By Makinen, Navarro and Ukkonen

Abstract

Let A and B be two run-length encoded strings of encoded lengths m’ and n’, respectively. we will show an O(m’n+n’m) time algorithmthat compute their edit distance.

Let A be a short pattern, a B be a long text and a threshold parameter K. We will show an algorithm that will report all the approximate occurrences of A in B Which are at an edit distance K or less from the pattern.

Example of simple edit distance:

A= aaabbcB= abbdb

aaabbc

0123456

a1

b2

b3

d4

b5

Example of simple edit distance:

A= aaabbcB= abbdb

aaabbc

0123456

a1012345

b2

b3

d4

b5

Example of simple edit distance:

A= aaabbcB= abbdb

aaabbc

0123456

a1012345

b2112233

b3

d4

b5

Example of simple edit distance:

A= aaabbcB= abbdb

aaabbc

0123456

a1012345

b2112233

b3222223

d4333334

b5444334

Example of simple edit distance:

A= aaabbcB= abbdb

aaabbc

0123456

a1012345

b2112233

b3222223

d4333334

b5444334

We distinct between three kinds of edit distance:

Levenshtein distance - DL (A,B) : Insertion =1 ,Deletion =1,Substitution=1.distance - DID (A,B) : Insertion =1 ,Deletion =1,Substitution=∞ (no Substitution).Global distance - DG (A,B) : Arbitrary coast for Insertion ,Deletion ,Substitution.

KeyWords

• Run-Length compressed

aaaabbcccaab = (a,4),(b,2),(c,3),(a,2),(b,1).

• White Box

• Black Box

aaaa

a

a

a

aaaa

B

B

B

Dividing the Edit Distance matrix into boxes

aaaabbbbbbcccccbb

a

a

a

b

b

b

b

b

a

a

a

a

a

a

b

b

Dividing the Edit Distance matrix into boxes

aaaabbbbbbcccccbb

a

a

a

b

b

b

b

b

a

a

a

a

a

a

b

b

An O(mn’ + m’n) Algorithm for DL

• Equal Letter Box (White):– “Copying” the values from the upper left borders to

the bottom right borders, using as much diagonal moves as possible.

78889

7

8

98 7 7 8

8

8

•In an Equal letter box, DID = DL because no substitutions are needed.

An O(mn’ + m’n) Algorithm for DL

• Different Letter Box (Black):– Filling only the borders: – 1 + min (t-1 + min relevant upper border ,

s-1 + min relevant left border ).

(3,5)

t > s

(5,3)

t < s

(5,5)

t = s

78889

7

8

99 9 10

11

10

9

1+min( 3-1 + min(7,8) , 1-1 + min(8,9)) = 91+min( 3-1 + min(7,8,8) , 2-1 + min(7,8,9)) = 91+min( 3-1 + min(7,8,8,8) , 3-1 + min(7,7,8,9)) = 101+min( 3-1 + min(8,8,8,9) , 4-1 + min(7,7,8,9)) = 111+min( 2-1 + min(8,8,9) , 4-1 + min(7,7,8)) = 101+min( 1-1 + min(8,9) , 4-1 + min(7,7)) = 9

Three different points along our computation:

t > s

t = s

t < st

s

Extending the Algorithm to Global Edit Distance.

Inside a given box there are only three different costs involved :

CI - insertion cost.CD - deletion cost.CS substitution cost.

Since the triangle inequity holds : CS < CI + CD we will not differentiate between white box and black box.

We assume without loss of generality that the cost CI and CD are the same in all boxes

Filling the borders:For each cell (s,t) in the border :

cell(s,t) = min(upper triangle values, leftmost triangle values)

triangle values = min( relevant border cells + each cell’s path to

(s,t) )path = CS * number of diagonal moves +

CI * number of insertions +CD * number of deletions.

