Exact Computation of Graph Edit Distance for Uniform and Non-Uniform - PowerPoint PPT Presentation

. . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs David B. Blumenthal & Johann Gamper GbRPR, Anacapri, 18 May 2017 D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 1/17 DF - GED u CSI _ GED nu

. . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Overview graphs costs (Abu-Aisheh et al. 2015) only (Gouda and Hassaan 2016) non-uniform edit costs D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . . . 2/17 . . . . DF - GED u CSI _ GED nu ▶ graph edit distance: flexible distance measure for labelled ▶ supports uniform and non-uniform edit costs ▶ exact computation is NP -hard ▶ existing exact algorithms ▶ A ⋆ - GED (Riesen, Fankhauser, and Bunke 2007) ▶ BLP - GED (Lerouge et al. 2016) ▶ DF - GED : node-based DFS, designed for non-uniform edit ▶ CSI _ GED : edge-based DFS, supports uniform edit costs ▶ contributions (1) DF - GED u : speed-up of DF - GED for uniform edit costs (2) CSI _ GED nu : generalised version of CSI _ GED that supports

. . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Two Communities CIKM D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . . . . . . 3/17 DF - GED u CSI _ GED nu ▶ Pattern Recognition ▶ Database Technologies ▶ a lot of work o graph edit distance exists ▶ publications in venues such as VLDB, ICDE, SIGMOD, TKDE, ▶ main focus: filtering and lower bounds ▶ slightly different definitions ▶ main difference: restriction on uniform edit costs

. . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Graph Edit Distance edges D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . . 4/17 . . . . DF - GED u CSI _ GED nu ▶ labelled undirected graph: 4-tuple G = ( V G , E G , ℓ G V , ℓ G E ) V : V G → Σ V for nodes, ℓ G E : E G → Σ E for ▶ label functions: ℓ G ▶ edit path between G and H : sequence of edit operations starting at G and ending at H ′ ≃ H ▶ edit operations: deleting, inserting, relabelling ▶ edit costs: c V : Σ V × Σ V → R for operations on nodes, c E : Σ E × Σ E → R for operations on edges { 1 α ̸ = β ▶ uniform edit costs: c V ( α, β ) , c E ( α, β ) = 0 α = β ▶ graph edit distance λ ( G, H ) : minimum cost of edit path between G and H

. . . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Node Maps D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . 5/17 . . . . DF - GED u CSI _ GED nu ▶ V G + | H | : V G plus | V H | isolated dummy nodes ▶ node map: injective partial function π : V G + | H | → V H + | G | with V G ⊆ dom ( π ) and V H ⊆ img ( π ) ▶ edit path induced by node map: let i ∈ V G , k ∈ V H , ij ∈ E G , kl ∈ E H ▶ π ( i ) = k ⇝ change node label from ℓ G V ( i ) to ℓ H V ( k ) ▶ π ( i ) = k ε ⇝ delete node i ▶ π − 1 ( k ) = i ε ⇝ insert node k ▶ π ( i ) π ( j ) = kl ⇝ change edge label from ℓ G E ( ij ) to ℓ H E ( kl ) ∈ E H ⇝ delete edge ij ▶ π ( i ) π ( j ) / ▶ π − 1 ( k ) π − 1 ( l ) / ∈ E G ⇝ insert edge kl ▶ alternative definition of λ ( G, H ) : minimum cost g ( π ) of edit path induced by a node map π

. Background . . . . . . . . . . Experiments . Conclusions and Future Work References . dummy node incomplete node maps complete node maps dummy node leaf, i. e., complete node to be computed at each inner node of the DFS D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . 6/17 . . . . . . . . . . . . . . . . . . . . . . . . . . DF - GED u CSI _ GED nu DF - GED : Node-Based DFS ▶ g ( π ) : cost of partial edit ∅ path induced by π ▶ h ( π ) : lower bound for V H V H induced cost from π to a j 1 j 2 j 3 j ε map rooted at π ⇝ has π = i 1 { i 1 �→ j 1 } heuristically sorted V G π = inner nodes � = i 2 { i 1 �→ j 1 , i 2 �→ j ε } g ( π 1 ) + g ( π 2 ) + g ( π 3 ) + i 3 ≤ ≤ > UB h ( π 1 ) h ( π 2 ) h ( π 3 ) leafs � = i 4 ⇝ UB = g ( π )

