Approximate Labeled Subtree Homeomorphism

1
Approximate Labeled Subtree Homeomorphism

• Ron Pinter
• Oleg Rokhlenko
• Dekel Tsur
• Michal Ziv-Ukelson

2
Subtree Homeomorphism
Pattern
Text
3
Subtree Homeomorphism
Pattern
Text
4
Subtree Homeomorphism
Pattern
Text
5
Subtree Homeomorphism
Pattern
Text
6
Subtree Homeomorphism
Pattern
Text
7
Best Homeomorphism
Pattern
Text
2 deletions
1 deletion
8
Approximate Labelled Subtree Homeomorphism (ALSH)
?i,j
T1
-1 -1
2 -2
-2 2
T2
LSH score 12
LSH score 5
9
Subtree Homeomorphism Complexity
m - of vertices in P n - of vertices in T

• Rooted trees nm1.5/logm Shamir and Tsur, 99
• Unrooted trees nm1.5/logm Shamir and Tsur, 99

10
Our Results
ALSH Rooted Unrooted
Unordered Trees
Ordered Trees
m - of vertices in P n - of vertices in T
11
Applications

• Analysis of metabolic pathways
• Semantic queries against semistructured databases
  (represented as e.g. XML documents)
• Natural language processing
• trees represent sentences
• nodes are labeled by words and sentential forms

12
Agenda

• ALSH for unordered trees
• rooted unordered
• unrooted unordered

• Improving time complexity by graph compression
  techniques
• rooted and unrooted unordered

• ALSH for ordered trees
• rooted ordered
• unrooted ordered

13
Related Work

• M.J. Chung. O(N2.5) time algorithms for the
  subgraph homeomorphism problem on trees. 1987.
• R. Shamir and D. Tsur. Faster subtree
  homeomorphism. 1999.
• G. Valiente. Constrained tree inclusion. 2003.
• P. Kilplelainer and H. Mannila. Ordered and
  unordered tree inclusion. 1995.
• M.A. Steel and T. Warnow. Kaikoura tree
  theorems Computing the maximum agreement
  subtree. 1993
• M. Farach and M. Thorup. Fast comparison of
  evolutionary trees. 1995
• M.Y. Kao, T.W. Lam, W.K. Sung, and H.F. Ting.
  Cavity matchings, label compressions, and
  unrooted evolutionary trees. 2000

14
ALSH on Rooted Unordered Trees
x1 x2 u
y1 w11 w12 w1u
y2 w21 w22 w2u
y3 w31 w32 w3u

v

15
ALSH on Rooted Unordered Trees
x1 x2 u
y1 w11 w12 w1u
y2 w21 w22 w2u
y3 w31 w32 w3u

v

16
ALSH on Rooted Unordered Trees
P
T
Weighted Assignment on G
u
v
x1
x2
y1
y2
y3
y1
x1 x2 u
y1 w11 w12 w1u
y2 w21 w22 w2u
y3 w31 w32 w3u

v

x1
y2
x2
y3
17
Time Complexity of ALSH Rooted Unordered Trees
P
T
ui
vj


x1
xki
y1
ylj

• The algorithm computes an assignment for each
  pair , where and
  .
• The assignment for ui and vj is computed using
  the bipartite graph , where
  .
• Fredman Tarjan 87 show how to compute
  AssignmentScore(G) in
  .

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Time Complexity of ALSH Rooted Unordered Trees

• Observation 1

and
Summing up all (ui,vj) node pairs
Observation 1
Observation 1
Under the similarity assumption weighted
assignment can be solved in O(V0.5Elog(VC))
Gabow and Tarjan, 89. And ALSH can be solved
in O(m1.5nlog(nC)) .
19
Unrooted Unordered Trees (naïve approach)
Pattern (P)
Text (T)
u
v
Weighted Assignment 2x4
20
Unrooted Unordered Trees (naïve approach)
Pattern (P)
Text (T)
u
v
Weighted Assignment 2x4
21
Unrooted Unordered Trees (naïve approach)
O(nm3nm2logn) !!!
Pattern (P)
Text (T)
u
v
Weighted Assignment 2x4
22
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
x3
y4
u
v
X
Y
Weighted Assignment 3x4
23
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
xq
x3
y4
u
v
X
Y
24
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
y1
x1
y2
y2
x2
y3
y3
xq
x3
y4
y4
u
v
X
Y
One augmentation !
25
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
x3
y4
u
v
X
Y
xq
26
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
x3
y4
u
v
X
Y
xq
One augmentation !
27
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
xq
x3
y4
u
v
X
Y
28
Unrooted Unordered Algorithm
Pattern (P)
Text (T)
y1
x1
y2
x2
y3
xq
x3
y4
u
v
X
Y
One augmentation !
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Decremental Property of Weighted Assignment

