Lowest common ancestor (LCA) algorithm

1. Biological Sciences faculty
Biophysics Department
Bioinformatics Algorithms Present
Presented By
Alireza Doustmohammadi
Graduate Student in Bioinformatics
December 2018
BA Present Tarbiat Modares University 1 of 27

2. Contents
Introduction
Donald Knuth
LCA Time Complexity
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3. Introduction
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LCP Problem:
longest common prefix of two suffixes
BA Present Tarbiat Modares University
Suffix i Suffix j

4. History
Donald Knuth
[https://alchetron.com/Donald-Knuth]
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 An American computer scientist, mathematician
 Popularized the asymptotic notation
[http://tcs.unige.ch/doku.php]
BA Present Tarbiat Modares University
LCA

5. Most Popular Activities
Donald Knuth
[https://www.amazon.in/Computer-
Programming-Volumes-1-4A
-Boxed/dp/0321751043]
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LCA
1. The Art of Computer Programming
2. TEX Machine

6. Efficient LCA Algorithm
Donald Knuth
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LCA
Use Euler Walking

7. Euler Walking
Donald Knuth
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LCA
1

8. Euler Walking
Donald Knuth
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LCA
2
1

9. Euler Walking
Donald Knuth
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LCA
4
2
1

10. Euler Walking
Donald Knuth
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LCA
2
4
2
1

11. Euler Walking
Donald Knuth
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LCA
5
2
4
2
1

12. Euler Walking
Donald Knuth
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LCA
2
5
2
4
2
1

13. Euler Walking
Donald Knuth
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LCA
6
2
5
2
4
2
1

14. Euler Walking
Donald Knuth
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LCA
2
6
2
5
2
4
2
1

15. Euler Walking
Donald Knuth
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LCA
3
1
2
6
2
5
2
4
2
1

16. Euler Walking
Donald Knuth
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LCA
7
3
1
2
6
2
5
2
4
2
1

17. Euler Walking
Donald Knuth
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LCA
3
7
3
1
2
6
2
5
2
4
2
1

18. Euler Walking
Donald Knuth
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LCA
8
3
7
3
1
2
6
2
5
2
4
2
1

19. Euler Walking
Donald Knuth
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LCA
3
8
3
7
3
1
2
6
2
5
2
4
2
1

20. Euler Walking
Donald Knuth
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LCA
1
3
8
3
7
3
1
2
6
2
5
2
4
2
1

21. Efficient LCA Algorithm
Donald Knuth
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[https://www.geeksforgeeks.org/lca-n-ary-tree-constant-query-o1/]
LCA

22. Efficient LCA Algorithm
Donald Knuth
22 of 27BA Present Tarbiat Modares University
[https://www.geeksforgeeks.org/lca-n-ary-tree-constant-query-o1/]
RMQ
LCA
Min depth in this range

23. Efficient RMQ Algorithm (Sparse table)
Donald Knuth
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LCA
1 2 4 2 5 2 6 2 1 3 7 3 8 3 1
0 1 2 1 2 1 2 1 0 1 2 1 2 1 0
Size of window

24. Efficient RMQ Algorithm (Sparse table)
Donald Knuth
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LCA
1 2 4 2 5 2 6 2 1 3 7 3 8 3 1
0 1 2 1 2 1 2 1 0 1 2 1 2 1 0
0 1 1 1 1 1 1 0 0 1 1 1 1 0 -
0 1 1 1 1 1 0 0 0 0 1 1 0 - -
0 0 0 0 0 0 0 0 - - - - - - -
0 - - - - - - - - - - - - - -
s[i][j]=Min(s[i-1][j],S[i-1][j+1])
Size of window

25. Efficient RMQ Algorithm (Sparse table)
Donald Knuth
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LCA
6 2 1 3 7
Result is min between this windows

26. Efficient RMQ Algorithm (Sparse table)
Donald Knuth
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LCA
1 2 4 2 5 2 6 2 1 3 7 3 8 3 1
0 1 2 1 2 1 2 1 0 1 2 1 2 1 0
0 1 1 1 1 1 1 0 0 1 1 1 1 0 -
0 1 1 1 1 1 0 0 0 0 1 1 0 - -
0 0 0 0 0 0 0 0 - - - - - - -
0 - - - - - - - - - - - - - -
Size of window

27. Calculate Time Complexity
Donald Knuth
27 of 27BA Present Tarbiat Modares University
LCA