The Essence of Levenshtein Edit Distance: A Mathematical Lens

The Birth of the Concept

The Levenshtein edit distance algorithm was introduced by Vladimir Levenshtein in 1965. It's a mathematical formula that calculates the least number of single-character edits needed to change one string into another.

The Intimidating Equation

Upon first glance, the formula for Levenshtein Distance may seem complex:

Breaking Down the Formula

The Min Function

The min function is key to understanding the formula. What we're doing is considering three options at each step:

1. Insertion: This is represented by $\text{Levenshtein}(a-1,b)$
2. Deletion: This is represented by $\text{Levenshtein}(a,b-1)$
3. Substitution: This is represented by $\text{Levenshtein}(a-1,b-1)$

We take the minimum of these three options because we want to find the least number of edits needed.

The Cost Factor

The "cost" is either 0 or 1, depending on whether the current characters in the strings a and b are equal or not. If they are equal, the cost is 0; otherwise, it's 1.

A More Intuitive View

Think of this as a journey where you start at one corner of a grid. Your destination is the opposite corner. The grid cells represent the characters in the strings you're comparing. Each move from one cell to another corresponds to an edit operation:

• Moving horizontally represents insertion.
• Moving vertically represents deletion.
• Moving diagonally represents substitution.

You're trying to find the shortest path from one corner to another, making the least number of moves (edits). That's your Levenshtein Distance.