MACE: MAximal Clique Enumerator, ver 2.0

10/July/2005 Takeaki Uno e-mail:uno@nii.jp,
homepage: http://research.nii.ac.jp/~uno/index.html

This program is available for only academic use, basically. Anyone can modify this program, but he/she has to write down the change of the modification on the top of the source code. Neither contact nor appointment to Takeaki Uno is needed. If one wants to re-distribute this code, do not forget to refer the newest code, and show the link to homepage of Takeaki Uno, to notify the news about codes for the users. For the commercial use, please make a contact to Takeaki Uno.

ProblemDefinition
Usage
Input/OutputFileFormat
ransformingOtherGraphFileFormat
Performance
AlgorithmandImplementationIssue
Introduction:GraphsandCliques
Acknowledgments
References

Problem Definition

For a graph G=(V,E) of vertex set V and edge set E, a clique is a subset of V such that any two its vertices are connected by an edge of E. A clique is called maximal if it is included in no other clique. This program enumerates all maximal cliques of the given graph, or enumerates all cliques of the give graph. A subgraph is called bipartite clique if it is a complete bipartite graph, i.e., the vertex set of the subgraph can be partitioned into two groups such that no edge connect two vertices from the same group, and there is always an edge connecting two vertices from one group and the other group. The bipartite clique enumeration problem can also be solved by clique enumeration, but one should use LCM for the problem to get better performance.

Usage

The program is written in C code (gcc). It uses only the basic library, so you can compile it in any environment. To compile the program, first put all the source files in a directly, and just execute

% make

Then, you can execute "mace". The format of the input parameter is,

mace MCqSs [options] input-filename [output-filename]

MACE is executed with at least two parameters. The first parameter is composed of commands, given by a combination of letters. The meaning of the letters are:

C: enumerate cliques of the give graph
M: enumerate maximal cliques of the given graph
q: do not output the reports of the execution, such as size of input graph,
to the standard output
S: show progress, by outputting "-----" for each 10,000 cliques
s: terminate after finding 1,000,000 cliques

The second parameter is the name of the input file, and the third parameter is the name of output file. You can omit the output file name. In this case, the program only counts the numbers of cliques to be output, classified by their sizes. Note that the name of the input file can not start with '-'. If the output file name is "-", the solutions will be output to the standard output.

Between the first parameter and the second parameter, you can give some options as follows.

-l [num] :output cliques with size at least [num]
-u [num] :output cliques with size at most [num]
-S [num]:stop after outputting [num] solutions
-, [char]: give the separator of the numbers in the output
the numbers in the output file are separated by the given
character [char].
-Q [filename]: replace the output numbers
according to the permutation table written in the file of
[filename], replace the numbers in the output. The numbers in the
file can be separated by any non-numeric character such as newline
character.

For example, by executing

mace C -l 5 -u 7 g100.grh clq.out

the program outputs all cliques of sizes from 5 to 7 (including both size 5 and size 7) in the "g100.grh" to the file "clq.out"

Example of the execution)

mace MS g15.grh clq.out
(output maximal cliques in "g15.grh" to "clq.out". Show the progress
during the execution)

mace Cqs -l 5 g15.grh clq.out
(output cliques in "g15.grh" of at least size 5 to "clq.out". Stop
if over 1,000,000 cliques are found.)

Input/Output File Format

The input graph, with n vertices, is considered to be composed of vertices from 0 to n-1. The cliques found are output to the output file specified by user, represented by a sequence of numbers corresponding to the vertices of the clique. One line of the output file is for one clique. The numbers in each line is separated by " ", and by giving "-," option we can change the separator. At the termination of the program, it outputs the number of cliques found, and the number of cliques classified by their sizes. For example, if there are cliques {0,1}, {2}, {0,1,3}, {1,2}, the output to the standard output will be

-----------
#vertices=??? #edges ???? <= numbers of vertices and edges
4 <= total number
0 <= number of cliques of size 0
1 <= number of cliques of size 1
2 <= number of cliques of size 2
1 <= number of cliques of size 3
-----------

and the output file will be

-----------
0,1
2
0,1,3
1,2
[EOF]
-----------

If q is given in the parameter, then "#vertices, #edges" is not printed. If output file name is not given, then no output file is generated.

