Understanding the longest common substring problem

Many problems that involve strings are really about finding patterns that appear in more than one input. Sometimes those patterns can have gaps (subsequences) or allow characters to appear in a different order (subsets). Other times, the pattern has to stay completely continuous (substrings). That difference might seem small, but it can change the way a problem behaves and how we solve it. A classic example is the longest common substring problem, where the goal is to find the longest sequence of characters that appears continuously in both strings.

At first glance, comparing all substrings of two strings feels straightforward, but the number of possible substrings grows quickly as the strings get longer. Naively checking all possibilities becomes impractical. This problem has practical applications in fields such as bioinformatics for DNA sequence analysis, plagiarism detection, file comparison tools, and data compression algorithms.

In this lesson, we will explore the longest common substring problem and see how dynamic programming helps us solve it in a structured and efficient way.

The longest common substring problem

Consider we are given two strings s1 and s2 of lengths m and n respectively. A substring is a contiguous sequence of characters within a string. Unlike a subsequence, the characters must appear in the exact same consecutive order as they do in the original string.

The goal is to find the length of the longest substring that appears in both s1 and s2 without breaking character order or continuity.