Ring Based Approximation of Graph Edit Distance Presented at S+SSPR - PowerPoint PPT Presentation

Background Rings as Local Structures Experiments References Ring Based Approximation of Graph Edit Distance Presented at S+SSPR 2018, Beijing, China, August 17–19, 2018 D. B. Blumenthal 1 , S. Bougleux 2 , J. Gamper 1 , L. Brun 2 1 Faculty of Computer Science, Free University of Bozen-Bolzano, Bolzano, Italy 2 Normandie Université, UNICAEN, ENSICAEN, CNRS, GREYC, Caen, France August 18, 2018 Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 1/19

Background Rings as Local Structures Experiments References Background Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 2/19

Background Rings as Local Structures Experiments References Graph Edit Distance (Definition) ◮ idea: distance between labeled graphs G and H = minimal amount of distortion needed for transforming G into H ◮ edit operations and edit costs: ◮ substituting a node u ∈ V G by a node v ∈ V H � c V ( u , v ) ◮ deleting an isolated node u ∈ V G � c V ( u , ε ) ◮ inserting an isolated node v ∈ V H � c V ( ε, v ) ◮ substituting an edge e ∈ E G by an edge f ∈ E H � c E ( e , f ) ◮ deleting an edge e ∈ E G � c E ( e , ε ) ◮ inserting an edge f ∈ E H � c E ( ε, f ) ◮ sequence P = ( o i ) r i = 1 of edit operations is edit path between G and H iff ( o r ◦ . . . ◦ o 1 )( G ) = H � c ( P ) = � r i = 1 c ( o i ) ◮ GED ( G , H ) := min { c ( P ) | P is edit path between G and H } ◮ computing GED is NP -hard � approximative techniques needed Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 3/19

Background Rings as Local Structures Experiments References Graph Edit Distance (Example) ◮ real-valued, positive node and edge labels: V : V G → R ≥ 0 , ℓ H V : V H → R ≥ 0 ◮ ℓ G E : E G → R ≥ 0 , ℓ H E : E H → R ≥ 0 ◮ ℓ G ◮ edit costs: ◮ c V ( u , v ) = |ℓ G V ( u ) − ℓ H V ( v ) | , c V ( u , ε ) = ℓ G V ( u ), c V ( ε, v ) = ℓ H V ( v ) ◮ c E ( e , f ) = |ℓ G E ( e ) − ℓ H E ( f ) | , c E ( e , ε ) = ℓ G E ( e ), c E ( ε, f ) = ℓ H E ( f ) ◮ c ( P ) = 3 . 0 + 5 . 0 + 2 . 0 = 10 o 1 o 2 o 3 G 1 . 0 1 . 0 1 . 0 3 . 0 H 1 1 1 1 edit operation edit operation edit operation 2 . 0 2 . 0 2 . 0 2 . 0 del E (( 2 , 3 )) del V ( 3 ) sub V ( 1 , 1 ) 2 2 . 0 2 2 . 0 2 2 . 0 2 2 . 0 edit cost edit cost edit cost 3 . 0 5 . 0 | 1 . 0 − 3 . 0 | = 2 . 0 3 . 0 3 5 . 0 3 5 . 0 Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 4/19

Background Rings as Local Structures Experiments References Graph Edit Distance (Computation) ◮ if we know how to edit nodes u 1 , u 2 ∈ V G , then we know how to edit the edge e = ( u 1 , u 2 ) ∈ E G ⇒ complete set of node operations induces edit path V G plus dummy node ε 1 2 3 e 1 e 2 induced edge edit operations: ◮ edge e 1 is substituted by edge f 1 ◮ edge e 2 is deleted f 1 ε 1 2 V H plus dummy node ◮ task: find complete set of node edit operations that induces cheap edit path Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 5/19

Background Rings as Local Structures Experiments References Linear Sum Assignment with Error Correction ◮ π ∈ { 1 , . . . , n + 1 } × row to column assignments { 1 , . . . , m + 1 } is solution for LSAPE instance C iff: ◮ each row except for m = 2 m + 1 1 n + 1 is covered exactly • 1 once ◮ each column except for row • • m + 1 is covered 2 deletions C = exactly once ◮ solution π minimizing • n = 3 C ( π ) = � n + 1 � k ∈ π [ i ] c i , k can i = 1 • 0 n + 1 be computed in O ( min { n , m } 2 max { n , m } ) time, greedy suboptimal solutions in O ( nm ) time column insertions Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 6/19

Background Rings as Local Structures Experiments References Linear Sum Assignment with Error Correction ◮ π ∈ { 1 , . . . , n + 1 } × row to column assignments { 1 , . . . , m + 1 } is solution for LSAPE instance C iff: ◮ each row except for m = 2 m + 1 1 n + 1 is covered exactly • 1 once ◮ each column except for row • • m + 1 is covered 2 deletions C = exactly once ◮ solution π minimizing • n = 3 C ( π ) = � n + 1 � k ∈ π [ i ] c i , k can i = 1 • 0 n + 1 be computed in O ( min { n , m } 2 max { n , m } ) time, greedy suboptimal solutions in O ( nm ) time column insertions Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 6/19

