1. Subtree Isomorphism in O(n2.5)
丁文韬
丁基伟

2. Graph Matching ProblemsV1 V2 V3 V5 V4 V6 V7 H1 G H2
Graph isomorphism
Bijection 𝜙: 𝑉 𝐺 → 𝑉 𝐻1 , 𝑢,𝑣 ∈ 𝐸 𝐺 iff. 𝜙 𝑢 ,𝜙 𝑣 ∈ 𝐸 𝐻1
Subgraph isomorphism
𝐺 is isomorphic to a 𝐻 𝑠 , which is a induced subgraph of 𝐻2
Graph monomorphism
𝐻 𝑠 is subgraph of 𝐻2
Graph homomorphism
Two vertices in 𝐺 can be mapped to one vertex in 𝐻1.(𝜙 𝑉 1 𝐺 =𝜙 𝑉 3 𝐺 =𝑉 3 𝐻1 )

3. Subtree Isomorphism
Determining if tree 𝑆 is isomorphic to any subtree of tree 𝑇.
Find a mapping function from 𝑉 𝑆 to 𝑉(𝑇)
Enumerate each vertex costs exponential time.
Find a sub structure to decompose the problem S T

4. Decompose the Problem Decompose to vertices? A 5 1 2 3 4 6 7 8 D C B ED E
𝐴,𝐶 ∈𝑆 and 1,4 ∉𝑇

5. Decompose the Problem Limb
𝑙[𝑢,𝑣] denotes the maximal subtree containing (𝑢,𝑣) where 𝑣 is an end vertex of of it and is designated as the root of 𝑙 𝑢,𝑣
In a rooted tree 𝑇 𝑣 , 𝑙[𝑢,𝑣] is a limb of 𝑇[𝑣] if 𝑣 is farther than 𝑢 from root. 5 1 2 3 4 6 7 8 4 5 6 7 8
𝑙[4,5] in 𝑇

6. Limb 𝑙 𝑇 ≔ the set of limbs in 𝑇 Recursive Limited
𝑙 𝐷,𝐶 can be matched to 𝑇 unless 𝑙 𝐶,𝐴 , 𝑙 𝐶,𝐵 and 𝐷,𝐶 can be matched to 𝑇
Limited
𝑙 𝑇 =2 𝑛 𝑡 −1
𝑙 𝑇 𝑣 =( 𝑛 𝑡 −1)
𝑙(𝑇)= 𝑣∈𝑇 𝑙(𝑇[𝑣])
(Each edge corresponding to 2 limbs) D C A B 5 1 2 3 4 6 7 8

7. Matching Between Limbs
Limb imbedding matrix R D C A B 5 1 2 3 4 6 7 8 E F 9
𝑙[5,4]
𝑙 5,6
𝑙 5,7 …
(𝑅,𝐷) 1
𝑙[𝐷,𝐶]
𝑙[𝐷,𝐸]
Lim imbedding submatrix of 𝑙 𝑅,𝐷 to 𝑙 9,5

8. Matching Between Limbs
Convert to bipartite matching problem
Bipartite matching problem can be solved within 𝑂( 𝑉 |𝐸|) time.
Hopcraft-Karp algorithm
𝑙[𝐷,𝐶]
𝑙[𝐷,𝐸]
𝑙[5,4]
𝑙[5,6]
𝑙[5,7]
𝑅,𝐷
𝑙[5,4]
𝑙 5,6
𝑙 5,7 …
(𝑅,𝐷) 1
𝑙[𝐷,𝐶]
𝑙[𝐷,𝐸]

9. Algorithm
Select 𝑟∈𝑙𝑒𝑎𝑣𝑒𝑠 𝑆 as the root vertex, decompose 𝑆[𝑟] into limbs
Set the limbs imbedding submatrix for leaves in 𝑆[𝑟]
(All leaves can be matched to any limbs in 𝑇)
For 𝑙 𝑢,𝑣 ∈𝑆 𝑟 in a topological order(e.g. ordered by non-decreasing height)
For each 𝑙 𝑥,𝑦 ∈𝑇
Set the limbs imbedding submatrix by previous result.
Build the corresponding bipartite graph
Run H-K algorithm to determine whether 𝑙[𝑥,𝑦] contains 𝑙[𝑢,𝑣]
Check whether 𝑙[𝑟,𝑣] can be matched to 𝑇