The output cliques are not sorted. If you want to sort it, use the script "sortout.pl". The usage is just,

% sortout.pl < input-file > output-file

"input-file" is the name of file to which mace output, and the sorted output will be written in the file of the name "output-file". The vertices of each clique will be sorted in the increasing order of indices, and all the cliques (lines) will be also sorted, by the lexicographical order (considered as a string). (Actually, you can specify separator like sortout.pl ",").

==== The format of the input file =====

Each following i-th line is the list of vertices adjacent to vertex i-1, so each line is: (vertex), (vertex), ... Any non-numeric letter (except for newline and end-of-file) is allowed to be used for the separator. Each vertex has to be ranged from 0 to (#vertices-1).

For an edge connecting vertices u and v, we do not need to write both "v in the (u-1)th line" and "u in the (v-1)th line". We need just one, "v in the (u-1)th line" or "u in the (v-1)th line If you write both, an edge will be counted doubly, and mace does not work correctly.

Example) a graph with edges (1,0), (2,0), (1,3), (2,3), (3,4), (0,4):

1,2
3

2 4
0
[EOF]

ransforming Other Graph File Format

For the use of other formats for input graph files, we have several scripts. We explain the functions of the Perl scripts in the following.

- compgrh.pl [b] [separator] < input-file > output-file
Write to output-file the complement graph of the graph read from the input-file. If you specify "b" option, then the input-file is regarded as a bipartite graph.

Ex.)
% compgrh.pl b "," < test15.grh > test2.grh

- transnum.pl table-file [separator] < input-file > output-file
Read file from standard input, and give a unique number to each name written by general strings (such as ABC, ttt), and transform every string name to a number, and output it to standard output. The mapping from string names to numbers is output to table-file. The default character for the separator of the string names is " "(space). It can be changed by giving a character for the option [separator]. For example, A,B is a string name, if the separator is space, but if we set the separator to ",", it is regarded as two names A and B. This is executed by "transnum.pl table-file "," < input-file...".

- untransnum.pl table-file < input-file > output-file
According to the table-file output by transnum.pl, un-transform numbers to string names. The output of the program is composed of numbers, thus it is used if we want to transform to the original string names. It reads file from standard output, and output to the standard output.

- transgrh.pl [Bb] [separator] < input-file > output-file
Transform a file in the format of that every line writes two end vertices of an edge, to the format of this program. For example, the file

0,2
0,1
1,4
3,4
1,3
[EOF]

, representing the graph with edge (0,2), edge (0,1) and..., is transformed to

1,2
3,4

4
[EOF]

If the name of vertices are written in general strings, transform them to numbers by transnum.pl before the execution. When we give parameter D or d, the input graph is regarded as a directed graph. If we give d, the first number is the origin of a directed edge, and if we give D, the second number is the origin. Bipartite graph can be transformed by giving options b or B. Suppose that we have a bipartite graph with vertex sets A and B both indices start from 0. If we transform it by the above way, vertex i in A and i in B are considered to be the same vertex. By giving b, the second number in each row is added by a constant so that no two vertices have the same index. In the case of B, the first number is added. When we transform the above graph by (transgrh.pl b "," < input > output), we have a graph in the format,

5,6
7,8

8
[EOF]

- sortgrhid
Transform a graph file format such that each row corresponds to a vertex. The first number of each row is the ID of the vertex, and the following numbers are the vertex ID's to which the vertex adjacent. The script sorts the row according to the ID's, and remove the ID from the file. For example, the file,

-------------
4,2,3
1,2
0,1,3,4
3
2,3
[EOF]
-------------

is transformed to

-------------
1,3,4
2
3

2,3
[EOF]
-------------

If the ID's are given in general strings, use transnum.pl before the execution.

