This document provides an overview of graph edit distance, including its definition, history, and algorithms. It begins by defining an edit path as a sequence of node/edge insertions, deletions, and substitutions that transforms one graph into another. The graph edit distance is the cost of the lowest cost edit path. It describes tree search algorithms used to explore the space of possible edit paths efficiently. It also explains how edit paths can be modeled as assignment problems that are solved using techniques like the Hungarian algorithm to find approximations of the graph edit distance.
DirectLiNGAM is a new non-Gaussian estimation method for the LiNGAM model that directly estimates the variable ordering without using ICA. It iteratively identifies exogenous variables using independence tests between each variable and its residuals from regression on other variables. This allows it to estimate the ordering in a fixed number of steps, with no algorithmic parameters, convergence issues, or scale dependence like previous ICA-based methods. It was shown to estimate the correct causal ordering on a real-world socioeconomic dataset, matching domain knowledge better than alternative methods like ICA-LiNGAM, PC algorithm, and GES.
Variational이라는 단어로는 아무것도 안떠오릅니다.
그래서, '꿩 대신 닭'이라고 표현해 봤습니다.
초반 독자적인 그림을 통해 개념잡기가 쉬워요.
설명부분은 초록색으로 표시했습니다.
확률변수(random variable)부터 막히면, 아래 블로그 글을 읽어 보세요.
https://blog.naver.com/nonezerok/221428251262
DirectLiNGAM is a new non-Gaussian estimation method for the LiNGAM model that directly estimates the variable ordering without using ICA. It iteratively identifies exogenous variables using independence tests between each variable and its residuals from regression on other variables. This allows it to estimate the ordering in a fixed number of steps, with no algorithmic parameters, convergence issues, or scale dependence like previous ICA-based methods. It was shown to estimate the correct causal ordering on a real-world socioeconomic dataset, matching domain knowledge better than alternative methods like ICA-LiNGAM, PC algorithm, and GES.
Variational이라는 단어로는 아무것도 안떠오릅니다.
그래서, '꿩 대신 닭'이라고 표현해 봤습니다.
초반 독자적인 그림을 통해 개념잡기가 쉬워요.
설명부분은 초록색으로 표시했습니다.
확률변수(random variable)부터 막히면, 아래 블로그 글을 읽어 보세요.
https://blog.naver.com/nonezerok/221428251262
This document summarizes a presentation on SAT/SMT solving in Haskell. It discusses two main Haskell libraries for SMT solving - sbv, which provides a high-level DSL for specifying SMT problems and interfaces with multiple solvers, and toysolver, which contains the toysat SAT solver and toysmt SMT solver implemented natively in Haskell. It then demonstrates solving the "send more money" puzzle using sbv and running a simple problem on toysmt.
The document discusses the SATYSFI Conf 2021 conference which will take place on June 26, 2021. It provides details on recent updates to the SATYSFI typesetting system including the addition of linear-transform-graphics, improvements to page breaking for multicolumn content, and adding debugging information for overfull/underfull boxes. Version 0.0.6, 0.0.7, and planned future updates are summarized. The document also discusses using domain specific languages for describing typesetting definitions and structures and provides examples using amidakuji diagrams.
This document discusses methods for summarizing Lego-like sphere and torus maps. It begins by introducing the concept of ({a,b},k)-maps, which are k-valent maps with faces of size a or b. It then discusses several challenges in enumerating and drawing such maps, including enumerating all possible Lego decompositions. Specific enumeration methods are described, such as using exact covering problems or satisfiability problems. The document also discusses challenges in graph drawing representations, and suggests using primal-dual circle packings as a promising approach.
This document summarizes a presentation on SAT/SMT solving in Haskell. It discusses two main Haskell libraries for SMT solving - sbv, which provides a high-level DSL for specifying SMT problems and interfaces with multiple solvers, and toysolver, which contains the toysat SAT solver and toysmt SMT solver implemented natively in Haskell. It then demonstrates solving the "send more money" puzzle using sbv and running a simple problem on toysmt.
