This document discusses algorithms for finding the lowest common ancestor (LCA) in trees. It presents the Euler tour technique, which works by performing an Euler walk of the tree and storing ancestors of nodes. It also discusses using a sparse table to solve the range minimum query (RMQ) problem in O(1) time to help determine the LCA. The document is a presentation about bioinformatics algorithms given by a graduate student in December 2018.

1. Biological Sciences faculty
Biophysics Department
Bioinformatics Algorithms Present
Presented By
Alireza Doustmohammadi
Graduate Student in Bioinformatics
December 2018
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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
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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]
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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
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8. Euler Walking
Donald Knuth
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9. Euler Walking
Donald Knuth
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2
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10. Euler Walking
Donald Knuth
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11. Euler Walking
Donald Knuth
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2
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12. Euler Walking
Donald Knuth
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13. Euler Walking
Donald Knuth
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14. Euler Walking
Donald Knuth
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15. Euler Walking
Donald Knuth
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16. Euler Walking
Donald Knuth
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17. Euler Walking
Donald Knuth
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18. Euler Walking
Donald Knuth
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19. Euler Walking
Donald Knuth
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20. Euler Walking
Donald Knuth
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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
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[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
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LCA