Background Rings as Local Structures Experiments References Linear Sum Assignment with Error Correction ◮ π ∈ { 1 , . . . , n + 1 } × row to column assignments { 1 , . . . , m + 1 } is solution for LSAPE instance C iff: ◮ each row except for m = 2 m + 1 1 n + 1 is covered exactly • 1 once ◮ each column except for row • • m + 1 is covered 2 deletions C = exactly once ◮ solution π minimizing • n = 3 C ( π ) = � n + 1 � k ∈ π [ i ] c i , k can i = 1 • 0 n + 1 be computed in O ( min { n , m } 2 max { n , m } ) time, greedy suboptimal solutions in O ( nm ) time column insertions Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 6/19

Background Rings as Local Structures Experiments References LSAPE Based Heuristics for GED (Paradigm) ◮ solution for LSAPE node substitutions instance C � = complete set of | V H | + 1 1 2 node operations � = edit path between • 1 d S ( S G ( u i ) , S H ( v k )) d S ( S G ( u i ) , S ( ε )) G and H c i , | V H | + 1 = c i , k = node � upper bound for • 2 deletions GED C = ◮ S G | H ( · ): local • 3 structure rooted at c | V G | + 1 , k = node of one of the | V G | + 1 0 d S ( S ( ε ) , S H ( v k )) input graphs ◮ d S : distance measure node insertions for local structures Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 7/19

Background Rings as Local Structures Experiments References LSAPE Based Heuristics for GED (Instantiations) S G ( u ) � = node u and its incident edges [2, 5]: ◮ baseline instantiation, yields rather loose upper bound ◮ construction time for C cubic or quadratic in maximum degrees (depending on distance measure for local structures) S G ( u ) � = subgraph of radius L rooted at u [3]: ◮ yields tighter upper bound than baseline ◮ construction time for C polynomially bounded only for graphs with constantly bounded maximum degrees S G ( u ) � = walks of length L rooted at u [4]: ◮ yields tighter upper bound than baseline ◮ construction time for C bounded by polynomial of degree 2 + ω ( ω is matrix multiplication complexity exponent) ◮ suffers from tottering and supports only constant edit costs Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 8/19

Background Rings as Local Structures Experiments References Rings as Local Structures Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 9/19

Background Rings as Local Structures Experiments References Towards a New LSAPE Based Heuristic for GED Shortcomings of Local Structures Used in Existing Instantiations ◮ baseline (root plus incident edges): considers only very local information � loose upper bound on some datasets ◮ subgraph of fixed radius around root: construction of C prohibitively expensive ◮ walks of fixed length starting at root: tottering, supports only constant edit costs � loose upper bound on some datasets Desiderata ◮ define ring structures S G ( u ) = R G L ( u ) and distance measure d S = d R , such that: ◮ R G L ( u ) considers more information than the baseline and contains nodes and edges at most once to avoid tottering ◮ d R supports general edit costs and can be evaluated quickly to ensure reasonable construction time for C Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 10/19

Background Rings as Local Structures Experiments References Definition of Rings ◮ rings are sequences of layers: R G L ( u ) = ( L G l ( u )) L − 1 l = 0 , R L ( ε ) = (( ∅ , ∅ , ∅ ) l ) L − 1 l = 0 ◮ layers contain nodes, inner, and outer edges at distance l l ( u ) , OE G l ( u ) , IE G from root: L G l ( u ) = ( V G l ( u )) ◮ nodes at distance l from root are reachable from root on path l ( u ) = { u ′ ∈ V G | d G of length l : V G V ( u , u ′ ) = l } ◮ inner edges at distance l from root connect two nodes at � � l ( u ) = E G ∩ distance l : IE G V G l ( u ) × V G l ( u ) ◮ outer edges at distance l from root connect nodes at distance � � l ( u ) = E G ∩ l and l + 1 : OE G V G l ( u ) × V G l + 1 ( u ) L G 0 ( u ) R G L G 3 ( u ) 1 ( u ) u L G 2 ( u ) Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 11/19

Background Rings as Local Structures Experiments References Definition of Distance Measure for Rings ◮ distance measure for rings: L − 1 � � � � � R G L ( u ) , R H L G l ( u ) , L H d R L ( v ) = λ l d L l ( v ) l = 0 ◮ λ ∈ R L ≥ 0 : weights for distances between layers ◮ distance measure for layers: � � � � � � L G l ( u ) , L H V G l ( u ) , V H OE G l ( u ) , OE H d L l ( v ) = α 0 φ V l ( v ) + α 1 φ E l ( v ) � � IE G l ( u ) , IE H + α 2 φ E l ( v ) ◮ φ V : P ( V G ) × P ( V H ) → R ≥ 0 : distance measure for node sets ◮ φ E : P ( E G ) × P ( E H ) → R ≥ 0 : distance measure for edge sets ◮ α ∈ R 3 ≥ 0 : weights for distances between nodes, outer, and inner edges Blumenthal et al.: Ring Based Approximation of Graph Edit Distance 12/19