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Big-O notation is a way to describe the upper bound of an algorithm's running time. It measures the worst-case scenario, helping us understand how an algorithm scales. The idea is to see how the running time increases with the size of the input.
For example, if an algorithm has a running time of O(n), it means the running time grows linearly with the size of the input. If the input doubles, the running time doubles.
When comparing different algorithms, Big-O notation helps us choose the one that performs better as the input size increases. In our case with the naive string matcher, understanding its Big-O notation tells us how efficiently it can find patterns in a text, especially for large texts or patterns.

The naive string matcher is a straightforward and simple string matching algorithm. It finds all occurrences of a pattern in a given text by checking every possible starting position in the text.
Here’s how it works:

• Start from the first character of the text.
• Check if the pattern matches the text, character by character.
• If the pattern does not match, shift the starting position one character to the right and repeat.

This process is repeated until the end of the text is reached. While the naive string matcher is easy to understand, it can be slow for large texts and patterns due to the repeated character comparisons.

Character comparisons are at the heart of the naive string matcher. For each position in the text, up to m comparisons may be made, where m is the length of the pattern. Let’s break it down:

• If we have a text of length n and a pattern of length m, there are n - m + 1 possible starting positions in the text.
• For each starting position, the algorithm checks each character of the pattern against the corresponding character in the text.

This leads to a total of \((n - m + 1) \times m\) comparisons in the worst case.
When estimating this in Big-O notation, we focus on the term that grows the fastest as the input size increases. In this case, it simplifies to \O(nm)\, indicating that the number of comparisons grows proportional to the product of the lengths of the text and the pattern.