Graph Edit Distance

Exact Parallel

Graph Edit Distance

Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel and Patrick

Martineau

Content

• Introduction

• GED problem

• Solving GED with Branch and Bound

• State of the art on Parallel Branch and Bound

• GED computation through Parallel Branch and Bound

• Results and Demo

• Conclusion

Introduction

• Motivation

– GED is a NP-Hard problem

– There are no parallel algorithms for GED solving – Exact methods can help to assess accuracy of

heuristics

– Hypothesis : Parallel = fast convergence to optimal

solution

GED problem

Transforming G1 into G2 by means of edit operations : Insertion, Deletion

and substitution.

Solving GED with Branch and Bound

• Solution space organised as an ordered tree search

• Decision variables

• 𝑦 _𝑖𝑘 = 1 𝑠𝑖 𝑖 𝑎𝑛𝑑 𝑘 𝑎𝑟𝑒 𝑚𝑎𝑡𝑐ℎ𝑒𝑑.

• ∀𝑖 ∈ 𝑉 ₁ ∪ 𝜀, 𝑘 ∈ 𝑉 ₂ ∪ 𝜀

j i

l k

Matching cost = c(i,j,k,l)

G1 G2

Graph matching

Solving GED with Branch and Bound

 B&B algorithm: [Abu-Aisheh 2015]

 p : partial solution

 g(p) : total cost of p

 h(p) : estimated cost of p

 f=g+h (a lower bound)

 Branching Scheme o Depth-first search

 Selection strategy

o Choose minimum cost p

 Initial upper bound o d_ub

A B

C D

A-A’ A-B’ A-C’ A-ϵ

root

B-A’ B-C’ B-ϵ

C-C’ C-ϵ

D--ϵ

g=1 h=4 f=5 g=2

h=5 f=7

g=3 h=2 f=5 g=1

h=5 f=6 g=1

h=5 f=6 g=1.5

h=4 f=5.5

g=3 h=3 f=6 g=1.5

h=4 f=5.5

g=2 h=3.8 f=5.8 g=5.7

h=0*

f=5.7*

Solving GED with Branch and Bound

 B&B algorithm: [Abu-Aisheh 2015]

 p : partial solution

 g(p) : total cost of p

 h(p) : edtimated cost of p

 f=g+h

 Branching Scheme o Depth-first search

 Selection strategy

o Choose minimum cost p

 Reduction Strategy o Prune

A B

C D

A-A’ A-B’ A-C’ A-ϵ

root

B-A’ B-C’ B-ϵ

C-C’ C-ϵ

D--ϵ

g=1 h=4 f=5 g=2

h=5 f=7

g=3 h=2 f=5 g=1

h=5 f=6 g=1

h=5 f=6 g=1.5

h=4 f=5.5

g=3 h=3 f=6 g=1.5

h=4 f=5.5

g=2 h=3.8 f=5.8 g=5.7

h=0*

f=5.7*

 Then

 expand p

 Else

 choose another p

Solving GED with Branch and Bound

 B&B algorithm: [Abu-Aisheh 2015]

 p : partial solution

 g(p) : total cost of p

 h(p) : estimated cost of p

 f=g+h

 Branching Scheme o Depth-first search

 Selection strategy

o Choose minimum cost p

 Bounding Strategy o Prune

A B

C D

A-A’ A-B’ A-C’ A-ϵ

root

B-A’ B-C’ B-ϵ

C-C’ C-ϵ

D--ϵ

g=1 h=4 f=5 g=2

h=5 f=7

g=3 h=2 f=5 g=1

h=5 f=6 g=1

h=5 f=6 g=1.5

h=4 f=5.5

g=3 h=3 f=6 g=1.5

h=4 f=5.5

g=2 h=3.8 f=5.8 g=5.7

h=0*

f=5.7*

 Then

 Update UB

 Else

 Do nothing

State of the art on Parallel Branch and Bound

• What can be parallelized ?

– g(p) computation [Chakroun 2013]

– h(p) computation [Chakroun 2013]

– Search tree exploration [Rao 1987] [Neary 2005]

• Exploration of partial solutions is more time consuming than g(p) and h(p)

• Task = Exploration of a tree node (p).

• One thread can have many tasks

Parallel Graph Edit Distance (PDFS)

G

G

Generate N tree nodes

Thread

1

(BnB)

Thread

2

(BnB)

Thread

M

(BnB) Branch and

Bound Search Tree Decomposition

10

A-A’ A-B’ A-C’ A-ϵ

root

B-A’ B-C’ B-ϵ B-A’ B-B’ B-ϵ

g=1 h=4 f=5 g=2

h=5 f=7

g=3 h=2 f=5 g=1

h=5 f=6

g=4 h=2 f=6

g=4 h=3 f=7

g=4 h=3 f=7 g=1

h=5 f=6 g=1.5

h=4 f=5.5

g=3 h=3

f=6 B-B’

g=4 h=3 f=7

Tree node to be given

to a thread

The Parallel Branch and Bound issue

– Expanding a partial solution can be more or less time consuming :

• Is p close to a leaf node (a complete solution) ? Close to the end.

• Is p far from the upper bound ? f(p) ≈ UB  Easy to cut.

– Search tree is irregular

Thread1 Thread2 Thread³

Thread 2 has nothing left to do

How to keep all threads busy ?

The Parallel Branch and Bound issue

• How to keep all threads busy ?

• Answer : Ensure that at anytime t all threads endorse the same amount of work.

– How to estimate to work load of a partial solution ?

• w(p) = g(p)+h(p)

• w(p) = distance to a leaf node

– How to assign partial solutions to threads ?

• Answer : Minimizing the gap between the maximum load of a thread and the average workload of the threads (Wmax –

Wmean)

• This problem is NP-Hard [ref] when the number of threads is >=

2.

• Good heuristics exist : The greedy algorithm called "Graham's

Rule"

Parallel Graph Edit Distance (PDFS)

G

G

Generate N tree nodes

Thread

1

(BnB)

Thread

2

(BnB)

Thread

M

(BnB) Branch and

Bound Search Tree

Decomposition

Tree nodes assignement :

Static load balancing

Dynamic load balancing

Demo

PDF convergence

Demo

PDF vs BP

Results

PDF vs DF

Conclusion

• A first attempt toward parallel graph edit distance.

• Exploring different parts of the search tree

• It can help to assess accuracy of heuristics

• From parallel to distributed GED

Bibliography

• Imen Chakroun and Nordine Melab. Operator-level gpu- accelerated branch and bound algorithms. In ICCS, volume 18, 2013.

• V N Rao and V Kumar. Parallel depth-rst search on

multiprocessors part i : Implementation. International journal on Parallel Programming, 16(6), 1987.

• Michael O. Neary and Peter R. Cappello. Advanced eager scheduling for java-based adaptive parallel computing.

Concurrency - Practice and Experience, 17 :797819, 2005.

• Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel,

Patrick Martineau: An Exact Graph Edit Distance Algorithm for Solving Pattern Recognition Problems. ICPRAM (1) 2015:

271-278

Static load balancing formulation