10. Algorithm
Complexity
Each limb in 𝑆[𝑣] corresponding to 𝑂 𝑙 𝑇 mbedding submatrix.
A bipartite maximum matching for each limb imbedding submatrix.
𝑇𝑖𝑚𝑒=𝑂 𝑙 𝑥,𝑦 ∈𝑇 𝑙 𝑢,𝑣 ∈𝑆 𝑇𝑖𝑚𝑒 𝐻−𝐾
Submatrix of 𝑙[𝑢,𝑣] has deg⁡(𝑢) rows and 𝑙 𝑇 columns, the corresponding bipartite graph has at most 𝑂 𝑙 𝑇 vertices and 𝑂 deg 𝑢 ⋅ 𝑙 𝑇 edges.
𝑇𝑖𝑚𝑒=𝑂 𝑙 𝑇 𝑙 𝑢,𝑣 ∈𝑆 deg 𝑢 ⋅ 𝑙 𝑇
=𝑂 𝑙 𝑇 ⋅ 𝑢∈𝑆 deg 𝑢 =𝑂( n t 𝑛 𝑠 )

11. Improve
𝑢∈𝑇 |𝑙 𝑇 𝑢 |≫|𝑙(𝑇)|
Consider a rooted set 𝑇 𝑢,⋅

12. Improve
𝑆[ 𝑎 0 ,𝑏] can be matched to 𝑇[𝑢,𝑣] only if the corresponding edge can be contained in a maximum matching.

13. Improve
Alternating path
图片来源：ws.nju.edu.cn/courses/gt/

14. Improve
An edge can be used in a maximum matching if it is in an alternating path of even length on any maximum matching V1 V3 V2 V4 V6 V5 V7
原始图片来源：ws.nju.edu.cn/courses/gt/

15. Improved Algorithm
Select 𝑟∈𝑙𝑒𝑎𝑣𝑒𝑠 𝑆 as the root vertex, decompose 𝑆[𝑟] into limbs
Set the limbs imbedding submatrix for leaves in 𝑆[𝑟]
(All leaves can be matched to any limbs in 𝑇)
For 𝑙 𝑢,𝑣 ∈𝑆 𝑟 in a topological order(e.g. ordered by non-decreasing height)
For each 𝑇 𝑢,⋅
Set the limbs imbedding submatrix by previous result.
Build the corresponding bipartite graph
Run H-K algorithm to find a maximum matching
Search the graph to determine the set of vertices that the first row can reach by alternating path
Check whether 𝑙[𝑟,𝑣] can be matched to 𝑇

16. Impreovd Algorithm Complexity
Each limb in 𝑆[𝑣] corresponding to a rooted imbedding submatrix.
A bipartite maximum matching for each limb imbedding submatrix.
𝑇𝑖𝑚𝑒=𝑂 𝑙 𝑢,𝑣 ∈𝑆 𝑇𝑖𝑚𝑒 𝐻−𝐾
Submatrix of 𝑙[𝑢,𝑣] has deg⁡(𝑢) rows and no more than 𝑙 𝑇 columns, the corresponding bipartite graph has at most 𝑂 𝑙 𝑇 vertices and 𝑂 deg 𝑢 ⋅ 𝑙 𝑇 edges.
𝑇𝑖𝑚𝑒=𝑂 𝑙 𝑢,𝑣 ∈𝑆 𝑇 𝑢,⋅ deg 𝑢 ⋅deg 𝑥 3 2
=𝑂 𝑢∈𝑆 deg 𝑢 ⋅ 𝑥∈𝑇 deg⁡(𝑥) =𝑂( n t 𝑛 𝑠 )

17. Reference
Matula D W. Subtree isomorphism in O (n 5/2)[J]. Annals of Discrete Mathematics, 1978, 2:
Conte D, Foggia P, Sansone C, et al. Thirty years of graph matching in pattern recognition[J]. International journal of pattern recognition and artificial intelligence, 2004, 18(03):

18. Thanks
Q&A

19. Tree Isomorphism For unrooted tree:
来源：

20. Isomorphism via Sub-isomorphism
Ordering via Structure Detection