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Algorithm building blocks

Monday, 28 January 2019
We consider the structures you will use as you build algorithms.
This section has no prerequisites.


As you may recall, an algorithm is a set of instructions to accomplish a problem. As you might expect, the way we express an algorithm depends, in large part, on the audience for the algorithm. Consider the problem of making a set of chocolate chip cookies. For a beginning cook, we would have to carefully explain what it means to sift flour, the steps involved in creaming together butter and sugar, and how to determine whether a cookie is done, and such. For an intermediate cook, we might not need to provide those details, but we would likely need to give an order in which ingredients are combined. For an expert cook, we might only list the ingredients and the recommended oven temperature and could assume that they would be able to figure out the rest.

In spite of the wide variety of audiences and kinds of algorithms, you will find that there are a few basic “building blocks” that you will use as you construct algorithms, whether in a natural language or in a programming language. As you learn a new programming language, make sure to pay attention to how you express these blocks.

In this section, we will introduce these blocks conceptually. You will learn how to express each in the chapters that follow.

“Built in” values and operations: Your basic building blocks

Even when you are writing an algorithm for a novice, you have to build upon some basic assumptions. In teaching someone to make a cookie, you can assume that they know what the different ingredients are (e.g., flour, sugar) and that they know some basic actions to do with those ingredients (e.g., stir). In teaching someone how to compute a square root, you assume that they know about numbers (including, for example, zero and one) and that they know how to work with those numbers (e.g., add, multiply). In teaching someone how to analyze a text computationally, you might assume that they know the parts of speech and, perhaps, how to parse a sentence.

Behind the scenes, computers “know” very little. But most programming languages and environments provide a rich infrastructure of values and operations on those values. We call these the “built in” values and operations.

In most programming languages, you will be able to work not only with numbers, but also strings (sequences of characters), files, images, ordered collections of values, and much more. Of course, not all operations are applicable to all kinds of values: You would not expect to sift water or to compute the square root of a word. Hence, we often group similar kinds of values into what computer scientists refer to as a type, a collection of values and valid operations on those values.

Sequencing: ordering the operations

Most algorithms involve multiple steps. As you might expect, the order in which we perform steps can have a significant impact on the outcome of our algorithm. If we bake the ingredients of a cookie before mixing them together, we are likely to get something much less satisfactory than what we get when we mix before baking. Hence, a core aspect of most algorithms is the way we sequence the steps in the algorithm.

In some cases, you will find that you explicitly sequence the steps, telling the reader (often, the computer) to do one task, then then next, then the next. For example, you might tell a baker to shift together the dry ingredients before adding them to the liquid ingredients.

In other cases, you will express the order implicitly. For example, if you ask someone to compute 3+4*5, they should know to mulitply 4*5 before adding three, even if you don’t make that explicit.

You will discover that most programming languages have multiple ways to express both explicit and implicit ordering. Frequently, you make ordering explicit by placing the instructions in order; the computer executes them from first to last. In other cases, the ordering will be less explicit, as in the computation of 3+4*5.

Variables: naming values

As you write algorithms, you will find it convenient to name things. In natural language algorithms, names are often implicit, such as “the dry ingredients” in a baking recipe. In computer languages, you will find that you more frequently need to explicitly name things, as in “let ‘dry ingredients’ refer to the combination of flour, baking soda, and salt”. And, just as we care about the type of the built-in values, we might also specify the type of variables, using some variables to represent numbers and others to represent strings.

Computer scientists tend to refer to named values as variables, even though they don’t always vary. We try to choose descriptive names for our variables to help the human reader of our programs understand their purpose. For example, a reader might be momentarily confused if we used my-name to refer to the number 5 or if we tried to multiple my-name by 2.

Subroutines: naming helper algorithms

In writing longer algorithms, you will often find that you have to explain similar processes again and again. For example, to teach someone how to make a nut butter and preserve sandwich, you might need to explain how to open a jar twice, once for the nut butter and once for the jar of preserves. Rather than repeating the instructions, we will find it helpful to make a separate algorithm that we can refer to in our main algorithm. For that particular example, we might have an “open jar” subroutine that we can use in our sandwich algorithm.

