Maximum clique algorithm python

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Maximum clique algorithm python

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Algorithms/Find maximum/Python method 1

If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again. CLIQUE is a subspace clustering algorithm using a bottom up approach to find all clusters in all subspaces. It starts examining one dimensional subspaces and merges them to compute higher dimensional ones. It uses the downward-closure property to achieve better performance by considering subspaces only if all of its k-1 dimensional projection contains cluster s.

In the context of the algorithm, clusters are dense regions. It partitions the feature space into xsi equal parts in each dimension, where the intersection of one interval from each dimension is called unit. If a unit contains more than tau parts of all the data points, then it is a dense unit. Clusters are the maximal sets of connected dense units. CLIQUE is used not only to detect clusters but to identify subspaces which contain clusters at the same time.

It is a fast and uses a relatively simple approach of finding clusters. It was introduced in motivated by finding an automatic subspace algorithm without requiring the user to guess subspaces which might contain interesting clusters. The quality of the results are highly dependent of the input parameters xsi and tau. As the density threshold is constant over all dimensionality, it would require the data points to have the same density in high and low dimensionality to find interesting clusters in all subspaces with the same tau.

The location of the grid lines also make a huge difference to the resulting clusters. The implementation is in Python language, all additional libraries used are listed in requirements. To install them in an environment, use the following command:. Running the script without parameters runs a default clustering on one of datasets in the folder. To run it with your settings you can use the following parameters:.

Running Clique. In all used evaluation methods higher means better performance. Detailed description of evaluation scores is not in scope of this document, for further information please visit scikit-learn documentation. Skip to content. Dismiss Join GitHub today GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. Sign up.

Python Branch: master. Find file. Sign in Sign up. Go back. Launching Xcode If nothing happens, download Xcode and try again.This post addresses the following questions:. High dimensional data consists in input having from a few dozen to many thousands of features or dimensions. This is a context typically encountered for instance in bioinformatics all sorts of sequencing data or in NLP where the size of the vocabulary if very high.

High dimensional data is challenging because:. Subspace clustering is a technique which finds clusters within different subspaces a selection of one or more dimensions. The underlying assumption is that we can find valid clusters which are defined by only a subset of dimensions it is not needed to have the agreement of all N features. For example, if we consider as input patient data observing the gene expression level we can have more than featuresa cluster of patients suffering from Alzheimer can be found only by looking at the expression data of a subset of genes, or stated differently, the subset exists in D.

Stated differently, subspace clustering is an extension of traditional N dimensional cluster analysis which allows to simultaneously group features and observations by creating both row and column clusters. The resulting clusters may be overlapping both in the space of features and observations. Another example is shown in the figures below, taken from the paper. We can notice that points from 2 clusters can be very close which can confuse many traditional clustering algorithms analyzing the entire feature space.

Further more, we can see that subspace clustering manages to find a subspace dimensions a and c where the expected clusters are easily identifiable.

Based on the search strategy, we can differentiate 2 types of subspace clustering, as shown in the figure below: bottom up approaches start by finding clusters in low dimensional 1 D spaces and iteratively merging them to process higher dimensional spaces up to N D. Top down approaches find clusters in the full set of dimensions and evaluate the subspace of each cluster.

The figure below, taken from the same paper provides an overview of the most common subspace clustering algorithms. In order to better understand subspace clustering, I have implemented the Clique algorithm in python here. In a nutshell, the algorithm functions as follows: for each dimension feature we split the space in nBins input parameter and for each bin we compute the histogram number of counts. We only consider dense unitsthat is the bins with a count superior to a threshold given as second input parameter.

A dense unit is characterized by the following:. In my implementation I have generated 4 random clusters in a 2D space and I have chosen 8 bins and 2 points as minimal density threshold. The figure below shows the resulting grid applied to the input space. The intuition behind the clique algorithm is that clusters existing in a k dimensional space can also be found in k We start from 1D and for each dimension we try to find the dense bins.

If 2 or more dense bins are neighbors, we merge them into one bigger bin.In an undirected graph, a clique is a complete sub-graph of the given graph.

Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph. The Max-Clique problem is the computational problem of finding maximum clique of the graph.

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Max clique is used in many real-world problems. In this graph, a clique represents a subset of people who all know each other. To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming for networks comprising more than a few dozen vertices. Max-Clique problem is a non-deterministic algorithm. In this algorithm, first we try to determine a set of k distinct vertices and then we try to test whether these vertices form a complete graph.

