Lowest%20Common%20Ancestors

1
Lowest Common Ancestors
Two vertices (u, v) Lowest common ancestors, lca
(u, v)
Example
lca (5, 6) 4 lca (3, 7) 2 lca (7, 8) 1
l(v) left most appearance of v r(v) right most
appearance of v If r(u) lt l(v) then lca (u,v) is
the vertex with minimum level over the interval
r(u), l(v) How to find r(u), l(v) efficiently?
2
Range Minima Problem
Let s1,s2,s3,s4,s5,s6,s7,s8
5,10,3,4,7,1,8,2 Given i, j, how to find min
ai...aj in constant time?
v1
Approach Prefix min Suffix min
v3
min ak..al 1. Find lca w of ak and al
in a complete binary tree Let x left
child, y right child of w 2. suffix
min of k in x 3. prefix min of l in y 4. take
min of 2 and 3 Example a3..a5 w v1 suffix
min of 3rd in v2 3 prefix min of 5th in v3
7 min a3..a5 3
v2
v4
v6
v7
v5
3
Applications Computing the minimum of its
descendants
Example
For each vi, compute xi minimum aj among all
its descendants vj of vi Then xi min ak .. aj
, where k is preorder (vi) and j k
descendants of (vi) -1
4
Complexity of Range Minima

• Two part
• Initial construction
• Searching

Merging P i.. j P j1.. k
Construction
O ( log n ) time with O ( n ) PEs
P i .. k parent copy
?
Each level of three O (1) total time O (log n)
time for constructing the P, S lists
min (?, pj1)
Searching Constant time
5
Breadth First Traversal
level 0
level 1

• BFS 1, 2, 8, 9, 3, 4, 5, 6, 7

Eulerian Tour ET (v1,v2), (v2,v3), (v3,v2),
(v2,v4), (v4,v5), (v5,v4), (v4,v6), (v6,v4),
(v4,v7), (v7,v4), (v4,v2), (v2,v1), (v1,v8),
(v8,v1), (v1,v9), (v9,v1)
Remove the left most edges, and right most edges
from ET ET (v1,v2), (v2,v3), (v3,v2), (v2,v4),
(v4,v5), (v5,v4), (v4,v6), (v6,v4), (v4,v7),
(v7,v4), (v4,v2), (v2,v1), (v1,v8), (v8,v1),
(v1,v9), (v9,v1) How to detect? Prefix maxima,
and suffix maxia of edge level If prefix max has
ai - ai-1 gt 0 then i-th edge is a leftmost
edge Back Edge ( Advance edge
) find mate of each parenthesis Mostly well
defined If (vi,vj) is the right most edge at
level k, then its mate is leftmost edge at level
k1 (if it exists) (vk,vl) is a mate of (vi,vj)
then make a link vk -gt vi (vi,vj) is the first
edge, vi -gt vj
6
Generalized Prefix Computation

• Next class read section 4.9