. . . . . . . . . . . . Background . Experiments Conclusions and Future Work References multiset with unassigned labels multiset with unassigned labels Lemma 1. at initialisation, sort node and edge labels D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . 7/17 . . . . . . . . . . . . DF - GED u CSI _ GED nu Our Speed-Up DF - GED u for Uniform Edit Costs from nodes in V G + | H | � �� � V ( V G + | H |− π ) × ℓ H ℓ G V ( V H + | G |− π ) , c V ) + ▶ h ( π ) : defined as MLA ( MLA ( ℓ G E ( E G − π ) E ( V H − π ) , c E ) × ℓ H � �� � from edges in E G ▶ computation for non-uniform edit costs requires cubic time For uniform edit costs, h ( π ) can be computed in linear time. 2. compute MLA ( A × B, c ) as Γ( A, B ) = max {| A | , | B |} − | A ∩ B |

. . . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Valid Edge Maps (I) D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . . . . . . . 8/17 . . . . DF - GED u CSI _ GED nu ▶ − → E G : one oriented edge ( i, j ) for each undirected ij ∈ E G ▶ ← → E H : both ( k, l ) and ( l, k ) for each kl ∈ E H − → ← → E G → E H ∪ { e ε } ▶ edge map: mapping ϕ : ▶ induces relation π ϕ on V G × V H : if ϕ ( i, j ) = ( k, l ) , then ( i, k ) ∈ π ϕ and ( j, l ) ∈ π ϕ ▶ valid edge map: ϕ is valid iff π ϕ is partial injective function

. . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Valid Edge Maps (II) Theorem maps D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . 9/17 . . . . . . . . . . . DF - GED u CSI _ GED nu ▶ partial edit path induced by valid edge map: let i ∈ V G , − → ← → k ∈ V H , ( i, j ) ∈ E G , ( k, l ) , ( l, k ) ∈ E H ▶ ϕ ( i, j ) = ( l, k ) ⇝ change edge label from ℓ G E ( ij ) to ℓ H E ( kl ) ▶ ϕ ( i, j ) = e ε ⇝ delete edge ij ▶ ϕ − 1 [ { ( k, l ) , ( l, k ) } ] = ∅ ⇝ insert edge kl ▶ π ϕ ( i ) = k ⇝ changed node label from ℓ G V ( i ) to ℓ H V ( k ) λ ( G, H ) = min { g ( ϕ ) + Γ( V G − π ϕ , V H − π ϕ ) | ϕ is valid edge map } holds for uniform edit costs, where g ( ϕ ) is the cost of the partial edit path induced by edge map ϕ . ▶ can compute λ ( G, H ) by traversing space of all valid edge

. . . . . . . . . . . . . . Background Experiments . References incomplete edge maps complete edge maps dummy edge induced cost of complete D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . Conclusions and Future Work . . . . . . . . . . . . 10/17 . . . . . . . . . . . . DF - GED u CSI _ GED nu CSI _ GED : Edge-Based DFS ∅ ▶ g ( ϕ ) : cost of partial edit ← → path induced by ϕ E H ▶ g ′ ( ϕ ) : lower bound for ( j 1 , j 2 ) ( j 2 , j 1 ) ( j 2 , j 3 ) ( j 3 , j 2 ) e ε edge map rooted at ϕ π ϕ = ( i 1 , i 2 ) { i 1 �→ j 1 , i 2 �→ j 2 } π ϕ = inner nodes � = ( i 1 , i 4 ) { i 1 �→ j 1 , − → i 2 �→ j 2 } E G ( i 2 , i 3 ) g ′ ( ϕ 1 ) g ′ ( ϕ 2 ) ≤ > UB leafs � = ( i 3 , i 4 ) ⇝ UB = g ( ϕ ) + Γ( V G − π ϕ , V H − π ϕ )

. . . . . . . . . . . . . . . Background Experiments Conclusions and Future Work References Theorem costs D. B. Blumenthal & J. Gamper: Exact Computation of Graph Edit Distance for Uniform and Non-Uniform Metric Edit Costs . . . . . . . . . . . . . . 11/17 . . . . . . . . . . . DF - GED u CSI _ GED nu Our Generalisation CSI _ GED nu V ( V G + | H |− π ϕ ) × ℓ H V ( V H + | G |− π ϕ ) , c V ) | λ ( G, H ) = min { g ( ϕ ) + MLA ( ℓ G ϕ is valid edge map } holds for non-uniform metric edit costs. ▶ can use CSI _ GED ’s DFS framework for non-uniform edit ▶ at leafs, use MLA instead of Γ to compute UB ▶ increased complexity at leafs (cubic instead of linear) ▶ no increased complexity at inner nodes of search tree