Edge Connectivity through Boost Graph Library

After two weeks, we have managed to interface Boost and Sagemath!

However, the interface was not as simple as it seemed. The main problem we found is the genericity of Boost: almost all Boost algorithms work with several graph implementations, which differ in the data structures used to store edges and vertices. For instance, the code that implements breadth-first search works if the adjacency list of a vertex v is a vector, a list, a set, etc. This result is accomplished by using templates [1]. Unfortunately, the only way to interface Sagemath with C++ code is Cython, which is not template-friendly, yet. In particular, Cython provides genericity through fused types [2], whose support is still experimental, and which do not offer full integration with templates [3-5].

After a thorough discussion with David, Nathann, and Martin (thank you very much!), we have found a solution: for the input, we have defined a fused type "BoostGenGraph", including all Boost graph implementations, and all functions that interface Boost and Sagemath use this fused type. This way, for each algorithm, we may choose the most suitable graph implementation. For the output, whose type might be dependent on the input type, we use C++ to transform it into a "standard" type (vector, or struct).

We like this solution because it is very clean, and it allows us to exploit Boost genericity without any copy-paste. Still, there are some drawbacks:
1) Cython fused types do not allow nested calls of generic functions;
2) Boost graphs cannot be converted to Python objects: they must be defined and deleted in the same Cython function;
3) No variable can have a generic type, apart from the arguments of generic functions.

These drawbacks will be overcome as soon as Cython makes templates and generic types interact: this way, we will be able create a much stronger interface, by writing a graph backend based on Boost, so that the user might create, convert, and modify Boost graphs directly from Python. However, for the moment, we will implement all algorithms using the current interface, which already provides genericity, and which has no drawback if the only goal is to "steal" algorithms from Boost.

As a test, we have computed the edge connectivity of a graph through Boost: the code is available in ticket 18564 [6]. Since the algorithm provided by Sagemath is not optimal (it is based on linear programming), the difference in the running time is impressive, as shown by the following tests:

sage: G = graphs.RandomGNM(100,1000)
sage: %timeit G.edge_connectivity()
100 loops, best of 3: 1.42 ms per loop
sage: %timeit G.edge_connectivity(implementation="sage")
1 loops, best of 3: 11.3 s per loop

sage: G = graphs.RandomBarabasiAlbert(300,3)
sage: %timeit G.edge_connectivity(implementation="sage")
1 loops, best of 3: 9.96 s per loop
sage: %timeit G.edge_connectivity()
100 loops, best of 3: 3.33 ms per loop

Basically, on a random Erdos-Renyi graph with 100 vertices and 1000 edges, the new algorithm is 8,000 times faster, and on a random Barabasi-Albert graph with 300 nodes and average degree 3, the new algorithm is 3,000 times faster! This way, we can compute the edge connectivity of much bigger graphs, like a random Erdos-Renyi graph with 5,000 vertices and 50,000 edges:

sage: G = graphs.RandomGNM(5,000, 50,000)
sage: %timeit G.edge_connectivity()
1 loops, best of 3: 16.2 s per loop

The results obtained with this first algorithm are very promising: in the next days, we plan to interface several other algorithms, in order to improve both the number of available routines and the speed of Sagemath graph library!

[1] https://en.wikipedia.org/wiki/Template_%28C%2B%2B%29
[2] http://docs.cython.org/src/userguide/fusedtypes.html
[3] https://groups.google.com/forum/#!topic/cython-users/qQpMo3hGQqI
[4] https://groups.google.com/forum/#!searchin/cython-users/fused/cython-users/-7cHr6Iz00Y/Z8rS03P7-_4J
[5] https://groups.google.com/forum/#!searchin/cython-users/fused$20template/cython-users/-7cHr6Iz00Y/Z8rS03P7-_4J
[6] http://trac.sagemath.org/ticket/18564

