Edit distance

BoundedMinkowskiCost(μ₁, μ₂)

Return value similar to MinkowskiCost, but ensure costs smaller than 2τ.

Optional Arguments

p=1: the p value for p-norm calculation. τ=1: value specifying half of the upper limit of the Minkowski cost.

MinkowskiCost(μ₁, μ₂; p::Real=1)

For labels μ₁ on the vertices of graph G₁ and labels μ₂ on the vertices of graph G₂, compute the p-norm cost of substituting vertex u ∈ G₁ by vertex v ∈ G₂.

Optional Arguments

p=1: the p value for p-norm calculation.

edit_distance(G₁::AbstractGraph, G₂::AbstractGraph)

Compute the edit distance between graphs G₁ and G₂. Return the minimum edit cost and edit path to transform graph G₁ into graph G₂. An edit path consists of a sequence of pairs of vertices(u,v) ∈ [0,|G₁|] × [0,|G₂|]` representing vertex operations:

• $(0,v)$: insertion of vertex $v ∈ G₂$
• $(u,0)$: deletion of vertex $u ∈ G₁$
• $(u>0,v>0)$: substitution of vertex $u ∈ G₁$ by vertex $v ∈ G₂$

Optional Arguments

• insert_cost::Function=v->1.0
• delete_cost::Function=u->1.0
• subst_cost::Function=(u,v)->0.5

By default, the algorithm uses constant operation costs. The user can provide classical Minkowski costs computed from vertex labels μ₁ (for G₁) and μ₂ (for G₂) in order to further guide the search, for example:

edit_distance(G₁, G₂, subst_cost=MinkowskiCost(μ₁, μ₂))

• heuristic::Function=DefaultEditHeuristic: a custom heuristic provided to the A*

search in case the default heuristic is not satisfactory.

Performance

• Given two graphs $|G₁| < |G₂|$, edit_distance(G₁, G₂) is faster to

compute than edit_distance(G₂, G₁). Consider swapping the arguments if involved costs are equivalent.

• The use of simple Minkowski costs can improve performance considerably.
• Exploit vertex attributes when designing operation costs.

References

• RIESEN, K., 2015. Structural Pattern Recognition with Graph Edit Distance: Approximation Algorithms and Applications. (Chapter 2)

Author

• Júlio Hoffimann Mendes (juliohm@stanford.edu)

Examples

julia> g1 = SimpleDiGraph([0 1 0 0 0; 0 0 1 0 0; 1 0 0 1 0; 0 0 0 0 1; 0 0 0 1 0]);

julia> g2 = SimpleDiGraph([0 1 0; 0 0 1; 1 0 0]);

julia> edit_distance(g1, g2)
(3.5, Tuple[(1, 2), (2, 1), (3, 0), (4, 3), (5, 0)])