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Advanced Algorithms COMS31900 Lowest Common Ancestor (with a bit on on Range Minimum Queries) Rapha el Clifford Slides by Benjamin Sach Advanced Algorithms COMS31900 Lowest Common Ancestor (with a bit on on Range Minimum Queries)

  1. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 (depth) D 0 1 2 3 2 3 2 1 2 1

  2. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0

  3. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1

  4. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0

  5. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1

  6. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2

  7. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1

  8. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2

  9. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1

  10. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0

  11. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1

  12. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  13. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  14. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  15. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  16. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . i j and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  17. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . i j and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  18. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . i j and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ how do we find LCA(i,j)?

  19. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . i j and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ Compute RMQ ( i ′ , j ′ ) in D how do we find LCA(i,j)?

  20. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . i j and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ Compute RMQ ( i ′ , j ′ ) in D how do we find LCA(i,j)?

  21. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  22. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? j i We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  23. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? j i We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  24. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? j i We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ how do we find LCA(i,j)?

  25. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? j i We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ Compute RMQ ( i ′ , j ′ ) in D how do we find LCA(i,j)?

  26. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? j i We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 1 0 1 0 1 2 1 2 1 0 1 0 i ′ j ′ Compute RMQ ( i ′ , j ′ ) in D how do we find LCA(i,j)?

  27. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  28. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit j i . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  29. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit j i . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N . . . (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  30. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit j i . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N . . . which copy of i ? (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  31. Solving LCAs using RMQs 0 0 Compute an Euler tour of T . . . (a depth first search with repeats) 4 1 3 1 2 Write down every node you visit j i . . . and its depth 6 7 8 5 2 How long is the tour? We follow each edge twice. . . and there are ( n − 1) edges 9 10 3 Find i and j in N . . . which copy of i ? any copy is fine (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 how do we find LCA(i,j)?

  32. Solving LCAs using RMQs 0 0 4 1 3 1 2 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  33. Solving LCAs using RMQs 0 0 Preprocessing Summary 1. Construct N and D from T 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 3. Preprocess D for RMQs 6 7 8 5 2 9 10 3 (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  34. Solving LCAs using RMQs 0 0 Preprocessing Summary 1. Construct N and D from T 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i 2. Find (any) j ′ st. N [ j ′ ] = j 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  35. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i 2. Find (any) j ′ st. N [ j ′ ] = j 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  36. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i 2. Find (any) j ′ st. N [ j ′ ] = j 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  37. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i 2. Find (any) j ′ st. N [ j ′ ] = j 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  38. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  39. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  40. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 O (?) 3. Compute RMQ ( i ′ , j ′ ) in D 3 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  41. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 O (?) 3. Compute RMQ ( i ′ , j ′ ) in D 3 O (1) 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0

  42. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 O (?) 3. Compute RMQ ( i ′ , j ′ ) in D 3 O (1) 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 Prep. time O ( n + prepRMQ ( n )) Space O ( n + spaceRMQ ( n )) Query time O (1 + queryRMQ ( n )) depends on the RMQ structure used

  43. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 O (?) 3. Compute RMQ ( i ′ , j ′ ) in D 3 O (1) 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 Prep. time O ( n log log n ) Space O ( n log log n ) Query time O (1) using the best result from last lecture

  44. Solving LCAs using RMQs 0 0 Preprocessing Summary O ( n ) 1. Construct N and D from T O ( n ) 2. Add a pointer from each node i to some N [ i ′ ] = i 4 1 3 1 2 O (?) 3. Preprocess D for RMQs 6 7 8 5 2 Query Summary - LCA(i,j) 1. Find (any) i ′ st. N [ i ′ ] = i O (1) 2. Find (any) j ′ st. N [ j ′ ] = j O (1) 9 10 O (?) 3. Compute RMQ ( i ′ , j ′ ) in D 3 O (1) 4. LCA ( i, j ) = N [ RMQ ( i ′ , j ′ )] why does this work? (node) N 0 1 5 9 5 10 5 1 6 1 0 2 0 3 7 3 8 3 0 4 0 2 n − 1 (depth) D 0 1 2 3 2 3 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 Prep. time O ( n log log n ) Space O ( n log log n ) Query time O (1) using the best result from last lecture

  45. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d depth = d + 1 depth > d + 1 S 1 S 2 S 3 S k (node) N x x x x x x (depth) D d d d d d d

  46. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d children depth = d + 1 depth > d + 1 S 1 S 2 S 3 S k (node) N x x x x x x (depth) D d d d d d d

  47. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d children depth = d + 1 depth > d + 1 S 1 S 2 S 3 S k subtrees (node) N x x x x x x (depth) D d d d d d d

  48. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d children depth = d + 1 depth > d + 1 S 1 S 2 S 3 S k subtrees (node) N x x x x x x (depth) D d d d d d d tour of S 1 tour of S 2 tour of S 3 tour of S k

  49. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d depth = d + 1 depth > d + 1 S 1 S 2 S 3 S k (node) N x x x x x x (depth) D d d d d d d tour of S 1 tour of S 2 tour of S 3 tour of S k

  50. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d depth = d + 1 y depth > d + 1 S 1 S 3 S k y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  51. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . x depth = d depth = d + 1 y depth > d + 1 S 1 S 3 S k y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  52. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  53. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k i j y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  54. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k i j imagine LCA ( i, j ) is not y y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  55. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k i j imagine LCA ( i, j ) is not y i ′ and j ′ are in here so RMQ does not return the location of a y y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  56. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k i j imagine LCA ( i, j ) is not y i ′ and j ′ are in here so RMQ does not return the location of a y (all of the y s are out of range) y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  57. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

  58. Solving LCA using RMQ - correctness We can also define a Euler tour of T recursively. . . Claim the RMQ reports the location of x depth = d some y in N iff LCA ( i, j ) = y depth = d + 1 y depth > d + 1 S 1 S 3 S k j i y y y y (node) N x x x x x x (depth) D d d d d d d d + 1 d + 1 d + 1 d + 1 tour of S 1 tour of S 2 tour of S 3 tour of S k

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