The document discusses the SATYSFI Conf 2021 conference which will take place on June 26, 2021. It provides details on recent updates to the SATYSFI typesetting system including the addition of linear-transform-graphics, improvements to page breaking for multicolumn content, and adding debugging information for overfull/underfull boxes. Version 0.0.6, 0.0.7, and planned future updates are summarized. The document also discusses using domain specific languages for describing typesetting definitions and structures and provides examples using amidakuji diagrams.
This document discusses methods for summarizing Lego-like sphere and torus maps. It begins by introducing the concept of ({a,b},k)-maps, which are k-valent maps with faces of size a or b. It then discusses several challenges in enumerating and drawing such maps, including enumerating all possible Lego decompositions. Specific enumeration methods are described, such as using exact covering problems or satisfiability problems. The document also discusses challenges in graph drawing representations, and suggests using primal-dual circle packings as a promising approach.
Small updates of matrix functions used for network centralityFrancesco Tudisco
Many relevant measures of importance for nodes and edges of a network are defined in terms of suitable entries of matrix functions $f(A)$, for different choices of $f$ and $A$. Addressing the entries of $f(A)$ can be computationally challenging and this is particularly prohibitive when $A$ undergoes a perturbation $A+\delta A$ and the entries of $f(A)$ have to be updated. Given the adjacency matrix $A$ of a graph $G=(V,E)$, in this talk we consider the case where $\delta A$ is a sparse matrix that yields a small perturbation of the edge structure of $G$.
In particular, we present a bound showing that the variation of the entry $f(A)_{u,v}$ decays exponentially with the distance in $G$ that separates either $u$ or $v$ from the set of nodes touched by the edges that are perturbed. Our bound depends only on the distances in the original graph $G$ and on the field of values of the perturbed matrix $A+\delta A$. We show several numerical examples in support of the proposed result.
Talk presented at the IMA Numerical Analysis and Optimization conference, Birmingham 2018
The talk is based on the paper:
S. Pozza and F. Tudisco, On the stability of network indices defined by means of matrix functions, SIMAX, 2018
The document summarizes research on the parameterized complexity of the graph MOTIF problem. It begins by defining the problem and providing an example. It then discusses how graph MOTIF can be solved efficiently using different parameters, such as cluster editing, distance to clique, and vertex cover number. The document also analyzes parameters for which graph MOTIF remains NP-hard, such as the deletion set number parameter. In conclusion, it provides references for the algorithms and results discussed.
Informed-search TECHNIQUES IN ai ml data sciencedevvpillpersonal
Informed search algorithms use problem-specific heuristics to improve search efficiency over uninformed methods. The most common informed methods are best-first search, A* search, and memory-bounded variants like RBFS and SMA*. A* is optimal if the heuristic is admissible for tree searches or consistent for graph searches. Heuristics provide an estimate of the remaining cost to the goal and can significantly speed up search. Common techniques for generating heuristics include Manhattan distance.
Gaps between the theory and practice of large-scale matrix-based network comp...David Gleich
This document discusses gaps between theory and practice in large scale matrix computations for networks. It provides an overview of representing networks as matrices and canonical problems like PageRank that can be modeled as matrix computations. It then discusses different methods for solving these problems, like Monte Carlo methods, relaxation methods, and Krylov subspace methods. It analyzes the computational complexity of these approaches and identifies open problems, such as developing unified convergence results for different algorithms and handling "top k" convergence. The talk concludes by identifying more structured problems on networks that could leverage matrix computations.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
This document discusses algorithms for solving the all pairs shortest path problem. It introduces the all pair distance problem for unweighted graphs, which can be solved by multiplying the adjacency matrix. For weighted graphs, it describes the randomized Boolean product witness matrix algorithm, which reduces the all pairs shortest path problem to matrix multiplication. Deterministic algorithms like Floyd-Warshall are also discussed.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document proposes an approach to improve geographic information (GI) interoperability through emergent semantics. It describes using structure preserving semantic matching (SPSM) to find correspondences between semantically related nodes in graph-like representations (e.g. schemas, ontologies) while preserving structural properties. An example matching geo-services requests is provided. Evaluation on synthesized datasets showed average precision and recall of 0.78, demonstrating the potential of the approach. Future work will include extensive evaluation and extending the approach to fully developed spatial data infrastructure ontologies.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by defining dynamic programming as avoiding recalculating solutions by storing results in a table. It then describes Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, calculating shortest paths that pass through each intermediate node. It takes O(n3) time for a graph with n nodes. Finally, it discusses the multistage graph problem and provides forward and backward algorithms to find the minimum cost path from source to destination in a multistage graph in O(V+E) time, where V and E are the numbers of vertices and edges.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Conference talk at the SIAM Conference on Financial Mathematics and Engineering, held in virtual format, June 1-4 2021, about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model".
- Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
I am Charles S. I am a Design & Analysis of Algorithms Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, York University, Canada. I have been helping students with their assignments for the past 15 years. I solve assignments related to the Design & Analysis Of Algorithms.
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The document presents a new iterative method (M2 method) for determining the exact solution to a parametric linear programming problem where the objective function and constraints contain parameters. The M2 method exploits the concept of a p-solution to a square linear interval parametric system and iteratively reduces the parameter domain while maintaining upper and lower bounds on the optimal objective value. A numerical example is given to illustrate the new iterative approach for solving parametric linear programming problems.
A Parallel Branch And Bound Algorithm For The Quadratic Assignment ProblemMary Calkins
This document summarizes a parallel branch and bound algorithm for solving the quadratic assignment problem (QAP). Key points:
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1. Graph Edit Distance: Basics and History
Luc Brun
Special Thanks to:
Benoit Ga¨uz`ere and
S´ebastien Bougleux
May 12, 2017
2. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
1 Definition
2 Tree search algorithms
Depth first search algorithm
3 From edit paths to assignment problems
4 Solving assignment problems
5 Conclusion and Future Work
6 Bibliography
(GREYC) Graph Edit Distance May 12, 2017 2 / 34
3. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Introduction
Recognition implies the definition of similarity or dissimilarity
measures.
Different measures between graphs:
Distance based on maximum common subgraphs,
Distance based on spectral characteristics,
Graph kernels (simmilarities),
Graph Edit distance.
yq q
Not definite negative,
yq q
Very fine.
(GREYC) Graph Edit Distance May 12, 2017 3 / 34
4. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Edit Paths
Edit Paths
Definition (Edit path)
Given two graphs G1 and G2 an edit path between G1 and G2 is a
sequence of node or edge removal, insertion or substitution which
transforms G1 into G2.
x
x x
h
E
Removal of two
edges and one node
x x
h
E
insertion of
an edge
x x
h
d
d
E
substitution of
a node
x x
h
d
d
A substitution is denoted u → v, an insertion → v and a removal
u → .
Alternative edit operations such as merge/split have been also
proposed[Ambauen et al., 2003].
(GREYC) Graph Edit Distance May 12, 2017 4 / 34
5. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Costs
Let c(.) denote the cost of any elementary operation. The cost of an
edit path is defined as the sum of the costs of its elementary
operations.
All cost are positive: c() ≥ 0,
A node or edge substitution which does not modify a label has a
0 cost: c(l → l) = 0.
x
x x
h
G1
E
e1 = | →
e2 = − →
e3 = • →
x x
h
E
e4 = →
x x
h
d
d
E
e5 = sEs
x x
h
d
d
G2
If all costs are equal to 1, the cost of this edit path is equal to 5.
(GREYC) Graph Edit Distance May 12, 2017 5 / 34
6. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Graph edit distance
Definition (Graph edit distance)
The graph edit distance between G1 and G2 is defined as the cost of
the less costly path within Γ(G1, G2). Where Γ(G1, G2) denotes the
set of edit paths between G1 and G2.
d(G1, G2) = min
γ∈Γ(G1,G2) e∈γ
c(e)
Suggests an exploration of Γ(G1, G2),
Tree based algorithms.
(GREYC) Graph Edit Distance May 12, 2017 6 / 34
7. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Tree search algorithms
Let us consider a partial edit path p between G1 and G2. Let:
g(p) the cost of the partial edit path.
h(p) a lower bound of the cost of the remaining part of the path
required to reach G2.
