gory mathematical details of big-Oh, the lower the term inside the parenthesis for large values of n, the better the data structure's operation's performance. For example, an operation that runs in O(log n) is more efficient than one that runs in O(n) since log n < n.

Note   In case you need a quick mathematics refresher, log_a b = y is just another way to write a^y = b. So, log₂ 4 = 2, since 2² = 4. Similarly, log₂ 8 = 3, since 2³ = 8. Clearly, log₂ n grows much slower than n alone, because when n = 8, log₂ n = 3. In Part 3 we'll examine binary search trees whose search operation provides an O(log₂ n) running time.

Throughout this article series, each time we examine a new data structure and its operations, we'll be certain to compute its asymptotic running time and compare it to the running time for similar operations on other data structures.