See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/337890015 — Presentation Slides — Scott : Structure Canonisation using Ordered-Tree Translation Presentation · December 2019 CITATIONS READS 0 29 3 authors: Nicolas Bloyet Pierre-Francois Marteau Université Bretagne Sud Université Bretagne Sud, Vannes, France 4 PUBLICATIONS 0 CITATIONS 129 PUBLICATIONS 887 CITATIONS SEE PROFILE SEE PROFILE Emmanuel Frénod Université Bretagne Sud 83 PUBLICATIONS 630 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Gesture Analysis, SYnthesis and Recognition View project Phd project: Modeling neutrino radiative transfer in Type II core-collapse supernovae View project All content following this page was uploaded by Nicolas Bloyet on 11 December 2019. The user has requested enhancement of the downloaded file.
A Method for Representing Graphs as Rooted Trees for Graph Canonization Scott : Structure Canonisation using Ordered-Tree Translation Nicolas Bloyet 1, 2, 3 Pierre-François Marteau 1 Emmanuel Frénod 2, 3 1 IRISA - Université Bretagne Sud - Vannes, France 2 LMBA - Université Bretagne Sud - Vannes, France 3 See-d - Parc d’Innovation Bretagne Sud - Vannes, France Complex Networks 2019, Lisbon, December 2019
Outline .. 0/15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. State of the Art . . . . . . 1/15 Outline Graph Isomorphism Graph Canonization Scott . Implementation Outline We present Scott , a method to handle Graph Canonization (hence Graph Isomorphism) on fully labeled graphs (edges and vertices). The method produces : • a string representation of the input graph (trace), unique for an isomorphism class, which can be used to identify any (sub)structure (hash) • a canonical adjacency matrix derived from that trace, which can be used to standardize (up to an isomorphism) any graph computing We provide an open-source Python implementation. Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod
Graph Isomorphism .. 1/15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. g 2/15 Outline . Graph Canonization State of the Art Scott Implementation Graph Isomorphism problem between their respective vertices sets preserving edges. f a e b c . h d i 1 3 4 2 5 6 7 8 H G Scott : Structure Canonisation using Ordered-Tree Translation . Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod . . . . . . . Let G ∈ G and H ∈ G be two graphs. They are said to be isomorphic if there exists a bijection f : V G → V H f ( a ) = 1 f ( b ) = 6 f ( c ) = 7 f ( d ) = 4 f ( e ) = 5 f ( g ) = 3 f ( h ) = 2 f ( i ) = 8
. Graph Canonization . . . . . . . 3/15 Outline Graph Isomorphism State of the Art . Scott Implementation Isomorphism class Isomorphism is an equivalence relation, and so partitions the ensemble of graphs into equivalence classes, where all graphs belonging to a class represent the same structure. C 1 C 2 In many graph-related tasks and applications, we want to consider graphs belonging to the same isomorphism class as equals. Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod ... G / ≃
. Outline . . . . . . . . 4/15 Graph Isomorphism . Graph Canonization State of the Art Scott Implementation Problem Complexity While determining if two graphs are isomorphic seems trivial for small graphs, it is actually a problem which remains unresolved in polynomial time (very costly). In some applications where we have to make a lot of isomorphism testing, we would prefer a re-usable method. Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod
Graph Canonization .. 4/15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Outline . . . . . . . . 5/15 Graph Isomorphism . Graph Canonization State of the Art Scott Implementation Graph Canonization Problem The graph canonization is a related problem, consisting in fjnding for a graph a canonical representant, usually a graph, unique for its isomorphism class. Two graphs are isomorphic if and only if they have the same canonical representant. 1 and actually more diffjcult in many cases Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod G ≃ H ⇐ ⇒ Canon ( G ) = Canon ( H ) This problem is at least as diffjcult 1 as graph isomorphism, as it answers to it explicitly, but provides a re-usable result.
. Graph Isomorphism . . . . . . . . 6/15 Outline Graph Canonization . State of the Art Scott Implementation Graph Canonization Problem For example, Smiles notation [Weininger, 1988] maps any molecular graph to a canonical string encoding. CN1C=NC2=C1C(=O)N(C(=O)N2C)C SMILES Once the canonical representant is computed, subsequent tasks like comparisons are trivial. Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod
State of the Art .. 6/15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Scott . . . . . . 7/15 Outline Graph Isomorphism Graph Canonization State of the Art Implementation . State of the Art Isomorphism testing • conauto [López-Presa et al., 2011] • saucy [Darga et al., 2008] Canonization • nauty [McKay et al., 1981, McKay and Piperno, 2014] • bliss [Junttila and Kaski, 2007] • traces [Piperno, 2008, McKay and Piperno, 2014] an edge-unlabeled way, due to their utilization of equitable vertice-coloration, undefjned for edges-labeled graphs. Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod None of them is able to natively deal with labeled edges otherwise than rewriting the graph in
Scott .. 7/15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Graph Canonization . . . . . . . 8/15 Outline Graph Isomorphism State of the Art . Scott Implementation Key Idea We propose here an algorithm based on graph rewriting instead of equitable coloration, to handle natively edges-labeled graphs. Scott execution follows three main steps, illustrated below. • Levelling of vertices, according to an elected root • Re-writing of cycles without information loss • Exploitation of the resulting tree (as trace or as matrix) Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod
. B . . . 9/15 Outline Graph Isomorphism Graph Canonization State of the Art Scott Implementation Tree encoding It is known [Neveu, 1986] that we can formally encode a tree as a string of symbols. C . D E level N order relation t r a b c d e Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod . . . . . . . . . . . . . We propose a canonic notation σ T : T → Σ ∗ , encoding any tree into a string. We can thus compress any tree t into a unique vertice of label σ T ( t ) . ≤ A Neveu ( t r ) = { r , A : a , B : b , BD : d , BE : e , C : c } σ T ( t r ) = ( A : a , ( D : d , E : e ) B : b , C : c ) r r
. Implementation . . . . . 10/15 Outline Graph Isomorphism Graph Canonization State of the Art Scott Vertices levelling . The fjrst step consists in ordering each vertex in levels, according to their minimum distance In the best case, the root identity is obvious (label, degree, etc.), otherwise it is determined a posteriori . r r 1 1 2 2 3 Scott : Structure Canonisation using Ordered-Tree Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Bloyet , P-F. Marteau , E. Frénod with a vertex identifjed as root.
. b p s a A a A RHS c LHS c c a RHS s c b a c p c . A B LHS s A We want to remove any form of cycle in the levelled graph, without any loss of information. c Scott : Structure Canonisation using Ordered-Tree Translation p i c C c RHS i c C b B a B A LHS i C c b a A B Cycle rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation Scott State of the Art Graph Canonization Graph Isomorphism Outline 11/15 N. Bloyet , P-F. Marteau , E. Frénod . . . . . . . Tree → Vertex compression : if N fjrst levels are computed, then any cycle encountered at level N + 1 belongs to one of the three following cases. In − bound Self − bound Co − bound σ A σ B σ A σ B ∗ 1 ∗ 1 # 1 # 1 & σ C σ C σ C
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