If g(p) + h(p) > UB we have:
∀γ ∈ Γ(G1, G2), γ = p.q, dγ(G1, G2) ≥ g(p) + h(p) > UB
eeu
d
d
d
p
γ
r
h
h
h
q
In other terms, all the sons of p will provide a greater
approximation of the GED and correspond thus to unfruitful
nodes.
(GREYC) Graph Edit Distance May 12, 2017 7 / 34
8. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Depth first search algorithm
1: Input: Two graphs G1 and G2 with V1 = {u1, . . . , un} and V2 = {v1, . . . , vm}
2: Output: γUB and UB a minimum edit path and its associated cost
3:
4: (γUB, UBCOST) = GoodHeuristic(G1, G2)
5: initialize OPEN = {u1 → } ∪ w∈V2
{u1 → w}
6: while OPEN = ∅ do
7: p = OPEN.popFirst()
8: if p is a leaf then
9: update (γUB, UBCOST) if required
10: else
11: Stack into OPEN all sons q of p such that g(q) + h(q) < UBCOST.
12: end if
13: end while
yq q
Bounded number of edit paths
(GREYC) Graph Edit Distance May 12, 2017 8 / 34
9. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Depth first search algorithm
1: Input: Two graphs G1 and G2 with V1 = {u1, . . . , un} and V2 = {v1, . . . , vm}
2: Output: γUB and UB a minimum edit path and its associated cost
3:
4: (γUB, UBCOST) = GoodHeuristic(G1, G2)
5: initialize OPEN = {u1 → } ∪ w∈V2
{u1 → w}
6: while OPEN = ∅ do
7: p = OPEN.popFirst()
8: if p is a leaf then
9: update (γUB, UBCOST) if required
10: else
11: Stack into OPEN all sons q of p such that g(q) + h(q) < UBCOST.
12: end if
13: end while
yq q
Bounded number of edit paths
yq q
Anytime, Parallel.
(GREYC) Graph Edit Distance May 12, 2017 8 / 34
10. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Depth first search algorithm
1: Input: Two graphs G1 and G2 with V1 = {u1, . . . , un} and V2 = {v1, . . . , vm}
2: Output: γUB and UB a minimum edit path and its associated cost
3:
4: (γUB, UBCOST) = GoodHeuristic(G1, G2)
5: initialize OPEN = {u1 → } ∪ w∈V2
{u1 → w}
6: while OPEN = ∅ do
7: p = OPEN.popFirst()
8: if p is a leaf then
9: update (γUB, UBCOST) if required
10: else
11: Stack into OPEN all sons q of p such that g(q) + h(q) < UBCOST.
12: end if
13: end while
yq q
Bounded number of edit paths
yq q
Anytime, Parallel.
yq q
Compution of the optimum may be long
(GREYC) Graph Edit Distance May 12, 2017 8 / 34
11. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Restricted edit paths
An element γ ∈ Γ(G1, G2) is potentially infinite by just doing and
undoing a given operation (e.g. insert and then delete a node). All
cost being positive such an edit path can not correspond to a
minimum:
Definition (Restricted Edit path)
An independent edit path between two labeled graphs G1 and G2 is
an edit path such that:
1 No node nor edge is both substituted and removed,
2 No node nor edge is simultaneously substituted and inserted,
3 Any inserted element is never removed,
4 Any node or edge is substituted at most once,
5 Any edge cannot be removed and then inserted.
(GREYC) Graph Edit Distance May 12, 2017 9 / 34
12. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
-assignments
Let v1 and V2 be two sets, with n = |v1| and m = |V2|.
Consider V1 = V1 ∪ { } and V2 = V2 ∪ { }.
Definition
An -assignment from V1 to V2 is a mapping ϕ : V1 → P(V2 ),
satisfying the following constraints:
∀i ∈ V1 |ϕ(i)| = 1
∀j ∈ V2 |ϕ−1
(j)| = 1
∈ ϕ( )
An assignment encodes thus:
1 Substitutions: ϕ(i) = j with (i, j) ∈ V1 × V2.
2 Removals: ϕ(i) = with i ∈ V1 .
3 Insertions: j ∈ ϕ−1
( ) with j ∈ V2.
