# Damerauâ€“Levenshtein Distance, Lua Implementation

I stumbled across Levenshtein distance today and had to try my hand at writing an implementation in Lua. I choose the slightly more complex Damerauâ€“Levenshtein distance, and I think it turned out pretty well.

Some notes of interest:

• Complexity is O( (#t+1) * (#s+1) ) when lim isn’t specified.
• This function can be used to compare array-like tables as easily as strings.
• This function is case sensitive when comparing strings.
• Using this function to compare against a dictionary of 250,000 words took about 0.6 seconds on my machine for the word “Teusday”, around 10 seconds for very poorly spelled words. Both tests used lim.

[ccn_lua]–[[
Function: EditDistance

Finds the edit distance between two strings or tables. Edit distance is the minimum number of
edits needed to transform one string or table into the other.

Parameters:

s – A *string* or *table*.
t – Another *string* or *table* to compare against s.
lim – An *optional number* to limit the function to a maximum edit distance. If specified
and the function detects that the edit distance is going to be larger than limit, limit
is returned immediately.

Returns:

A *number* specifying the minimum edits it takes to transform s into t or vice versa. Will
not return a higher number than lim, if specified.

Example:

:EditDistance( “Tuesday”, “Teusday” ) — One transposition.
:EditDistance( “kitten”, “sitting” ) — Two substitutions and a deletion.

returns…

:1
:3

Notes:

* Complexity is O( (#t+1) * (#s+1) ) when lim isn’t specified.
* This function can be used to compare array-like tables as easily as strings.
* The algorithm used is Damerauâ€“Levenshtein distance, which calculates edit distance based
off number of subsitutions, additions, deletions, and transpositions.
* Source code for this function is based off the Wikipedia article for the algorithm
.
* This function is case sensitive when comparing strings.
* If this function is being used several times a second, you should be taking advantage of
the lim parameter.
* Using this function to compare against a dictionary of 250,000 words took about 0.6
seconds on my machine for the word “Teusday”, around 10 seconds for very poorly
spelled words. Both tests used lim.

Revisions:

v1.00 – Initial.
]]
function EditDistance( s, t, lim )
local s_len, t_len = #s, #t — Calculate the sizes of the strings or arrays
if lim and math.abs( s_len – t_len ) >= lim then — If sizes differ by lim, we can stop here
return lim
end

— Convert string arguments to arrays of ints (ASCII values)
if type( s ) == “string” then
s = { string.byte( s, 1, s_len ) }
end

if type( t ) == “string” then
t = { string.byte( t, 1, t_len ) }
end

local min = math.min — Localize for performance
local num_columns = t_len + 1 — We use this a lot

local d = {} — (s_len+1) * (t_len+1) is going to be the size of this array
— This is technically a 2D array, but we’re treating it as 1D. Remember that 2D access in the
— form my_2d_array[ i, j ] can be converted to my_1d_array[ i * num_columns + j ], where
— num_columns is the number of columns you had in the 2D array assuming row-major order and
— that row and column indices start at 0 (we’re starting at 0).

for i=0, s_len do
d[ i * num_columns ] = i — Initialize cost of deletion
end
for j=0, t_len do
d[ j ] = j — Initialize cost of insertion
end

for i=1, s_len do
local i_pos = i * num_columns
local best = lim — Check to make sure something in this row will be below the limit
for j=1, t_len do
local add_cost = (s[ i ] ~= t[ j ] and 1 or 0)
local val = min(
d[ i_pos – num_columns + j ] + 1, — Cost of deletion
d[ i_pos + j – 1 ] + 1, — Cost of insertion
d[ i_pos – num_columns + j – 1 ] + add_cost — Cost of substitution, it might not cost anything if it’s the same
)
d[ i_pos + j ] = val

— Is this eligible for tranposition?
if i > 1 and j > 1 and s[ i ] == t[ j – 1 ] and s[ i – 1 ] == t[ j ] then
d[ i_pos + j ] = min(
val, — Current cost
d[ i_pos – num_columns – num_columns + j – 2 ] + add_cost — Cost of transposition
)
end

if lim and val < best then best = val end end if lim and best >= lim then
return lim
end
end

return d[ #d ]
end[/ccn_lua]
Gist of the same source code.