Levenshtein

 (Edit) 

DISTANCE

Algorithm



ABDULLAH ALSHAMMRI

overview

  • What is it !?
  • Applications and Usages.
  • How it works?
  • Example and demo.
  • Complexity.
  • Implementation.
  • References.

EDIT DISTANCE

  • Measures the similarity between two strings.
  • Test is similar to Text 
    • By replacing ONE character s with x , it becomes the same

  • Levenshtein is the name of a Russian scientist Vladimir Levenshtein who invented the algorithm in 1965.

  • It is a Dynamic Programming Algorithm.   

Applications and usages

  • Spell Checkers
    • TRST => Do you mean Test?
  • DNA Analysis
  • Used among other algorithms
    • It used in LCS, Longest Common Subsequence.
    • I was going to discuss LCS, however, I found LCS depends on Edit Distance.
    • LCS useful with File Comparison.
    • Refer to the reference in the last slide for more information about LCS.
      • Linux diff app depends on LCS witches depends on Edit Distance.

How IT WORKS?

  • S1 is the first string and m is its length.
  • S2 is the second string and n is its length
  • if n==0 return m and exit
  • if m==0 return n and exit
  • create new 2d array called distance
  • fill first row (0,1,2...to n) and fill first column (0,1,2,.. to m)
  • and for each distance[i][j] do : 
  • The distance ( minimal operation needed ) is distance[n][m]

Example

 

j=0

j=1

j=2

j=3

j=4

j=5

 

 

#

E

X

I

T

i=0

#

0

1

2

3

4

i=1

E

1

 

 

 

 

i=2

X

2

 

 

 

 

i=3

I

3

 

 

 

 

i=4

S

4

 

 

 

 

i=5

T

5

 

 

 

 

i=6

S

6

 

 

 

 

Example

 

j=0

j=1

j=2

j=3

j=4

j=5

 

 

#

E

X

I

T

i=0

#

0

1

2

3

4

i=1

E

1

 0

1

2 

3 

i=2

X

2

 

 

 

 

i=3

I

3

 

 

 

 

i=4

S

4

 

 

 

 

i=5

T

5

 

 

 

 

i=6

S

6

 

 

 

 

Example

 

j=0

j=1

j=2

j=3

j=4

j=5

 

 

#

E

X

I

T

i=0

#

0

1

2

3

4

i=1

E

1

0 

1 

2 

3 

i=2

X

2

1 

0 

1 

2 

i=3

I

3

 

 

 

 

i=4

S

4

 

 

 

 

i=5

T

5

 

 

 

 

i=6

S

6

 

 

 

 

Example

 

j=0

j=1

j=2

j=3

j=4

j=5

 

 

#

E

X

I

T

i=0

#

0

1

2

3

4

i=1

E

1

0 

1 

2 

3 

i=2

X

2

1 

0 

1 

2 

i=3

I

3

 2

1 

0 

1 

i=4

S

4

3 

2 

1 

1 

i=5

T

5

4 

3 

2 

1 

i=6

S

6

 5

4 

3 

2 

Example


So The minimum operations needed to convert EXISTS to EXIT is 2



2 here is  two  deletion operations 
EXISTS => EXIT

COMPLEXITY

  • It's a Dynamic Programming paradigm , each iteration depends on the last operation to solve the current one.

  • Time Complexity is Big O(MN) , or O(MN+M+N)
  • Space Complexity is a 2D array O(MN)
    • N and M is the length of tow strings  S1 and S2

  • Brute Force version is very complex
    • HELLO and XXXXX needs more than  2523 STEPS to finish
    • However , the Dynamic Programming version need +35 STEPS.

IMPLEMENTATION


REFERENCES

  •  Levenshtein, Vladimir I. (February 1966). "Binary codes capable of correcting deletions, insertions, and reversals". Soviet Physics Doklady.

  •  Hirschberg, D. S. (1975). "A linear space algorithm for computing maximal common subsequences". Communications of the ACM .
  • Levenshtein distance , http://en.wikipedia.org/wiki/Levenshtein_distance







THANKS
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