1. Complement to lecture 11 : Levenshtein distance algorithm

2. Levenshtein distance Also called “Edit distance”
Purpose: to compute the smallest set of basic operations
Insertion
Deletion
Replacement
that will turn one string into another
Intro. to Programming, lecture 11 (complement): Levenshtein 2

3. M I I C H H A A E L J A C K S O N E N D S H Levenshtein distance 1 2 3
“Michael Jackson” to “Mendelssohn” M I I C H H A A E L J A C K S O N E N D S H S D S S S D D D D I
Operation
Distance 1 2 3 4 5 6 7 8 9 10
Intro. to Programming, lecture 11 (complement): Levenshtein 3

4. Levenshtein distance algorithm
levenshtein (source, target : STRING): INTEGER
-- Minimum number of operations to turn source into target
local
distance : ARRAY_2 [INTEGER]
i, j, del, ins, subst : INTEGER do
create distance.make (source.count, target.count)
from i := 0 until i > source.count loop
distance [i, 0] := i ; i := i + 1 end
from j := 0 until j > target.count loop distance [0, j ] := j ; j := j + 1 end
-- (Continued)
Indexed from zero
Intro. to Programming, lecture 11 (complement): Levenshtein 4

5. Levenshtein, continued
from i := 1 until i > source.count loop
from j := 1 until j > target.count invariant
loop
if source [i ] = target [ j ] then
distance [i, j ] := distance [ i -1, j -1] else
deletion := distance [i -1, j ] insertion := distance [i , j - 1] substitution := distance [i - 1, j - 1]
distance [i, j ] := minimum (deletion, insertion, substitution) + 1
end
j := j + 1
i := i + 1
Result := distance (source.count, target.count)
-- For all p : 0 .. i, q : 0 .. j –1, we can turn source [1 .. p ]
-- into target [1 .. q ] in distance [p, q ] operations
s [m .. n ]: substring of s with items at positions k such that m  k  n (empty if m > n)
Intro. to Programming, lecture 11 (complement): Levenshtein 5

6. Target B E A T L E S
Source  1 2 3 4 5 6 7 I I I I I I I 1 2 3 4 5 6 7 D K I I I I I I B 1 1 1 2 3 4 5 6 D D K K I I I I E 2 2 1 1 2 3 4 5 D D K S S S K I I I E 3 3 2 1 1 2 3 3 ? 4 D D D S D K S I I I T 4 4 3 2 2 1 2 3 4 D D D S D D S S S I I H 5 5 4 3 3 2 2 3 4 I K D S
Keep
Insert
Delete
Substitute

7. B E A T L E S B E E T H 1 2 3 4 5 6 7 Keep B,1 1 1 2 3 4 5 6 Keep E,21 2 3 4 5 6 7
Keep B,1 B 1 1 2 3 4 5 6
Keep E,2 E 2 1 1 2 3 4 5
Subst EA,3 E 3 2 1 1 2 3 3 4
Keep T,4 T 4 3 2 2 1 2 3 4
Subst HS,7
Ins L,5
Ins E,6 H 5 4 3 3 2 2 3 4
Keep
Insert
Delete
Substitute