- example of the usage: when transform the file test.grh which is edge list

format with general string vertex names with separator ",", and enumerate
chordless cycles:

transnum.pl table "," < test.grh > tmp.grh transgrh.pl < tmp.grh > tmp2.grh
mace M tmp2.grh out
untransnum.pl table < out > out2

Performance

This algorithm is polynomial delay, taking at most O(|V|) time for each clique in the case of clique enumeration, and O(|V||E|) time for each maximal clique in the case of maximal clique enumeration, where |V| and |E| are the number of vertices, and the number of edges in the input graph. Practical computation time is constant for each clique/maximal clique, and finds about 1,000,000 cliques per second in the case of clique enumeration, and fine about 100,000 maximal cliques per second in the case of maximal clique enumeration, if the graph is sparse. As increase the density of the graph, the computation time increases almost linearly.

Algorithm and Implementation Issue

Let G=(V,E) be a graph with vertex set V=(1,...,n) and edge set defined on V. Let tail(K) be the largest vertex in K.
Enumeration of cliques is actually easy. Since any subset of a clique is also a clique, we can find any clique by iteratively adding vertices to the emptyset. This yields a depth-first backtracking algorithm; In each iteration, for each vertex v not in the current clique K, if K\cup {v} is a clique then recursively call the iteration to enumerate the cliques including K\cup {v}. To avoid duplications, we add only vertices larger than tail(K).

==== squeeze the candidates to be add ====
To execute the above operations quickly, we maintain the vertex set CAND(K) which is the set of vertices adjacent to all the vertices in K. K\cup {v} is a clique if and only if v is in CAND(K), so we have to generate recursive calls for the vertices in CAND(K). When K is the emptyset, then CAND(K) is V, it means K\cup {v} is a clique for any vertex v. Denote by N(v) the set of vertices adjacent to v. Then CAND(K\cup {v}) is the intersection of CAND(K) and N(v). By having CAND(K) and N(v) in sorted order in the memory, we can take the intersection in linear time in the sizes of them. Moreover, in each iteration, we need only vertices larger than tail(K), thus in the bottom of the recursion, the vertices to be maintained are so few. The bottom of the recursion dominates the total computation time in usual enumeration algorithms, thus clique enumeration is usually done very quickly.

==== maximal clique enumeration by reverse search ====
If we use the above algorithm to enumerate maximal cliques, by outputting only the maximal cliques among all cliques, we need very long time compared to the number of maximal cliques. Thus, we use another algorithm, search method. For a clique K, let C(K) be the lexicographically minimum one among the maximal cliques including K. A set A={a_1,...,a_n} is said to be lexicographically smaller than a set B={b_1,...,b_m} if and only if a_1=b_1,...,a_k=b_k, and a_{k+1} < b_{k+1} holds for some k, or B\subseteq A. We note that k can be 0. We denote by K(i) the set of vertices in K less than i. For a maximal clique K, we define the core index core_i(K) of K by the minimum vertex satisfying C(K) = C(K(i)). Using these terms we define the parent of K, not equal to C(emptyset), by the maximal clique C(K(core_(i))). The parent is uniquely defined, and lexicographically smaller than K. Thus, the binary relation "parent-child" induces a directed tree, rooted at the the lexicographically minimum maximal clique. Thus, starting from the lexicographically minimum maximal clique, and perform a depth first search on the tree, we can enumerate all maximal cliques. The depth first search needs an algorithm for finding all children of a given maximal clique, and it is enough, since what we have to do is recursively find children of maximal cliques.

We can prove that K is obtained from the parent P(K) of K by adding core_i(K) to P(K), remove vertices not adjacent to core_i(K), i.e., K = C(P(K)\cap N(core_i(K))\cup {core_i(K)}). Conversely, for a maximal clique K', K is a child of K' only if K = C(K'\cap N(v)\cup {v}). To check whether K is a child of K' or not, we compute the parent of K, and if its parent is K', then K is a child of K'.

==== characterization of children ====
For a maximal clique K = C(K'\cap N(v)\cup {v}) obtained from K', the condition that K is a child of K' can be characterized in the following way so that we can check it efficiently.
(i) K(v) = K'(v)\cap N(v)
(ii) for any u in K(u), there is no vertex w<u not in K'(v) such that
u is adjacent to all vertices in K(v) \cup K'(u).
We can see that if either condition is violated, the parent of K is not K'.