There is no polynomial time deterministic algorithm to solve this problem. This problem is NP-Complete. Take a look at the following graph. Here, the sub-graph containing vertices 2, 3, 4 and 6 forms a complete graph. Hence, this sub-graph is a clique. Previous Page. Next Page. Previous Page Print Page.Summary MaxCliqueDyn is a fast exact algorithm for finding a maximum clique in an undirected graph described in Ref.

A clique is a fully connected subgraph of a graph and a maximum clique is the clique with the largest number of vertices in a given graph.

Maximum clique algorithms differ from maximal clique algorithms e. The maximal search is for all maximal cliques in a graph cliques that cannot be enlargedwhile the maximum clique algorithms find a maximum clique a clique with the largest number of vertices. This makes maximum clique algorithms about an order of magnitude faster. The old source code is still available. The MaxCliqueDyn algorithm is described in Ref.

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GitLab project with the latest version of the source code is available here. Improvement The MaxCliqueDyn algorithm described in [1] and implemented in mcqd. Vertices with their colors below Kmin can remain in their original order.

This idea consistently reduces the number of steps needed to find a maximum clique as well as the time required to find a maximum clique. This heuristics increases overall performance of the algorithm on a large number of DIMACS and random graphs, and can be tuned to specific graphs by changing the Tlimit parameter. Installation Instructions Unzip: unzip mcqd. The maximum clique is now in the qmax array, and its size is in qsize.

For detailed example see mcqd. It was developed with purpose of quickly comparing protein structures. Protein surfaces are first represented as protein graphs and then an associative graph or product graph G1 X G2 is constructed from two compared protein graphs.

A maximum clique in such an associative graph corresponds to the maximum common subgraph of the graphs G1 and G2which translates to the biggest similarity in topology or properties of the compared proteins. Maximum clique algorithms are preffered to maximal clique algorithms when comparing protein structures, because knowing all smaller cliques is not necessary when comparing rigid 2D or 3D objects.

Only the largest one is important. The MaxCliqueDyn algorithm is also considerably faster than the algorithms of Tomita et al. Reference [1] Janez Konc and Dusanka Janezic. An improved branch and bound algorithm for the maximum clique problem. Maximum Clique Algorithm.The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. It is based on a basic algorithm MaxClique algorithm which finds a maximum clique of bounded size.

The bound is found using improved coloring algorithm. This algorithm was designed by Janez Konc and description was published in In comparison to earlier algorithms described in the published article [1] the MaxCliqueDyn algorithm is improved by an improved approximate coloring algorithm ColorSort algorithm and by applying tighter, more computationally expensive upper bounds on a fraction of the search space.

Both improvements reduce time to find maximum clique. In addition to reducing time improved coloring algorithm also reduces the number of steps needed to find a maximum clique. The pseudo code of the algorithm is:.

The ColorSort algorithm is an improved algorithm of the approximate coloring algorithm. In the approximate coloring algorithm vertices are colored one by one in the same order as they appear in a set of candidate vertices R so that if the next vertex p is non-adjacent to all vertices in the some color class it is added to this class and if p is adjacent to at least one vertex in every one of existing color classes it is put into a new color class.

The MaxClique algorithm returns vertices R ordered by their colors. Therefore, sorting those vertices by color is of no use to MaxClique algorithm. The improved coloring with ColorSort algorithm takes in consideration the above observation. Each vertex is assigned to a color class C k. ColorSort algorithm [1]. Set of vertices R can now be used as input for both approximate coloring algorithm and ColorSort algorithm.

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Using any of the two algorithms a table below is constructed. The MaxCliqueDyn algorithm is in basic MaxClique algorithm that uses ColorSort algorithm instead approximate coloring algorithm for determining color classes.

On each step of MaxClique algorithm the algorithm also recalculates the degrees of vertices in R regarding to the vertex the algorithm is currently in. These vertices are then sorted by decreasing order with respect to their degrees in graph G R. Then the ColorSort algorithm considers vertices in R sorted by their degrees in the induced graph G R rather than in G. By doing so the number of steps required to find the maximum clique is reduced to the minimum.

Even so, the overall running time of the MaxClique algorithm is not improved, because computational expense O R 2 of the determination of the degrees and sorting of vertices in R stays the same.

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MaxCliqueDyn algorithm [1]. Value T limit can be determined by experimenting on random graphs. From Wikipedia, the free encyclopedia. Categories : Graph algorithms. Hidden categories: CS1 errors: missing periodical Articles with example pseudocode.Using Pip its as easy as:.

