Optimal String Edit Metric

In computational linguistics and text processing, measuring the similarity or dissimilarity between two strings is a fundamental operation. The Optimal String Alignment (OSA) distance is an edit distance metric that quantifies the minimum number of elementary operations required to transform one string into another.

This metric is a refinement of the classic Levenshtein distance, enhanced with the ability to transpose adjacent characters—a common real-world error pattern in human typing and OCR systems.

Permitted Edit Operations: The OSA distance allows exactly four types of single-character operations, each incurring a cost of 1 unit:

1. Insertion: Add a character at any position in the string
2. Deletion: Remove a character from any position in the string
3. Substitution: Replace a character at any position with a different character
4. Transposition: Swap two adjacent characters in the string

Key Constraint: Unlike the Damerau-Levenshtein distance, the OSA distance does not allow a substring to be edited more than once. This means that once a transposition is applied to a pair of adjacent characters, those characters cannot be involved in subsequent edit operations. This "no substring editing twice" restriction makes OSA slightly different from the unrestricted Damerau-Levenshtein distance.

Your Task: Implement a function that computes the OSA distance between two input strings. The function should efficiently calculate the minimum number of edit operations needed to transform the source string into the target string using the four permitted operations described above.

Mathematical Formulation: The OSA distance can be computed using dynamic programming. Let OSA(i, j) represent the edit distance between the first i characters of the source string and the first j characters of the target string. The recurrence relation is:

$$OSA(i, j) = \min \begin{cases} OSA(i-1, j) + 1 & \text{(deletion)} \ OSA(i, j-1) + 1 & \text{(insertion)} \ OSA(i-1, j-1) + cost & \text{(substitution, cost=0 if chars match)} \ OSA(i-2, j-2) + 1 & \text{(transposition, if applicable)} \end{cases}$$

where the transposition case applies only when source[i-1] == target[j-2] and source[i-2] == target[j-1].