The words `computer' and `commuter' are very similar,
and a change of just one letter,
p->m will change the first word into the second.
The word `sport' can be changed
into `sort'
by the deletion of the `p',
or equivalently, `sort' can be changed into `sport'
by the insertion of `p'.
The edit distance of two strings, s1 and s2,
is defined as the minimum number of point mutations
required to change s1 into s2,
where a point mutation is one of:
change a letter,
insert a letter or
delete a letter
The following recurrence relations define the edit distance, d(s1,s2),
of two strings s1 and s2:
d('', '') = 0 -- '' = empty string
d(s, '') = d('', s) = |s| -- i.e. length of s
d(s1+ch1, s2+ch2)
= min( d(s1, s2) + if ch1=ch2 then 0 else 1 fi,
d(s1+ch1, s2) + 1,
d(s1, s2+ch2) + 1 )
The first two rules above are obviously true,
so it is only necessary consider the last one.
Here, neither string is the empty string,
so each has a last character, ch1 and ch2 respectively.
Somehow, ch1 and ch2 have to be explained in an edit of
s1+ch1 into s2+ch2.
If ch1 equals ch2, they can be matched for no penalty, i.e. 0,
and the overall edit distance is d(s1,s2).
If ch1 differs from ch2, then ch1 could be changed into ch2, i.e. 1,
giving an overall cost d(s1,s2)+1.
Another possibility is to delete ch1 and edit s1 into s2+ch2,
d(s1,s2+ch2)+1.
The last possibility is to edit s1+ch1 into s2 and then insert ch2,
d(s1+ch1,s2)+1.
There are no other alternatives.
We take the least expensive, i.e. min, of these alternatives.
The recurrence relations imply an obvious ternary-recursive routine.
This is not a good idea because it is exponentially slow,
and impractical for strings of more than a very few characters.
Examination of the relations reveals
that d(s1,s2) depends only on d(s1',s2') where
s1' is shorter than s1, or s2' is shorter than s2, or both.
This allows the dynamic programming technique to be used.
A two-dimensional matrix, m[0..|s1|,0..|s2|] is used to hold
the edit distance values:
m[,] can be computed row by row.
Row m[i,] depends only on row m[i-1,].
The time complexity of this algorithm is O(|s1|*|s2|).
If s1 and s2 have a `similar' length, about `n' say,
this complexity is O(n^{2}), much better than exponential!
The time-complexity of the algorithm is O(|s1|*|s2|),
i.e. O(n^{2}) if the lengths of both strings is about `n'.
The space-complexity is also O(n^{2})
if the whole of the matrix is kept for a trace-back
to find an optimal alignment.
If only the value of the edit distance is needed,
only two rows of the matrix need be allocated;
they can be "recycled",
and the space complexity is then O(|s1|), i.e. O(n).
Variations
The costs of the point mutations can be varied
to be numbers other than 0 or 1.
Linear gap-costs are sometimes used
where a run of insertions (or deletions) of length `x',
has a cost of `ax+b', for constants `a' and `b'.
If b>0, this penalises numerous short runs of insertions and deletions.
Longest Common Subsequence
The longest common subsequence (LCS)
of two sequences, s1 and s2, is a subsequence of both s1 and of s2
of maximum possible length.
The more alike that s1 and s2 are, the longer is their LCS.
Other Algorithms
There are faster algorithms for the edit distance problem,
and for similar problems.
Some of these algorithms are fast if certain conditions hold,
e.g. the strings are similar, or dissimilar, or the alphabet is large, etc..
Ukkonen (1983) gave an algorithm with worst case time complexity O(n*d),
and the average complexity is O(n+d^{2}),
where n is the length of the strings, and d is their edit distance.
This is fast for similar strings where d is small, i.e. when d<<n.
Applications
File Revision
The Unix command
diff f1 f2 finds the difference between
files f1 and f2, producing an edit script to convert f1 into f2.
If two (or more) computers share copies of a large file F,
and someone on machine-1 edits
F=F.bak, making a few changes, to give F.new,
it might be very expensive and/or slow to transmit the whole
revised file F.new to machine-2.
However, diff F.bak F.new
will give a small edit script which can be transmitted quickly
to machine-2 where the local copy of the file can be updated to equal
F.new.
diff treats a whole line as a "character"
and uses a special edit-distance algorithm that is fast
when the "alphabet" is large and there are few chance matches
between elements of the two strings (files).
In contrast, there are many chance character-matches in DNA where the
alphabet size is just 4, {A,C,G,T}.
Try `man diff' to see the manual entry for diff.
Remote Screen Update Problem
If a computer program on machine-1 is
being used by someone from a screen on (distant) machine-2,
e.g. via rlogin etc.,
then machine-1 may need to update the screen on machine-2 as
the computation proceeds.
One approach is for the program (on machine-1) to keep
a "picture" of what the screen currently is (on machine-2)
and another picture of what it should become.
The differences can be found (by an algorithm related to edit-distance)
and the differences transmitted...
saving on transmission band-width.
Spelling Correction
Algorithms related to the edit distance may be used in spelling correctors.
If a text contains a word, w, that is not in the dictionary,
a `close' word, i.e. one with a small edit distance to w,
may be suggested as a correction.
Transposition errors are common in written text.
A transposition can be treated as a deletion plus an insertion,
but a simple variation on the algorithm can treat
a transposition as a single point mutation.
Plagiarism Detection
The edit distance provides an indication of similarity
that might be too close in some situations ...
think about it.
Molecular Biology
Example
An example of a DNA sequence from `Genebank'
can be found
[here].
The simple edit distance algorithm would normally be run
on sequences of at most a few thousand bases.
The edit distance gives an indication of how `close' two strings are.
Similar measures are used to compute a distance between
DNA sequences (strings over {A,C,G,T}, or
protein sequences (over an alphabet of 20 amino acids),
for various purposes, e.g.:
to find genes or proteins that may have shared functions or properties
to infer family relationships and evolutionary trees over
different organisms
Speech Recognition
Algorithms similar to those for the edit-distance problem
are used in some speech recognition systems:
find a close match between a new utterance and one in
a library of classified utterances.
Notes
V. I. Levenshtein. Binary codes capable of correcting deletions,
insertions and reversals.
Doklady Akademii Nauk SSSR 163(4) p845-848, 1965,
also
Soviet Physics Doklady 10(8) p707-710, Feb 1966.
Discovered the basic DPA for edit distance.
S. B. Needleman and C. D. Wunsch.
A general method applicable to the search for similarities in the
amino acid sequence of two proteins.
Jrnl Molec. Biol. 48 p443-453, 1970.
Defined a similarity score
on molecular-biology sequences, with an O(n^{2}) algorithm
that is closely related to those discussed here.
Hirschberg (1975) presented a method of recovering an alignment (of an LCS)
in O(n^{2}) time but in only linear, O(n)-space; see
[here].
E. Ukkonen
On approximate string matching.
Proc. Int. Conf. on Foundations of Comp. Theory,
Springer-Verlag, LNCS 158 p487-495, 1983.
Worst case O(nd)-time, average case O(n+d^{2})-time
algorithm for edit-distance, where d is the edit-distance between
the two strings.
See also exact, as opposed to approximate, (sub-)string
[matching].
More research information on "the" DPA and Bioinformatics
[here].
If your programming language does not support 2-dimensional arrays,
and requires arrays or strings to indexed from zero upwards,
some home-grown address translation will be needed to program
the DPA defined above.
Exercises
Give a DPA for the longest common subsequence problem (LCS).
Modify the edit distance DPA to that it treats a transposition
as a single point-mutation.