1. Approximate Matching of Run-Length Compressed Strings
Veli Mäkinen*
Gonzalo Navarro**
Esko Ukkonen*
* Department of Computer Science, University of Helsinki, Finland
** Department of Computer Science, University of Chile, Santiago, Chile

2. Approximate Matching of Run-Length Compressed Strings, CPM'2001
The Problem
Run-length coding of a string is RL(aaaabbbbbbccaaabb)=a4b6c2a3b2.
Problem: Given two run-length encoded strings, calculate their edit distance.
The trivial solution is to decompress and then calculate, but can it be done faster...
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

3. Variations of the scheme
Different variations of edit distance.
Levenshtein distance: Minimum number of character insertions, deletions, and substitutions to convert a string into another string.
Longest common subsequence (LCS): Distance DID, minimum number of insertions and deletions, is a dual problem for LCS; 2·|LCS(A,B)|=m+n-DID(A,B).
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

4. Variations of the scheme (2)
Search problem: Search all approximate occurrences of a short pattern P inside a large text T, where both the pattern and the text are run-length encoded.
With approximate occurrence we mean that the (edit) distance between the pattern and a substring of the text is at most some treshold value k.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

5. Approximate Matching of Run-Length Compressed Strings, CPM'2001
Previous Results
Bunke & Csirik, 95: O(m’n+mn’) for calculating the LCS between two strings of lengths m and n, run-length encoded to lengths m’ and n’.
Apostolico & Landau & Skiena, 97: O(m’n’ log(m’n’)) for the LCS.
Mitchell, 97: O((m’+n’+p) log(m’+n’+p)) for the LCS, where p is the amount of matches between compressed characters.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

6. Approximate Matching of Run-Length Compressed Strings, CPM'2001
Our Results
The first algorithm for the Levenshtein distance in this context: O(m’n+mn’) by generalizing the result of Bunke & Csirik for the LCS.*
Search algorithm for Levenshtein and LCS distances: O(mm’n’).
* Independently Arbell & Landau & Mitchell found similar algorithm.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

7. Approximate Matching of Run-Length Compressed Strings, CPM'2001
Our Results (2)
O(min(d2min(m’,n’), m’n’max(m’,n’))) for the LCS, where d=m+n-2·|LCS(A,B)|.
Conjecture: O(m’n’) average case for the LCS.
Experimental results to support the conjecture.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

8. Bunke & Csirik algorithm for the LCS
a a a a a b b b b c c c c a a
Equal letter box: values from the diagonal. 1 2 3 4 5 6 7 8 9 10 11 1 2 6 10 2 1 5 9 3 4 8 4 1 5 7 5 4 3 2 6 6 3 5 7 7 4 8 8 5 3 7 9 8 7 6 5 4 3 2 10 7 3 11 8 4 12 9 5 13 12 11 10 9 8 7 6 14 11 7 9 15 14 13 12 11 10 9 8 7 6 a b
Different letter box: minimum of two values: (value from the left border+distance, value from the upper border +distance).
2·|LCS(A,B)|=m+n-DID(A,B) ==>
|LCS(aaaaabbbbccccaa,aaabbbbaaaa)| = ( )/2=9
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

9. O(m’n+mn’) for the Levenshtein distance
a a a a a b b b b c c c c a a 1 2 3 4 5 6 7 8 9 10 11 1 2 6 10 2 1 5 9 3 4 8 4 1 7 5 4 3 2 6 6 3 7 4 2 6 8 5 2 6 9 8 7 6 5 4 3 2 10 7 3 6 11 8 4 6 12 9 5 6 13 12 11 10 9 8 7 6 14 11 7 6 15 14 13 12 11 10 9 8 7 6
Equal letter boxes are calculated
as in the LCS. a b
Different letter boxes are more
difficult...
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

10. O(m’n+mn’) for the Levenshtein distance (2)
mintop 4 X 5 t
minleft
X=min(mintop+t,minleft+l) l
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

11. O(m’n+mn’) for the Levenshtein distance (3)
min=1
0: 0
1: 1
2: 1
3: 0
mintop=1 X 5 4
t=1
minleft=3
X=min(mintop+t,minleft+l)
= 2
l=5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

12. O(m’n+mn’) for the Levenshtein distance (4)
min=0
0: 1
1: 1
2: 1
3: 0
mintop=0 X 5 4
t=2
minleft=3
X=min(mintop+t,minleft+l)
= 2
l=5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

13. O(m’n+mn’) for the Levenshtein distance (5)
min=0
0: 1
1: 2
2: 1
3: 0
mintop=0 X 4
t=3
minleft=3
X=min(mintop+t,minleft+l)
= 3
l=5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

14. O(m’n+mn’) for the Levenshtein distance (6)
min=0
0: 1
1: 2
2: 2
3: 0
mintop=0 X
t=4
minleft=3
X=min(mintop+t,minleft+l)
= 4
l=5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

15. O(m’n+mn’) for the Levenshtein distance (7)
min=0
0: 1
1: 2
2: 1
3: 1
mintop=0
X 4
t=4
minleft=3
l=4
X=min(mintop+t,minleft+l)
= 4
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

16. O(m’n+mn’) for the Levenshtein distance (8)
min=0
0: 1
1: 1
2: 1
3: 1
mintop=0 X
t=4
minleft=4
l=3
X=min(mintop+t,minleft+l)
= 4
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

17. O(m’n+mn’) for the Levenshtein distance (9)
min=1
0: 0
1: 1
2: 1
3: 1
mintop=1 X
t=4
minleft=4
l=2
X=min(mintop+t,minleft+l)
= 5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