(GREYC) Graph Edit Distance May 12, 2017 10 / 34
13. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
From assignments to matrices
An −assignment can be encoded in matrix form:
X = (xi,k)(i,k)∈{1,...,n+1}×{1,...,m+1} with
∀(i, k) ∈ {1, . . . , n+1}×{1, . . . , m+1} xi,k =
1 if k ∈ ϕ(i)
0 else
We have:
∀i = 1, . . . , n,
m+1
k=1
xi,k = 1 (|ϕ(i)| = 1)
∀k = 1, . . . , m,
n+1
j=1
xj,k = 1 (|ϕ−1
(k)| = 1)
xn+1,m+1 = 1 ( ∈ ϕ( ))
∀(i, j) xi,j ∈ {0, 1}
(GREYC) Graph Edit Distance May 12, 2017 11 / 34
14. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
From functions to matrices
Example
u1
u2
u3
v1
v2
V1
V2
x =
v1 v2 |
u1 1 0 | 0
u2 0 1 | 0
u3 0 0 | 1
0 0 | 1
The set of permutation matrices encoding −assignments is
called the set of -assignment matrices denoted by An,m.
Usual assignments are encoded by larger (n + m) × (n + m)
matrices[Riesen, 2015].
(GREYC) Graph Edit Distance May 12, 2017 12 / 34
15. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Back to edit paths
Let us consider two simple graphs G1 = (V1, E1) and
G2 = (V2, E2) with |V1| = n and |V2| = m.
Proposition
There is a one-to-one relation between the set of restricted edit paths
from G1 to G2 and An,m.
Theorem
Any non-infinite value of 1
2
xT
∆x + cT
x corresponds to the cost of a
restricted edit path. Conversely the cost of any restricted edit path
may be written as 1
2
xT
∆x + cT
x with the appropriate x.
(GREYC) Graph Edit Distance May 12, 2017 13 / 34
16. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Costs of Node assignments
C =
c(u1 → v1) · · · c(u1 → vm) c(u1 → ε)
...
...
... c(ui → vj)
... c(ui → ε)
...
...
c(un → v1) . . . c(un → vm) c(un → ε)
c(ε → v1) c( → vi ) c( → vm) 0
c = vect(C)
Alternative factorizations of cost matrices
exist[Serratosa, 2014, Serratosa, 2015].
(GREYC) Graph Edit Distance May 12, 2017 14 / 34
17. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Cost of edges assignments
Let us consider a (n + 1)(m + 1) × (n + 1)(m + 1) matrix D
such that:
dik,jl = ce(i → k, j → l)
with:
(i, j) (k, l) edit operation cost ce(i → k, j → l)
∈ E1 ∈ E2 substitution of (i, j) by (k, l) c((i, j) → (k, l))
∈ E1 ∈ E2 removal of (i, j) c((i, j) → )
∈ E1 ∈ E2 insertion of (k, l) into E1 c( → (k, l))
∈ E1 ∈ E2 do nothing 0
∆ = D if both G1 and G2 are undirected and
∆ = D + DT
else.
Matrix ∆ is symmetric in both cases.
(GREYC) Graph Edit Distance May 12, 2017 15 / 34
18. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Let us approximate
GED(G1, G2) = min
x∈vect(An,m)
1
2
xT
∆x + cT
x
Matrix ∆ is non convex with several local minima. The problem
is thus NP-complete.
One solution to solve this quadratic problem consists in dropping
the quadratic term. We hence get:
d(G1, G2) ≈ min cT
x | x ∈ vec[An,m] = min
n+1
i=1
m+1
k=1
ci,kxi,k
This problem is an instance of a bipartite graph matching
problem also called linear sum assignment problem.
(GREYC) Graph Edit Distance May 12, 2017 16 / 34
19. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Bipartite Graph matching
Such an approximation of the GED is called a bipartite Graph
edit distances (BP-GED).
Munkres or Jonker-Volgenant[Burkard et al., 2012] allow to
solve this problem in cubic time complexity.