==== sweep pointer method ====
To check this conditions, we use sweep pointer method. (i) is violated when there is a vertex u<v adjacent to all vertices in K'(v)\cap N(v) \cup {v}. We trance N(v) in the increasing order of vertices, and when we meet a vertex u not in K'(v), check whether u is adjacent to all vertices in K'(v)\cap N(v) or not. This can be done by tracing N(w) for each vertex w in K'(v)\cap N(v). Notice that to check the next vertex in N(v), we can start the trance of the N(w)'s from the positions which we terminated the trace for checking it for u. This tracing method, called sweep pointer does not increase the computation time much since the time complexities are the same, and usually we check only few items and conclude that the maximal clique is not a child. Thus, we can check it very quickly. The second condition can be checked in the same way, but notice that we have to the vertices checked one-by-one to the vertex set to which the next vertex has to be adjacent to all of them. The two checks can be done simultaneously so that we do not have to scan the N(w)'s twice.

Moreover, when the size of clique is small, we can use bit operations so that we can operate 32 or 64 bits at once. Suppose that the size of the current maximal clique K is less than 32. We give ID's 0,1,2,4,...,2^{frq(I)} to each vertex. Let the bit-mark b(S) of a subset S of K be the sum of ID's of the vertices in S. For each vertex u adjacent to at least one vertex in K, we compute b(N(u)\cap K). Then, a vertex u is adjacent to all vertices in K\cap N(v), if and only if b(N(u)) \cap b(N(v)) = b(N(v)). Here \cap means the and operation for bit string, and two numbers are regarded as bit-strings. This operation can be done in very few steps, thus we can accelerate the computation for the checking of the two conditions. When we go to a child of K and check the conditions for finding the children of the child, we re-use the bit-mark by replacing the ID's of removed vertices and newly added vertices, and updates the bit-marks. In practice it accelerates the speed of the algorithm 2 or 3 times when the data set is sparse and the minimum support is small, say 10.

Introduction: Graphs and Cliques

A graph is an object, composed of vertices, and edges such that each edge connects two vertices. The following is an example of a graph, composed of vertices {0,1,2,3,4,5} and edges connecting 0 and 1, 1 and 2, 0 and 3, 1 and 3, 1 and 4, 3 and 4.

0-1-2
|/|
3-4

A clique is a set of vertex in the graph. For example, the vertices 0, 1, and 3 compose a clique. A vertex itself is a clique, and two vertices connected by an edge is also a clique. A maximal clique is a clique included in no other clique. For example, vertices 0 and 1 compose a clique, but it is not maximal clique. vertices 0, 1, and 3 compose a clique, and it is also a maximal clique.

Clique is used widely for modeling a kind of related objects. Enumerating all cliques is a popular method in database analysis and solving the models. We show some example as follows.

==== Finding Web Communities ===
Consider a web network graph, in which vertices are web sites, and two vertices are connected by an edge if the corresponding sites are linked. Then, a clique of this web network graph can be considered to compose a community, since they have link in any pair of its members. Finding web communities by hand is not an easy task. By enumerating the cliques in the web network graph, we can automatically find candidates and seeds of communities.

==== Finding Clusters ====
Suppose that we have data of collection of objects, and we want to find groups which we can consider that the members of the group are similar, or have common features. Such a group is called a cluster. Consider a relation graph such that two objects similar to each other is connected. Then, any pair of members of a clique are similar, thus a clique should be totally included in a cluster. Therefore, cliques are used as seeds of clusters. By joining cliques, or appending a clique iteratively, clusters can be found. For the purpose maximal cliques are good to append, and maximal clique enumeration does a big help.

Acknowledgments

We gratefully thank to Kazuhisa Makino of Tokyo University, the co-author of the paper of mace. We also thank to Zhiao Shi for bug reports. We also thank to Dr. Krister Swenson of University of Ottawa for giving an advice to improve the readme file.

References

 S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM Journal of Computing, 6, pp. 505-517, 1977.

 Kazuhisa Makino and Takeaki Uno. New algorithms for enumerating all maximal cliques. In Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT 2004), pp. 260-272. Springer-Verlag, 2004.

 Etsuji Tomita, Akira Tanaka, and Haruhisa Takahashi. The worst-case time complexity for generating all maximal cliques and computational experiments. Theoretical Computer Science, 363, pp. 28-42, 2006.