18. O(m’n+mn’) for the Levenshtein distance (10)
min=2
0: 0
1: 0
2: 1
3: 1
mintop=2
4 X
t=4
l=1
minleft=4
X=min(mintop+t,minleft+l)
= 5
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

19. O(m’n+mn’) for the Levenshtein distance (11)
Different letter boxes can be
calculated as fast as equal
letter boxes.
m’ rows with n cells +
n’ columns with m cells
==> O(m’n+mn’).
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

20. Approximate searching
Can be done by ”assigning first row to zero”, but time complexity is O(m’n+mn’).
If all runs in the text are shorter than 2m, then n<2mn’, and the time complexity is O(m’mn’).
If a run is longer than 2m-1, only the first 2m columns need to be calculated; the rest equals to the last column. ==> O(m’mn’) search algorithm.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

21. ”Greedy” algorithm for the LCS
Idea: Calculate only corners.
a a a a a b b b b c c c c a a
Different letter boxes are easy; a corner value can be calculated in constant time from corners above and on the left. 3 7 11 5 2 6 9 6 2 13 10 6 15 12 8 a b
Equal letter boxes: corner values can be traced.
Time complexity is
O(min(m’n’(m’+n’),m’n+mn’)).
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

22. Diagonal algorithm for the LCS
Calculate first inside a
diagonal band 0,...(n-m).
a a a a a b b b b c c c c a a 3 7 11 5 2 6 9 6 2 13 10 6 15 12 8 6 a b
Shortest path that goes outside
this band has cost > d’=(n-m)+1.
If dmnd’, then the band is wide
enough.
If dmn>d’, then double d’ and
increase the band so that the
shortest path that goes outside
this band has cost >d’.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

23. Diagonal algorithm for the LCS (2)
At the beginning, d’=4+1=5.
a a a a a b b b b c c c c a a
As dmn=6>5, we have to
double d’. 3 7 11 5 2 4 6 9 6 2 4 13 10 6 8 15 12 8 a b
After the first doubling, d’=10 and
the diagonal band is -3,...,+7.
As dmn=8<d’, the band is wide
enough.
==> We can stop the doubling.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

24. Diagonal algorithm for the LCS (3)
Time complexity is
O(min(d2min(m’,n’),
m’n’max(m’,n’))):
a a a a a b b b b c c c c a a 3 7 11 5 2 4 6 9 6 2 4 13 10 6 8 15 12 8 a b
The number of diagonals after the last doubling is  d+1.
The sum of diagonals before the last doubling is < d+1.
Each diagonal has at most O(min(m’,n’)) corners.
The length of each tracing path can be limited by 2d.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

25. Improving the ”greedy” algorithm (1): Skipping different letter boxes
a a a a a b b b b c c c c a a
Observation. Runs of different
letter boxes can be skipped in
tracing paths.
==> Average case
O(m’n’max(m’,n’)/||2). 3 7 11 5 2 6 9 6 2 13 10 6 15 12 8 a b
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

26. Improving the ”greedy” algorithm (2): Bridges
a a a a a b b b b c c c c a a
Observation. In different letter
boxes, all values are known on
the bottom or on the right border.
==> Tracing a corner value
can be stopped at a bridge. 1 2 3 4 5 7 8 9 10 11 5 2 3 4 6 9 6 2 3 4 5 13 10 6 7 8 9 15 12 8 a b
7 8 9
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

27. Approximate Matching of Run-Length Compressed Strings, CPM'2001
O(m’n’) average case?
Conjecture: Assuming that run-lengths are equally distributed in both strings with the same mean, the expected running time of the ”greedy” algorithm using the bridge property, is O(m’n’).
Experimental results support the conjecture: For randomly generated data the average tracing path length was <2.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

28. Experimental results (1): Lengths of the runs
m’=n’=2000,
||=2
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

29. Experimental results (2): m’ n’
||=2,
runs in [1,1000]
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

30. Experimental results (3): ||
m’=n’=2000,
runs in [1,1000]
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

31. Experimental results (4): B=”k random insertions/deletions on A”
m’=n’=2000,
runs in [1,1000]
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

32. Approximate Matching of Run-Length Compressed Strings, CPM'2001
Experimental results (5): Real data: All pair of lines in three black/white images
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

33. Approximate Matching of Run-Length Compressed Strings, CPM'2001
References:
A. Apostolico, G. Landau, and S. Skiena.
Matching for run-length encoded strings.
J. of Complexity, 15:4--16, 1999.
(Also at Sequences '97, Positano Italy, June , 1997).
O. Arbell, G. Landau, and J. Mitchell.
Edit distance of run-length encoded strings.
Submitted for publication, August 2000.
H. Bunke and J. Csirik.
An algorithm for matching run-length coded strings.
Computing, 50: , 1993.
An improved algorithm for computing the edit distance of run-length coded strings.
Information Processing Letters, 54(2):93--96, 1995.
V. Levenshtein.
Binary codes capable of correcting deletions, insertions and reversals.
Soviet Physics Doklady 6: , 1966.
J. Mitchell.
A geometric shortest path problem, with application to computing
a longest common subsequence in run-length encoded strings.
In Technical Report, Dept. of Applied Mathematics, SUNY Stony
Brook, 1997.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001

34. Approximate Matching of Run-Length Compressed Strings, CPM'2001
References...
P. Sellers.
The theory and computation of evolutionary distances: Pattern recognition.
J. of Algorithms, 1(4): , 1980.
E. Ukkonen.
Algorithms for approximate string matching.
Information and Control 64(1--3): , 1985.
Finding approximate patterns in strings.
J. of Algorithms 6(1--3): , 1985.
R. Wagner and M. Fisher.
The string-to-string correction problem.
J. of the ACM 21(1): , 1974.
July 1, 2001
Approximate Matching of Run-Length Compressed Strings, CPM'2001