Background: Algorithms

An algorithm specifies a series of steps that perform a particular computation or task. Algorithms were originally born as part of mathematics – the word “algorithm” comes from the Arabic writer Muḥammad ibn Mūsā al-Khwārizmī, – but currently the word is strongly associated with computer science. Throughout this book we’ll examine a number of different algorithms to perform a variety of tasks.

Algorithms resemble recipes. Recipes tell you how to accomplish a task by performing a number of steps. For example, to bake a cake the steps are: preheat the oven; mix flour, sugar, and eggs throughly; pour into a baking pan; and so forth.

However, “algorithm” is a technical term with a more specific meaning than “recipe”, and calling something an algorithm means that the following properties are all true:

  • An algorithm is an unambiguous description that makes clear what has to be implemented. In a recipe, a step such as “Bake until done” is ambiguous because it doesn’t explain what “done” means. A more explicit description such as “Bake until the cheese begins to bubble” is better. In a computational algorithm, a step such as “Choose a large number” is vague: what is large? 1 million, 1 billion, or 100? Does the number have to be different each time, or can the same number be used on every run?
  • An algorithm expects a defined set of inputs. For example, it might require two numbers where both numbers are greater than zero. Or it might require a word, or a list of zero or more numbers.
  • An algorithm produces a defined set of outputs. It might output the larger of the two numbers, an all-uppercase version of a word, or a sorted version of the list of numbers.
  • An algorithm is guaranteed to terminate and produce a result, always stopping after a finite time. If an algorithm could potentially run forever, it wouldn’t be very useful because you might never get an answer.
  • Most algorithms are guaranteed to produce the correct result. It’s rarely useful if an algorithm returns the largest number 99% of the time, but 1% of the time the algorithm fails and returns the smallest number instead. [1]
  • If an algorithm imposes a requirement on its inputs (called a precondition), that requirement must be met. For example, a precondition might be that an algorithm will only accept positive numbers as an input. If preconditions aren’t met, then the algorithm is allowed to fail by producing the wrong answer or never terminating.

Studying algorithms is a fundamental part of computer science. There are several different characteristics of an algorithm that are useful to know:

  1. Does an algorithm actually exist to perform a given task?
  2. If someone proposes an algorithm to solve a task, are we sure that the algorithm works for all possible inputs?
  3. How long does the algorithm take to run? How much memory space does it require?
  4. Once we know it’s possible to solve a problem with an algorithm, a natural question is whether the algorithm is the best possible one. Can the problem be solved more quickly?

Most of these questions will be discussed for the algorithms covered in this book.

An Example Algorithm

Let’s look at a very simple algorithm called find_max().

Problem: Given a list of positive numbers, return the largest number on the list.

Inputs: A list L of positive numbers. This list must contain at least one number. (Asking for the largest number in a list of no numbers is not a meaningful question.)

Outputs: A number n, which will be the largest number of the list.

Algorithm:

  1. Set max to 0.
  2. For each number x in the list L, compare it to max. If x is larger, set max to x.
  3. max is now set to the largest number in the list.

An implementation in Python:

def find_max (L):
    max = 0
    for x in L:
        if x > max:
            max = x
    return max

Does this meet the criteria for being an algorithm?

  • Is it unambiguous? Yes. Each step of the algorithm consists of primitive operations, and translating each step into Python code is very easy.
  • Does it have defined inputs and outputs? Yes.
  • Is it guaranteed to terminate? Yes. The list L is of finite length, so after looking at every element of the list the algorithm will stop.
  • Does it produce the correct result? Yes. In a formal setting you would provide a careful proof of correctness. In the next section I’ll sketch a proof for an alternative solution to this problem.

A Recursive Version of find_max()

There can be many different algorithms for solving the same problem. Here’s an alternative algorithm for find_max():

  1. If L is of length 1, return the first item of L.
  2. Set v1 to the first item of L.
  3. Set v2 to the output of performing find_max() on the rest of L.
  4. If v1 is larger than v2, return v1. Otherwise, return v2.

Implementation:

def find_max (L):
    if len(L) == 1:
        return L[0]
    v1 = L[0]
    v2 = find_max(L[1:])
    if v1 > v2:
        return v1
    else:
        return v2

Let’s ask our questions again.

  • Is it unambiguous? Yes. Each step is simple and easily translated into Python.

  • Does it have defined inputs and outputs? Yes.

  • Is it guaranteed to terminate? Yes. The algorithm obviously terminates if L is of length 1. If L has more than one element, find_max() is called with a list that’s one element shorter and the result is used in a computation.

    Does the nested call to find_max() always terminate? Yes. Each time, find_max() is called with a list that’s shorter by one element, so eventually the list will be of length 1 and the nested calls will end.

Finally, does it produce the correct result? Yes. Here’s a sketch of a proof. [2]

Consider a list of length 1. In this case the largest number is also the only number on the list. find_max() returns this number, so it’s correct for lists of length 1.

Now consider a longer list of length N+1, where N is some arbitrary length. Let’s assume that we’ve proven that find_max() is correct for all lists of length N. The value of v2 will therefore be the largest value in the rest of the list. There are two cases to worry about.

  • Case 1: v1, the first item of the list, is the largest item. In that case, there are no other values in the list greater than v1. We’re assuming find_max() is correct when executed on the rest of the list, so the value it returns will be less than v1. The if v1 > v2 comparison will therefore be true, so the first branch will be taken, returning v1. This is the largest item in the list, so in this case the algorithm is correct.
  • Case 2: v1, the first item of the list, is not the largest item. In that case, there is at least one value in the list that’s greater than v1. find_max() is correct for the shortened version of the rest of the list, returning the maximum value it contains, so this value must be greater than v1. The if v1 > v2 comparison will therefore be false, so the else branch will be taken, returning v2, the largest value in the rest of the list. This case assumes that v1 is not the largest value, so v2 is therefore the largest value, and the algorithm is also correct in this case.

With these two cases, we’ve now shown that if find_max() is correct for lists of length N, it’s also correct for lists of length N+1. In the first part of our argument, we’ve shown that find_max() is correct for lists of length 1. Therefore, it’s also correct for lists that are 2 elements long, and 3 elements, and 4, 5, 6, ... up to any number.

This may seem like a trick; we showed that it’s correct for the trivial case of the single-element list, and then showed that it’s correct on a problem of a certain size. Such proofs are called inductive proofs, and they’re a well-known mathematical technique for proving a theorem.

Carrying out an inductive proof of some property requires two steps.

  1. First, you show that the property is true for some simple case: an empty list or a list of length 1, an empty set, a single point. Usually this demonstration is very simple; often it’s obviously true that the property is true. This is called the basis case.
  2. Next, you assume the property is true for size N and show that it’s true for some larger size such as N+1. This is called the inductive step, and is usually the more difficult one.

Once you have both demonstrations, you’ve proven the property is true for an infinite number of values of N; correctness for N=1 implies that the N=2 case is also correct, which in turn implies correctness for N=3, 4, 5, and every other positive integer. Not every theorem can be put into a form where an inductive proof can be used.

References

XXX something on induction

Footnotes

[1]There are special situations where algorithms that are sometimes wrong can still be useful. A good example is testing whether a number is prime. There’s an algorithm called the Rabin-Miller test that’s always correct when it reports a number is composite, but has a 25% chance of being wrong when it reports a number is prime. One test therefore isn’t enough to conclude you’ve found a prime, but you can perform repeated tests and reduce the chance of being wrong to as low as you like (but never zero).
[2]It’s possible to write formal proofs of correctness for an algorithm, but the resulting proofs are lengthy even for short algorithms such as this one.
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