Performance Comparison of Different Graph Libraries

As promised in the last post, I have compared the performances of several graph libraries, in order to choose which ones should be deployed with Sagemath. Here, I provide the main results of this analysis, while more details are available on my website (see also the links below).
The libraries chosen are the most famous graph libraries written in Python, C, or C++ (I have chosen these languages because they are easier to integrate in Sagemath, using Cython). Furthermore, I have excluded NetworkX, which is already deployed with Sagemath.
First of all, I have to enforce that no graph library comparison can be completely fair, and also this comparison can be criticized, due to the large amount of available routines, to the constant evolution of libraries, and to many small differences in the outputs (for instance, one library might compute the value of a maximum s-t flow, another library might actually compute the flow, and a third one might compute all maximum flows). Despite this, I have tried to be as fair as possible, through a deeper and more detailed analysis than previous comparisons (https://graph-tool.skewed.de/performance, http://www.programmershare.com/3210372/, http://arxiv.org/pdf/1403.3005.pdf).
The first comparison deals with the number of algorithms implemented. I have chosen a set of 107 possible algorithms, trying to cover all possible tasks that a graph library should perform (avoiding easy tasks that are common to all libraries, like outputting the number of nodes, the number of edges, the neighbors of a node, etc). In some cases, two tasks were collapsed in one, if the algorithms solving these tasks are very similar (for instance, computing a maximum flow and computing a minimum cut, computing vertex betweenness and edge betweenness, etc).
The number of routines available for each library is plotted in the following chart, and a table containing all features is available in HTML or as a Google Sheet.

The results show that Sagemath has more routines than all competitors (66), closely followed by igraph (62). All other libraries are very close to each other, having about 30 routines each. Furthermore, Sagemath could be improved in the fields of neighbor similarity measures (assortativity, bibcoupling, cocitation, etc), community detection, and random graph generators. For instance, igraph contains 29 routines that are not available in Sagemath.

The second comparison analyzes the running-time of some of the algorithms implemented in the libraries. In particular, I have chosen 8 of the most common tasks in graph analysis: computing the diameter, computing the maximum flow between two vertices, finding connected components and strongly connected components, computing betweenness centrality, computing the clustering coefficient, computing the clique number, and generating a graph with the preferential attachment model. I have run each of these algorithms on 3 inputs, and I have considered the total execution time (excluding the time needed to load the graph). More details on this experiment are available here, and the results are also available in a Google Sheet.
In order to make the results more readable, I have plotted the ratio between the time needed by a given library and the minimum time needed by any library. If an algorithm was not implemented, or it needed more than 3 hours to complete, the corresponding bar is not shown.

Overall, the results show that NetworKit is the fastest library, or one of the fastest, in all routines that are implemented (apart from the generation of preferential attachment graphs, where it is very slow). Boost graph library is very close to NetworKit, and it also contains more routines. Also Sagemath is quite efficient in all tasks, apart from the computation of strongly connected components and the generation of a preferential attachment graph, where it needed more than 3 hours. However, in the latter case, the main problem was not speed but memory consumption.

In conclusion, Sagemath can highly benefit from the possibility of using algorithms from other libraries. First of all, it might improve the number of algorithms offered, especially by including igraph, and it also might improve its performance, by including Boost, NetworKit, or other fast graph libraries.

Comparison of Graph Libraries

Many times, people asked me "Which is the best available graph library?", or "Which graph library should I use to compute this, or that?".
Well, personally I love to use Sage, but there are also several good alternatives. Then, the question becomes "How could we improve Sage, so that people will choose it?".

In my opinion, graph libraries are compared according to the following parameters:

1. simplicity and documentation: people have little time, and the faster they learn how to use the library, the better;
2. number of routines available;
3. speed: sometimes, the input is very big, and the algorithms take much time to finish, so that a fast implementation is fundamental.

While it is very difficult to measure the first point, the others can be compared and improved. For this reason, in order to outperform other libraries, we should implement new features, and improve existing ones. You don't say!

However, this answer is not satisfactory: in principle, we could add all features available in other libraries, but this is a huge translational work, and while we are doing this work the other libraries will change, making this effort a never-ending story.

My project proposes an alternative: cooperating instead of competing. I will try to interface Sage with other libraries, and to use their algorithms when the Sage counterpart is not available, or less efficient. This way, with an affordable amount of work, we will be able to run all algorithms available in the best graph libraries!

As a first step, I have compared all the most famous C, C++, and Python graph libraries according to points 2 and 3, in order to choose which libraries should be included. The next posts will analyze the results of this comparison.

Google Summer of Code: let's start!

This blog will follow my Google Summer of Code project, entitled Performance Improvements for the Graph Module of Sagemath. The complete project is available here, and related documents with partial results will be available on the same website.
In this first post, I would like to thank my mentor David Coudert and Nathann Cohen, who helped me a lot in writing this project and understanding how the graph module of Sagemath works.
With their help, and with the help of the Sage community, I hope it will be a useful and funny work! Let's start!