A General Vertex Partition Refinement Algorithm for Graph Isomorphism

2013 ◽  
Vol 760-762 ◽  
pp. 1919-1924 ◽  
Author(s):  
Ai Min Hou ◽  
Chuan Fu Hu ◽  
Zhi Feng Hao
Keyword(s):  
Time Complexity ◽  
Polynomial Algorithm ◽  
Graph Isomorphism ◽  
Vertex Partition ◽  
Refinement Algorithm ◽  
Partition Refinement

A general depth-first backtracking algorithm for graph isomorphism with the vertex partition and refinement technique is presented in this paper. The time complexity of this nondeterministic polynomial algorithm is O(nα+3) where nα is the number of backtracking points and (h-1)/2α (h+1)/2 for h=logn in the worst cases. The tests on many types of graphs validated the efficiency of this algorithm for graph isomorphism.

scholarly journals Checking for Open Bisimilarity in the pi-Calculus

2001 ◽  
Vol 8 (8) ◽  
Author(s):  
Ulrik Frendrup ◽  
Jesper Nyholm Jensen
Keyword(s):  
User Interface ◽  
Iterative Method ◽  
Time Complexity ◽  
Source Code ◽  
Performance Tests ◽  
Running Time ◽  
Pi Calculus ◽  
Definition Of ◽  
Refinement Algorithm ◽  
Partition Refinement

<p>This paper deals with algorithmic checking of open bisimilarity in the pi-calculus. Most bisimulation checking algorithms are based on the partition refinement approach. Unfortunately the definition of open bisimulation does not permit us to use a partition refinement approach for open bisimulation checking directly, but in the paper 'A Partition Refinement Algorithm for the pi-Calculus' Marco Pistore and Davide Sangiorgi present an iterative method that makes it possible to check for open bisimilarity using partition refinement. We have implemented the algorithm presented by Marco Pistore and Davide Sangiorgi. Furthermore,<br />we have optimized this algorithm and implemented this optimized algorithm. The time-complexity of this algorithm is the same as the time-complexity for the first algorithm, but performance tests have shown that in many cases the running time of the optimized algorithm is shorter than the running time of the first algorithm. Our implementation of the optimized open bisimulation checker algorithm and a user interface have been integrated in a system called the OBC Workbench.The source code and a manual for it is available from http://www.cs.auc.dk/research/FS/ny/PR-pi/.</p>

Error-Correcting Graph Isomorphism Using Decision Trees

1998 ◽  
Vol 12 (06) ◽  
pp. 721-742 ◽  
Author(s):  
B. T. Messmer ◽  
H. Bunke
Keyword(s):  
Decision Tree ◽  
Fast Algorithm ◽  
Time Complexity ◽  
A Priori ◽  
Graph Isomorphism ◽  
Subgraph Isomorphism ◽  
Input Graph ◽  
Require Time ◽  
Original Algorithm ◽  
Model Graph

In this paper we present a fast algorithm for the computation of error-correcting graph isomorphisms. The new algorithm is an extension of a method for exact subgraph isomorphism detection from an input graph to a set of a priori known model graphs, which was previously developed by the authors. Similar to the original algorithm, the new method is based on the idea of creating a decision tree from the model graphs. This decision tree is compiled off-line in a preprocessing step. At run time, it is used to find all error-correcting graph isomorphisms from an input graph to any of the model graphs up to a certain degree of distortion. The main advantage of the new algorithm is that error-correcting graph isomorphism detection is guaranteed to require time that is only polynomial in terms of the size of the input graph. Furthermore, the time complexity is completely independent of the number of model graphs and the number of edges in each model graph. However, the size of the decision tree is exponential in the size of the model graphs and the degree of error. Nevertheless, practical experiments have indicated that the method can be applied to graphs containing up to 16 vertices.

Finite Characterization of the Coarsest Balanced Coloring of a Network

2020 ◽  
Vol 30 (14) ◽  
pp. 2050212
Author(s):  
Ian Stewart
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Universal Cover ◽  
Running Time ◽  
Set Up ◽  
Balanced Coloring ◽  
Refinement Algorithm ◽  
Partition Refinement

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

An Algorithm for Testing Isomorphism of Planer Graph Based on Distant Matrix

2012 ◽  
Vol 487 ◽  
pp. 317-321
Author(s):  
Yan Peng Wu ◽  
Shui Qiang Liu
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Time Complexity ◽  
Graph Isomorphism ◽  
Distance Matrix ◽  
Space Complexity ◽  
The Subject ◽  
The Many

The testing for graph isomorphism is one of the many problems in the subject of graph theory. This thesis proposes an algorithm for testing isomorphism of planer graph of polynomial time via structuring characteristics of planer graph based on distance matrix. The algorithm, with a time complexity of O (n^4) and a space complexity of O (n^2), has a great application value.

scholarly journals A Partition Refinement Algorithm for the π-Calculus

2001 ◽  
Vol 164 (2) ◽  
pp. 264-321 ◽  
Author(s):  
Marco Pistore ◽  
Davide Sangiorgi
Keyword(s):  
Π Calculus ◽  
Refinement Algorithm ◽  
Partition Refinement

scholarly journals From generic partition refinement to weighted tree automata minimization