The general procedure is as follows:
1 compute:
x = arg min
x∈vec(An,m)
cT
x
2 Return the cost of the edit path γx encoded by x.
(GREYC) Graph Edit Distance May 12, 2017 17 / 34
20. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Matching Neighborhoods
The matrix C defined previoulsy only encodes node information.
Idea: Inject edge information into matrix
C[Riesen and Bunke, 2009]. Let
di,k = minx
n+1
j=1
m+1
l=1
c(ij → kl)xj,l
the cost of the optimal mapping of the edges incident to i onto
the edge incident to j.
Let :
c∗
i,k = c(ui → vk) + di,k and x = arg min(c∗
)T
x
The edit path deduced from x defines an edit cost dub(G1, G2)
between G1 and G2.
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21. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Matching larger structure
The upper bound provided by dub(G1, G2) may be large,
especially for large graphs. So a basic idea consists in enlarging
the considered substructures:
1 Incident edges and adjacent nodes.
2 Bags of walks [Ga¨uz`ere et al., 2014].
3 Centered subgraphs [Carletti et al., 2015].
4 . . .
All these heuristics provide an upper bound for the Graph edit
distance.
But: up to a certain point the linear approximation of a
quadratic problem reach its limits.
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22. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
From linear to quadratic optimization
One idea to improve the results of the linear approximation of
the GED consists in applying a post processing stage.
Two ideas have been proposed [Riesen, 2015]:
1 By modifying the initial cost matrix and recomputing a new
assignment.
2 By swapping elements in the assignment returned by the linear
algorithm.
In order to go beyond these results, the search must be guided
by considering the real quadratic problem.
GED(G1, G2) = min
x∈vect(An,m)
1
2
xT
∆x + cT
x
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23. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Previous Quadratic methods
[Justice and Hero, 2006]
[Neuhaus and Bunke, 2007]
. . .
(GREYC) Graph Edit Distance May 12, 2017 21 / 34
24. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
From quadratic to linear problems
Let us consider a quadratic problem:
xT
Qx =
n
i=1
n
j=1
qi,jxi xj
Let us introduce yn∗i+j = xi xj we get:
xT
Qx =
n2
k=1
qkyk
with an appropriate renumbering of Q’s elements. Hence a
Linear Sum Assignment Problem with additional constraints.
Note that the Hungarian algorithm can not be applied due to
additional constraints. We come back to tree based algorithms.
This approach has been applied to the computation of the exact
Graph Edit Distance by[Lerouge et al., 2016]
(GREYC) Graph Edit Distance May 12, 2017 22 / 34
25. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Integer Projected Fixed Point (IPFP)
[Leordeanu et al., 2009]
Start with an good Guess x0:
x = x0
while a fixed point is not reached do
b∗
= arg min{(xT
∆ + cT
)b, b ∈ {0, 1}(n+1)(m+1)
}
t∗
= line search between x and b∗
x = x + t∗
(b∗
− x)
end while
b∗
minimizes a sum of:
(xT
∆ + cT
)i,k = ci,k +
n+1
j
m+1
l
di,k,j,l xj,l
which may be understood as the cost of mapping i to j given
the previous assignment x.
The edit cost decreases at each iteration. Moreover,
at each iteration we come back to an integer solution,
(GREYC) Graph Edit Distance May 12, 2017 23 / 34
26. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Graduated NonConvexity and Concativity Pro-
cedure (GNCCP)
Consider [Liu and Qiao, 2014]:
Sη(x) = (1 − |ζ|)S(x) + ζxT
x with S(x) =
1
2
xT
∆x + cT
x
where ζ ∈ [−1, 1].
ζ = 1 : Convex objective function
ζ = −1 : Concave objective function
The algorithm tracks the optimal solution from a convex to a
concave relaxation of the problem.