example

An O(mn’ + m’n) Algorithm for DG

Example 78889

7

8

9

CI = 7CD = 4CS = 10

2016 2120

13

17

min ( min(9+1*7 , 8+1*10) , min(7+1*10 + 2*4 , 8+3*4)) =16min (min(9+2*7 , 8+1*10+1*7 , 7+2*10) , min(7+2*10 + 1*4 , 8+2*4+1*10 , 8+3*4)) =20min (min(9+3*7 , 8+1*10+2*7 , 7+2*10+1*7 , 7+3*10) , min(7+3*10 , 8+2*10+1*4 , 8+1*10+2*4 , 8+3*4)) =20min (min(9+4*7 , 8+1*10+3*7 , 7+2*10+2*7 , 7+3*10+1*7) , min(8+3*10 , 8+2*10+1*4 , 8+1*10+2*4 , 9+3*4)) =21min(min(8+3*7 , 7+1*10+2*7 , 7+2*10+1*7) , min(8+2*10 , 8+1*10+1*4,9+2*4)) =17min (min(7+3*7 , 7+1*10+2*7) , min( 8+1*10 ,9+1*4)) =13

Complexity:

Stays O(m’n + n’m) same as the Levenshtein algorithm because we only add constant time calculations (multiplications)

Example:

aaaabbbbbbcccccbb

a

a

a

b

b

b

b

b

a

a

a

a

a

a

b

b

Example:

aaaabbbbbbcccccbb

012345678910

a1

a2

a3

b4

b5

b6

b7

b8

a

a

a

a

a

a

b

b

Example:

aaaabbbbbbcccccbb

012345678910

a13

a22

a32101

b4

b5

b6

b7

b8

a

a

a

a

a

a

b

b

Example:

aaaabbbbbbcccccbb

012345678910

a139

a228

a32101234567

b4

b5

b6

b7

b8

a

a

a

a

a

a

b

b

Example:

aaaabbbbbbcccccbb

012345678910

a139

a228

a32101234567

b41

b52

b63

b74

b87655

a

a

a

a

a

a

b

b

Example:

aaaabbbbbbcccccbb

012345678910

a139

a228

a32101234567

b416

b525

b634

b743

b87655432112

a

a

a

a

a

a

b

b

Notice There are two cases computing the borders:

1 .cell(s,t) gets its minimum value from the left border.2 .cell(s,t) gets its minimum value from the top border.

In case 1, the minimum path cost to cell(s,t) can be written as CellValue = BorderCellValue + PathCost(BorderCell, Cell) = BorderCellValue + Diagonal * CS + (Position – Diagonal) * CI = BorderCellValue + Diagonal * (CS – CI) + Position * CI.

BORDERCELLVALUE + DIAGONAL * (CS – CI) is not dependant on Position! Hence it can be calculated in advance, and kept in an array for all border cells, allowing each CellValue calculation to spend only constant time.

Same applies for case 2, by changing CI to CD.

Approximate SearchingGiven a string A (short pattern ) ,a string B ( long text ) and a threshold parameter K ,we are interested reporting all the “approximate occurrences “ of A in B using K errors or less.

)the position of substrings in B that are at distance K or less from the pattern A (.

A = aabca

B = aaeeeeeebbbbbbbaaaaaabbbbbbcccaaaccccabcdcdcdeeeaaab

K = 3

Classical algorithm to find the “matches”

Computes a matrix exactly like previous algorithm with the only difference that the first row of the matrix I initialized with zeros.

The last row of the matrix is examined ,and every text position which is smaller then K is reported as a match .

aabbbbaaacbbbbcca

a

a

b

Classical algorithm to find the “matches”

Computes a matrix exactly like previous algorithm with the only difference that the first row of the matrix I initialized with zeros.

The last row of the matrix is examined ,and every text position which is smaller then K is reported as a match .

aabbbbaaacbbbbcca

00000000000000000

a

a

b

Classical algorithm to find the “matches”

Computes a matrix exactly like previous algorithm with the only difference that the first row of the matrix I initialized with zeros.

The last row of the matrix is examined ,and every text position which is smaller then K is reported as a match .

aabbbbaaacbbbbcca

00000000000000000

a00111100011111110

a10122210012222221

b21012221111222332

More efficient algorithm – pattern and text are run-length compressed:Fill the matrix only at beginning of text runs.Complete the first m columns only.

aabbbbaaacbbbbcca

00000000000000000

a00111100011111110

a10122210012222221

b21012221111222332

Complexity = O(m2n’+R) For each run – m*m, there are n’ runs, R is the size of the output.

Improving the trivial algorithm

Problem: We would have wanted to apply DL, but if the text is very large, m’n may be a

lot bigger than m2n.’

Solution: Combine the two. Evaluating only the borders of the runs, and only the first m cells of each run, yields an O(m’m + m +m) per run of the text, multiplying by n’ to the final O(m’mn’ + R) complexity. דוגמה