Algorithms and Complexity in C#

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Algorithms and Complexity
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This chapter contains a brief introduction to a central topic in computer science: algorithms and complexity. In theory, modern computers can solve nearly any problem that can be expressed precisely. In practice, however, we encounter two considerable obstacles: No computer can solve problems without valid strategies or methods for solving them, and valid problem-solving strategies and methods are useful only if they yield answers within a reasonable amount of time. Strategies and methods for solving particular problems, given arbitrary input, are called algorithms. The computational complexity of a problem-solving algorithm measures the way in which the resources needed to execute the algorithm depend on the input for which the problem is to be solved. Some algorithms require for correctness, ef ciency, or both data to be organized in a particular way. A data structure is a scheme for organizing data to support ef cient algorithms, and most algorithms assume either implicitly or explicitly particular data structures. In some respects, database programmers need to know considerably less about algorithms and complexity than other programmers, such as systems programmers. Recall that SQL is a fourth-generation, declarative programming language. An SQL program describes the desired result, and the RDBMS implementation analyzes the description and then chooses and implements an ef cient algorithm to produce the result. The mere fact that correct implementations of SQL exist is remarkable; the fact, that there exist astoundingly good implementations, like Microsoft s, is nothing short of miraculous. The modern RDBMS is not only a testament to its creators; it s also a testament to the foundations of computer science, which provided the mathematical framework for conceiving, developing, and validating such a complex system. Many excellent books on algorithms and complexity are available, and they typically include a catalog of important algorithms and analyses of their complexity. In this chapter, I will instead describe some real-world problems that serve as good analogies to get you thinking about some of algorithms Microsoft SQL Server implements. These problems, which for small input are hand solvable, demonstrate some fundamental patterns of complexity, and they illustrate in a concrete way several factors that affect the running time and space requirements of important algorithms.
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Inside Microsoft SQL Server 2008: T-SQL Querying
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Many of you probably have a change jar somewhere a container full of coins. From time to time, you might dig into your change jar to nd a quarter,1 and the process of doing so is probably second nature. Partly because it s so familiar, the process of retrieving a quarter from a change jar will be a useful example for the discussion of algorithms and complexity. While a coin isn t exactly data, retrieving a quarter is much like executing this T-SQL SELECT query:
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SELECT TOP (1) Coin FROM ChangeJar WHERE Denomination = 0.25 ORDER BY (SELECT NULL);
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How to Retrieve a Quarter from a Coin Jar
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I m sure you know more than one algorithm for executing this task to retrieve a quarter from a coin jar. Most of the time, you look into the jar, spot a quarter at the top, and pull it out. Every now and then, however, there s no quarter at the top, and you have to dig deeper. When this happens, you might shake the jar or stick your hand into it and mix the coins up, expecting to nd a quarter at the top again after the mixing. If you still can t nd a quarter, you might empty the coins onto your kitchen counter and spread them out so that you can hunt through your coins more quickly than you can when they re all in the jar. This last strategy, of course, requires you have a kitchen counter (or other at surface) nearby that you can clear off before emptying the coins onto it. If you try to do this right before suppertime, you might have to wait a little while or abandon the strategy. You can see from this example that how and how quickly you can nd a quarter in a coin jar depends on many things: what s in the jar, how the jar s contents are distributed, how you go about looking, and what other tools (like a table) are at your disposal, just to name a few. More obscure factors, too, can affect both your strategy and its ef ciency: how bright the room lights are, how big your hands are compared to the size of the jar s mouth, how full the jar is (because shaking a full jar doesn t do a good job of mixing up its contents), and whether someone else is also retrieving a quarter from of the same jar (or preparing dinner) at the same time as you. How many other factors can you think of The various strategies for retrieving a quarter, as well as the factors that affect how well each strategy works, all have analogs both in the abstract study of algorithms and complexity and in the practical matter of executing queries in a SQL Server database. For example, the kitchen counter corresponds to both the abstract notion of space and the real SQL Server data cache. Shaking the coin jar corresponds to randomizing the distribution of values in the algorithm s input or changing the SQL Server statistics for an index or table.
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A quarter is the largest commonly circulating US coin, and it is worth 25 cents, or one-quarter dollar. If digging for quarters isn t something you do often enough to have a feel for it, use an analogous scenario, with any common coin instead of quarters.
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