• Lemma
• Let
• be a bipartite graph
• for

• Computing the weighted assignments for the
  series of bipartite graphs , for
  can be done in time

30
Compressed Graph

• Motivation
• Assuming a constant-sized label alphabet.
• Using the notion of clique partition of a
  bipartite graph.
• Feder and Motwani 1991, Shamir and Tsur 1999

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Compressed Graph
y1
P
T
x1
u
v
y2
x1
x2
y1
y2
y3
x2
y3

• Each node in bipartite graph represents the whole
  subtree.
• Bounded alphabet the number of distinct
  trees is bounded.
• Lemma
• The number of distinct labeled rooted trees in a
  forest of n vertices is

32
Compressed Graph
Graph G
X
2
7
3
2
4
4
4
5
7
Y
P
T
u
v
x1
x3
y1
y2
y4
x2
y3
X
Y
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Compressed Graph
Graph G
Graph G
X
2
7
3
0
2
0
0
4
4
4
5
C
7
7
2
3
5
4
4
Y
P
T
u
v
x1
x3
y1
y2
y4
x2
y3
X
Y
34
Compressed Graph

• Lemma
• The assignment between and can be computed
  in time
• where
• d(u) is the number of neighbors of node u.
• D(u) is the number of distinct trees in the
  forest of trees rooted at neighbors of u.
• c(v) is the number of children of node v.

E V logV
35
Compressed Graph

• Observation 2
• The sum of vertex degrees in an unrooted tree P
  is

• Summing up the work over all pairs we
  get

Observation 1
Observation 2
36
Time Complexity

• Lemma
• (similar to Shamir and Tsur 1999)
• Thus the algorithm computes the optimal ALSH
  solution for two rooted unordered trees in

37
ALSH on Rooted Ordered Trees
y1
x1
y2
x2
y3
P
T
u
v
x1
x2
y1
y2
y3
38
ALSH on Rooted Ordered Trees
y1
x1
y2
x2
y3
P
T
u
v
x1
x2
y1
y2
y3
39
ALSH on Rooted Ordered Trees
y1
The main property NO CROSS-EDGES in the
bipartite graph !!!
x1
y2
x2
y3
P
T
u
v
x1
x2
y1
y2
y3
40
ALSH on Rooted Ordered Trees
y1
The main property NO CROSS-EDGES in the
bipartite graph !!!
x1
y2
x2
y3
P
T
u
v
x1
x2
y1
y2
y3
41
ALSH on Rooted Ordered Trees
y1
x1
y2
x2
y3
42
ALSH on Rooted Ordered Trees
0
0
0
y1
x1
?(x1,y1)
-8
y2
ki1
x1
-8
?(x2,y2)
x2
y3
x2
y1
y2
y3
lj1
Observation 1
Observation 1
43
ALSH on Unrooted Ordered Trees(Cyclic order)
A
B
E
D
C
44
Cyclic String Comparison
ABCDE
P
n
m
Source
Destination Column ns Maxima
45
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
46
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
47
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
48
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
49
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
50
Cyclic String Comparison
A
B
E
C
D
T
T
P
P
P
P
51
Time Complexity
A
Real numbers score metric Maez 1990
B
E
Rational numbers score metric Schmidt
1998
C
D
T
P
P
P
P
52
Time Complexity
A
Real numbers score metric Maez 1990
B
E
Rational numbers score metric Schmidt
1998
C
D
Observation 1
Observation 1
53
ALSH on Unrooted Ordered Trees(Linear order)
A
B
E
D
C
54
ALSH on Unrooted Ordered Trees(Linear order)
y1 y2 . . . . . yl

A
B
C
D
E
Dynamic Programming Matrix
FORWARD
BACKWARD
55
ALSH on Unrooted Ordered Trees(Linear order)
y1 y2 . . . . . yl

searching the best i-1,j i1,j1

Forward
A
B
C
D
E
Backward
56
ALSH on Unrooted Ordered Trees(Linear order)
y1 y2 . yj yj1 . . yl

searching the best i-1,j i1,j1

Forward
A
B
C
D
E
Backward
57
Time Complexity

• Weighted assignment for each pair (u,v) can be
  computed in

• And summing up the work for all pairs (u,v)

Observation 1
Observation 1
58
Acknowledgments

• Seffi Naor
• Amihood Amir
• Gabriel Valiente
• Ydo Wexler
• Carmel Kent