# Lexicographic order

Martin McBride, 2021-11-02

Tags lexicographic

Categories statistics probability

Lexicographic ordering is similar to dictionary ordering, but it extends the idea to other items, not just letters of the alphabet. It allows us to arrange sequences of items, drawn from a particular set, in a well-defined order.

For example, a word is a sequence of items from the set of letters (a to z). Alphabetic ordering considers each word, letter by letter. So if we compare the words "cat" and "cart":

- Both words have a first letter "c", so we move on to the next letter.
- Both words have a second letter "a", so we move on to the next letter.
- The third letter of "cat" is "t", but the third letter of "cart" is "r", which comes earlier in the alphabet.

Therefore, of course, "cart" comes before "cat" in alphabetical (therefore lexicographical) order.

## Lexicographical ordering of numbers

A number is a sequence of items from the set of digits (0 to 9). Consider the two numbers 206 and 2035.

If we arrange those values in increasing numerical order, then clearly 206 comes before 2035.

In lexicographical ordering, we consider each digit in turn:

- Both numbers have the first digit 2, so we move on to the next digit.
- Both numbers have a second digit 0, so we move on to the next digit.
- The third digit of 206 is 6, but the third digit of 2035 is 3, which is smaller.

Therefore 2035 comes before 206 in lexicographical order, even though it is numerically larger.

## Lexicographical ordering of other types of data

Suppose we had a collection of counters with colours red, orange, yellow, green, blue, indigo and violet - the rainbow colours:

We can apply lexicographical ordering, provided we define the order of the items in the set. Unlike letters and digits, colours don't have a natural order. But we can specify that we are going to use the rainbow order defined above.

As an example, consider the three colour sequence OGB, and the four colours sequence OGYR.

The first 2 colours are the same in both sequences, but when we compare the third colour, yellow is before blue, therefore OGYR comes before OGB.