Difference in normalization of Levenshtein (edit) distance?

Question

If the Levenshtein distance between two strings, s and t is given by L(s,t),

what is the difference in the impact on the resulting heuristic of the following two different normalization approaches?

1. L(s,t) / [length(s) + length(t)]

2. L(s,t) / max[length(s), length(t)]

3. (L(s,t)*2) / [length(s) + length(t)]

I noticed that normalization approach 2 is recommended by the Levenshtein distance Wikipedia page but no mention is made of approach 1. Are both approaches equally valid? Just wondering if there is some mathematical justification for using one over the other.

Also, what is the difference between approach 1 and approach 3?

With the following example:

s = "Hi, my name is"
t = "Hello, my name is"
L(s,t) = 4
length(s) = 14 # (includes white space)
length(t) = 17 # (includes white space)

The Levenshtein distance given the three normalization algorithms above are:

[Approach 1]   4  /(14+17) = 0.129
[Approach 2]   4  /(17)    = 0.235
[Approach 3] (4*2)/(14+17) = 0.258

Answer 1

The effects of both variants should be nearly the same.

• The second approach covers a range from 0 (strings are equal) to 1 (completely different)...
• while the upper range in the first variant depends on the length of the strings: if the lengths are nearly equal the upper bound is 0.5, and increases on larger differences between the lengths.