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Once installed import the package and Initialize a graph object. Add the first two nodes and an edge between them. At this point our graph is just two connected nodes. Adding edges one at a time is pretty slow but luckily we can also add lists of nodes and lists of edges where each edge is represented by a node tuple.

Clique Problem - Intro to Algorithms

Our Graph should now look something like this. We can see a list of nodes or edges by printing these attributes of our graph. It is also possible to define nodes as strings. Most importantly each node can be assigned any number of attributes which are then stored in dictionaries. To make this data more manageable feed the output of nodes. To see the induced subgraph of that vertex set, we need to combine the above with the subgraph method.

Which will give us the following complete 3-vertex graph. Final Thoughts and Questions. There are a number of graph libraries out there for Python but I chose Networkx for its readability, ease of setup and above all for its excellent documentation. If you have any further question or wish to explore the library more please refer to the official documentation.

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maximum clique algorithm python

Creating Graphs in Python using Networkx. An intro to building your first Graph in Python. Jackson Gilkey Follow. Towards Data Science A Medium publication sharing concepts, ideas, and codes. Data Scientist. Studied at Flatiron in NYC. Towards Data Science Follow. A Medium publication sharing concepts, ideas, and codes. Write the first response. More From Medium. More from Towards Data Science. Rhea Moutafis in Towards Data Science. Taylor Brownlow in Towards Data Science.By James McCaffrey December The idea of the maximum clique problem is to find the largest group of nodes in a graph that are all connected to one another.

Take a look at the simple graph in Figure 1. The graph has nine nodes and 13 edges.

MaxCliqueDyn maximum clique algorithm

The graph is unweighted there are no priorities associated with the edges and undirected all edges are bidirectional. Nodes 2, 4 and 5 form a clique of size three. The maximum clique is the node set 0, 1, 3 and 4, which forms a clique of size four. The maximum clique problem is interesting for several reasons. It arises in many important practical problems, such as social network communication analysis, where nodes represent people and edges represent messages or relationships.

maximum clique algorithm python

And the problem uses techniques such as greedy algorithms and tabu algorithms, which are important advanced programming techniques. Having a solution to the maximum clique problem in your personal library can be a useful practical tool, and understanding the algorithm used can add new techniques to your skill set. This is the third column in a series that uses the maximum clique problem to illustrate advanced coding and testing techniques, but this column can be read without direct reference to the previous two.

In informal terms, a greedy algorithm is an algorithm that starts with a simple, incomplete solution to a difficult problem and then iteratively looks for the best way to improve the solution. The process is repeated until some stopping condition is reached. Tabu algorithms are designed to deal with this weakness.

The word tabu, sometimes spelled taboo, means forbidden. In simple terms, tabu algorithms maintain a list of forbidden data. Simple forms of tabu algorithms use a fixed prohibit time. Advanced tabu algorithms adapt the prohibit time dynamically while the algorithm executes. I have a console application that begins by loading into memory the graph corresponding to the one shown in Figure 1.

maximum clique algorithm python

Designing and coding an efficient graph data structure is a significant problem in itself. I explained the graph data structure code in my October column. The algorithm begins by selecting a single node at random and initializing a clique with that node. The algorithm then iteratively tries to find and add the best node that will increase the size of the clique. Behind the scenes, the algorithm remembers the last time each node was moved into the current solution clique or moved out from the clique.

The algorithm uses this information to prohibit adding or dropping recently used nodes for a certain amount of time. This is the tabu part of the algorithm. The algorithm restarts itself every so often when no progress has been made in finding a better larger clique.

Creating Graphs in Python using Networkx

The algorithm silently stores the solutions cliques it has previously seen. The algorithm uses that solution history information to dynamically adapt the prohibit time. If the algorithm keeps encountering the same solutions, it increases the prohibit time to discourage recently used nodes from being used. This is the adaptive part of the tabu algorithm. In most situations where a greedy algorithm is used, the optimal solution is not known, so one or more stopping conditions must be specified.

When the algorithm hits a stopping condition, the best solution found is displayed. The complete source code is available at msdn. This column assumes you have intermediate-level programming skills with a C-family language or the Visual Basic. NET language. The overall structure of the program shown in Figure 2 is presented in Figure 3. I decided to place all code in a single console application using static methods for simplicity, but you might want to modularize parts of the code into class libraries and use an object-oriented approach.

The program is less complicated than you might suspect in looking at Figure 3 because most of the methods are short helper methods.


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