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What bothers me a bit, is that in example where we trying to find Levenstein distance between two string are we 100% sure that calculated distance from end of string to the start will result in the same value as when we calculate distance from start to end?

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That's a very good question.

I think the simplest approach to prove it would be to show that both computations yield the global minimum. Since there is only one global minimum, then they must be equal.

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