2021 ◽  
Author(s):  
Thorsten Wißmann ◽  
Hans-Peter Deifel ◽  
Stefan Milius ◽  
Lutz Schröder
Keyword(s):  
Tree Automata ◽  
Weighted Tree ◽  
Transition Systems ◽  
Run Time Analysis ◽  
Deterministic Automata ◽  
Weighted Tree Automata ◽  
Run Time ◽  
The Given ◽  
Refinement Algorithm ◽  
Partition Refinement

AbstractPartition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed a partition refinement algorithm that is generic in the transition type of the given system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time analysis of our algorithm to cover additional instances, notably weighted automata and, more generally, weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular, and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors. Experiments show that even for complex system types, the tool is able to handle systems with millions of transitions.

scholarly journals A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines

2021 ◽  
Author(s):  
Nodari Vakhania ◽  
Frank Werner
Keyword(s):  
Time Complexity ◽  
Polynomial Algorithm ◽  
Delivery Time ◽  
Uniform Machine ◽  
Delivery Times ◽  
Uniform Machines ◽  
Complexity Status ◽  
Algorithmic Framework ◽  
Identical Machine ◽  
Job Delivery

The problem of sequencing $n$ equal-length non-simultaneously released jobs with delivery times on $m$ uniform machines to minimize the maximum job completion time is considered. To the best of our knowledge, the complexity status of this classical scheduling problem remained open up to the date. We establish its complexity status positively by showing that it can be solved in polynomial time. We adopt for the uniform machine environment the general algorithmic framework of the analysis of behavior alternatives developed earlier for the identical machine environment. The proposed algorithm has the time complexity $O(\gamma m^2 n\log n)$, where $\gamma$ can be any of the magnitudes $n$ or $q_{\max}$, the maximum job delivery time. In fact, $n$ can be replaced by a smaller magnitude $\kappa&lt;n$, which is the number of special types of jobs (it becomes known only upon the termination of the algorithm).

scholarly journals DEL-based Epistemic Planning for Human-Robot Collaboration: Theory and Implementation

2021 ◽  
Author(s):  
Thomas Bolander ◽  
Lasse Dissing ◽  
Nicolai Herrmann
Keyword(s):  
Theory Of Mind ◽  
Epistemic Logic ◽  
Autonomous Robots ◽  
Dynamic Epistemic Logic ◽  
Epistemic States ◽  
Planning Algorithm ◽  
Implicit Coordination ◽  
Human Robot Collaboration ◽  
Refinement Algorithm ◽  
Partition Refinement

Epistemic planning based on Dynamic Epistemic Logic (DEL) allows agents to reason and plan from the perspective of other agents. The framework of DEL-based epistemic planning thereby has the potential to represent significant aspects of Theory of Mind in autonomous robots, and to provide a foundation for human-robot collaboration in which coordination is achieved implicitly through perspective shifts. In this paper, we build on previous work in epistemic planning with implicit coordination. We introduce a new notion of indistinguishability between epistemic states based on bisimulation, and provide a novel partition refinement algorithm for computing unique representatives of sets of indistinguishable states. We provide an algorithm for computing implicitly coordinated plans using these new constructs, embed it in a perceive-plan-act agent loop, and implement it on a robot. The planning algorithm is benchmarked against an existing epistemic planning algorithm, and the robotic implementation is demonstrated on human-robot collaboration scenarios requiring implicit coordination.

scholarly journals Discovery of Large Disjoint Motif in Biological Network using Dynamic Expansion Tree

2018 ◽  
Author(s):  
Sabyasachi Patra ◽  
Anjali Mohapatra
Keyword(s):  
Biological Networks ◽  
Time Complexity ◽  
Motif Discovery ◽  
Large Scale ◽  
Graph Isomorphism ◽  
Network Motifs ◽  
Protein Protein Interaction ◽  
Target Network ◽  
Discovery Algorithms

AbstractNetwork motifs play an important role in structural analysis of biological networks. Identification of such network motifs leads to many important applications, such as: understanding the modularity and the large-scale structure of biological networks, classification of networks into super-families etc. However, identification of network motifs is challenging as it involved graph isomorphism which is computationally hard problem. Though this problem has been studied extensively in the literature using different computational approaches, we are far from encouraging results. Motivated by the challenges involved in this field we have proposed an efficient and scalable Motif discovery algorithm using a Dynamic Expansion Tree (MDET). In this algorithm embeddings corresponding to child node of expansion tree are obtained from the embeddings of parent node, either by adding a vertex with time complexity O(n) or by adding an edge with time complexity O(1) without involving any isomorphic check. The growth of Dynamic Expansion Tree (DET) depends on availability of patterns in the target network. DET reduces space complexity significantly and the memory limitation of static expansion tree can overcome. The proposed algorithm has been tested on Protein Protein Interaction (PPI) network obtained from MINT database. It is able to identify large motifs faster than most of the existing motif discovery algorithms.

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