Computer scientists use a wide variety of names for these helper algorithms. Some just call them algorithms. Many refer to them as subroutines to emphasize that they are subordinate to the primary algorithm. In programming languages, we usually call them procedures or functions.

Like the functions you’ve encountered in your study of mathematics, subroutines often take inputs (e.g., a closed jar) and return a newly computed value (e.g., an open jar). We tend to refer to those inputs as parameters or arguments.

Some computer scientists are careful to distinguish between functions (whose primary purpose is to compute a value) and procedures (whose primary purpose is something other than computing a value) and between parameters (what we call the inputs to a subroutine when we define the subroutine) and arguments (what we call the inputs to a subroutine when we use the subroutine). We will be a bit more casual in our usage.

Conditionals: making decisions

We’ve considered four basic components of algorithms: the built-in operations, sequencing, named values, and subroutines, . By themselves, those three components let you write some algorithms, but only fairly straightforward algorithms. To write more complex algorithms, you need more complex algorithm structures. One of the most important kinds of structures is the conditional, which lets you make decisions based on some condition. For example, if we are writing a program that generates sentences, even after choosing the verb, we’ll have to use a different form of that verb depending on whether the subject is singular or plural. Those of a more mathematical mindset might consider the problem of computing the absolute value of a number: If the number is negative, we return negative one times the number. If the number is not negative, we just return the number.

Most conditionals take one of two forms. The most basic conditional performs an action only if a condition holds.

if <some condition> holds then
  perform some actions        

For example,

if your baking powder is old then                 
  double the amount of baking powder in the recipe

More frequently, though, we use the conditional to choose between one of two options.

if <some condition> holds then      
  perform some actions              
  perform a different set of actions

For example,

if you are at an elevation below 3000 feet then     
  set the oven temperature to 325 degrees fahrenheit
  set the oven temperature to 350 degrees fahrenheit


To compute the absolute value of a real number, n
  if n < 0 then                                  
    return -1*n                                  
    return n                                     

Once in a while, you’ll find that you want to decide between more than two possibilites. In those situations, we often organize those into a sequence of conditions, which we often refer to as guards.

Racket has a variety of conditionals that we will explore once we have mastered the basics.

Repetition: repeating tasks

We’ve seen that subroutines provide one mechanism for repeating a series of actions. If you want to do the same thing again, but with a different input, you just call the subroutine again. However, we frequently find that we want to repeat actions a large number of times and would find it inconvenient, at best, to write the call to the subroutine again and again and again. For situations like this, most programming languages provide structures that let us concisely write algorithms that perform a task multiple times.

  • We might do work until we reach a desired state. For example, in baking, we often stir solids and liquids together until the solids are fully dissolved. And, in teaching me to make bagels, my mother taught me to knead the dough until it reaches a consistency close to that of an earlobe. Many computer scientists refer to these as while loops.

  • We might do work for a specified number of repetitions. For example, some recipes call for you to beat for 100 strokes. Many computer scientists refer to these as for loops.

  • We might do work for each element in a collection. For example, many text analysis routines process each word in a text in turn. Many computer scientists refer to these as for-each loops. Others refer to the process of doing something to each element in a collection as mapping.

You’ll discover that there are multiple ways to express each kind of repetition, as well as some more general forms of repetition.

Input and output: communicating with others

Finally, we need a way for our program to communicate with the outside world. Programs may take input from a variety of places (e.g., the keyboard, a file, a network connection) and may provide output (results) somewhere else (e.g., the screen, a file, a network connection).

Some programming systems, including DrRacket, let us run programs interactively, typing expressions and seeing the results. But others will explicitly prompt for input and explicitly generate output. You will find reasons to use both the “interactive computing” mode that DrRacaket provides and the more general mechanisms for input and output.

Self Checks

Check 1: Reflecting on basic algorithms

Pick a non-trivial algorithm, such as a recipe for making chocolate chip cookies, and identify examples from that algorithm for each of the different algorithmic building blocks. If you wrote an algorithm on the first day of class, use that algorithm.

Check 2: Conditionals, subroutines, and repetition

Suppose we want to have the computer “read” a text and decide whether it expresses a positive or negative world view. In what ways would conditionals, subroutines, and repetition help with this task?