The Normalized Edit Distance with Uniform Operation Costs is a Metric

TL;DR

This paper proves that the normalized edit distance with equal operation costs is a true metric, resolving a long-standing question and comparing it to alternative metrics in formal verification contexts.

Contribution

It establishes that the normalized edit distance with uniform costs satisfies the metric properties, closing a gap in the literature and providing insights for applications.

Findings

Normalized edit distance with uniform costs is a metric.

Comparison of this metric with alternative normalized metrics.

Key properties explaining the metric's suitability for applications.

Abstract

We prove that the normalized edit distance proposed in [Marzal and Vidal 1993] is a metric when the cost of all the edit operations are the same. This closes a long standing gap in the literature where several authors noted that this distance does not satisfy the triangle inequality in the general case, and that it was not known whether it is satisfied in the uniform case where all the edit costs are equal. We compare this metric to two normalized metrics proposed as alternatives in the literature, when people thought that Marzal's and Vidal's distance is not a metric, and identify key properties that explain why the original distance, now known to also be a metric, is better for some applications. Our examination is from a point of view of formal verification, but the properties and their significance are stated in an application agnostic way.