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28. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
GNCCP Algorithm
ζ = 1, d = 0.1, x = 0
while (ζ > −1) and (x ∈ An,m) do
Q = 1
2
(1 − |ζ|)∆ + ζI
L = (1 − |ζ|)c
x = IPFP(x, Q, L)
ζ = ζ − d
end while
(GREYC) Graph Edit Distance May 12, 2017 26 / 34
29. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Experiments
Datasets
Dataset Number of graphs Avg Size Avg Degree Properties
Alkane 150 8.9 1.8 acyclic, unlabeled
Acyclic 183 8.2 1.8 acyclic
MAO 68 18.4 2.1
PAH 94 20.7 2.4 unlabeled, cycles
MUTAG 8 × 10 40 2
(GREYC) Graph Edit Distance May 12, 2017 27 / 34
30. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Experiments
Algorithm
Alkane Acyclic
d t d t
A∗
15.5 1.29 17.33 6.02
[Riesen and Bunke, 2009] 35.2 0.0013 35.4 0.0011
[Ga¨uz`ere et al., 2014] 34.5 0.0020 32.6 0.0018
[Carletti et al., 2015] 26 2.27 28 0.73
IPFPRandom init 22.6 0.007 23.4 0.006
IPFPInit [Riesen and Bunke, 2009] 22.4 0.007 22.6 0.006
IPFPInit [Ga¨uz`ere et al., 2014] 19.3 0.005 20.4 0.004
[Neuhaus and Bunke, 2007] 20.5 0.07 25.7 0.42
GNCCP 16.8 0.12 19.1 0.07
(GREYC) Graph Edit Distance May 12, 2017 28 / 34
31. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Experiments
Algorithm
MAO PAH
d t d t
[Riesen and Bunke, 2009] 105 5 10−3
138 7 10−3
[Ga¨uz`ere et al., 2014] 56.9 2 10−2
123.8 3 10−2
[Carletti et al., 2015] 44 6.16 129 2.01
IPFPRandom init 65.2 0.031 63 0.04
IPFPInit [Riesen and Bunke, 2009] 59 0.031 62.2 0.04
IPFPInit [Ga¨uz`ere et al., 2014] 32.9 0.030 48.9 0.048
[Neuhaus and Bunke, 2007] 59.1 7 52.9 8.20
GNCCP 32.9 0.46 38.7 0.86
(GREYC) Graph Edit Distance May 12, 2017 29 / 34
32. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Graph Edit distance Contest
Several Algorithms (all limited to 30 seconds):
Algorithms based on a linear transformation of the quadratic
problem solved by integer programming:
F2 (•),
F24threads( ),
F2LP ((♦, relaxed problem)
Algorithms based on Depth first search methods:
DF( ),
PDFS( ),
DFUB( ).
Beam Search: BS ( )
IPFP[Ga¨uz`ere et al., 2014] : QAPE (+)
Bipartite matching[Ga¨uz`ere et al., 2014]: LSAPE(×)
(GREYC) Graph Edit Distance May 12, 2017 30 / 34
33. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Average speed-deviation scores on MUTA sub-
sets
Setting of editing costs
vertices edges
cs cd ci cs cd ci
Setting 1 2 4 4 1 2 2
(GREYC) Graph Edit Distance May 12, 2017 31 / 34
34. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Conclusion
Bipartite Graph edit distance has re initiated a strong interest on
Graph edit Distance from the research community
1 It is fast enough to process large graph databases,
2 It provides a reasonable approximation of the GED.
More recently new solvers for the GED have emerged.
1 They remain fast (while slower than BP-GED),
2 They strongly improve the approximation or provide exact
solutions.
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35. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Future Work
The Bipartite Matching remains a core element of several
quadratic algorithms. So any improvement of such algorithms
has many consequences.
Quadratic algorithms for GED are still immature, a lot of job
remains to be done.
Distances not directly related to GED should also be
investigated (Kernel Based, Hausdorff,. . . )
Related applications are also quite fascinating:
Median/mean computation,
Learning costs for GED,
. . .
(GREYC) Graph Edit Distance May 12, 2017 33 / 34
36. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Bibliography
(GREYC) Graph Edit Distance May 12, 2017 34 / 34
37. Definition
Tree search algorithms
From edit paths to assignment probl
Solving assignment problems
Conclusion and Future Work
Bibliography
Abu-Aisheh, Z. (2016). Anytime and distributed Approaches for Graph
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Solving assignment problems
Conclusion and Future Work
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Solving assignment problems
Conclusion and Future Work
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Solving assignment problems
Conclusion and Future Work
Bibliography
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