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Well, apparently nobody knows how to enumerate directed animals according to the number of edges. It is an open question of combinatorics. The following table from "Directed Animals on Two Dimensional Lattices" article by A. R. Conway, R. Brak and A. J. Guttmann presents results for n<40 **Number of bond animals on the square lattice...** ``` 1 1 2 2 3 5 4 14 5 42 6 130 7 412 8 1326 9 4318 10 14188 11 46950 12 156258 13 522523 14 1754254 15 5909419 16 19964450 17 67618388 18 229526054 19 780633253 20 2659600616 21 9075301990 22 31010850632 23 106100239080 24 363428599306 25 1246172974048 26 4277163883744 27 14693260749888 28 50516757992258 29 173812617499767 30 598455761148888 31 2061895016795926 32 7108299669877836 33 24519543126693604 34 84623480620967174 35 292204621065844292 36 1009457489428859322 37 3488847073597306764 38 12063072821044567580 39 41725940730851479532 40 144383424404966638976 ```
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How software verification paradigm deals with Thompson hack?
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##To take away:## - This paper is about a slight improvement of the $k$-clique Algorithm of Chiba and Nishizeki - The performance in practice on sparse graphs is impressive - The parallelization is non-trivial and the speedup is nearly optimal up to 40 threads - Authors generate a stream of k-cliques to compute "compact" subgraphs - A parallel C code is available here: https://github.com/maxdan94/kClist ##Suggestions to extend this work:## - Can we find a node ordering better than the core ordering? - Generate a stream of $k$-cliques to compute other quantities? - Generalize the algorithm to $k$-motifs? - Parallelization on higher order $k$-cliques if more threads are available?
Slides of the talk: https://drive.google.com/file/d/15MVJ2TzkdsHcyF6tE4VeYQqH8bU0kzDE/view
> Another extension: can we guarantee a given order on the output stream? that it is uniformly random, for instance? I think that this is a very interesting and open question! I have tried to generate a stream of k-cliques such that the order is random by modifying the kClist algorithm. But I was not able to do so. I wanted to do that in order to optimize a function depending on the k-cliques using stochastic gradient descent. I have found that using a random ordering lead to a faster convergence than using the order the k-cliques are outputed by the kClist algorithm. Here is what I've tried: - If you have enough RAM, then you can of course store all k-cliques and do a [random permutation](https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle). But, since you mention "steam", I do not think that this is the case for you. - You can use another node-ordering (different from the core-ordering) to form the DAG. You can use, for instance, a random node ordering. You may lose the theoretical upperbound on the running time, but you will see that, in practice, the algorithm is still very fast (say twice slower than with the core ordering (but this depends on the input graph and k, you may also find some settings where it is actually faster than with the core ordering)). The order the k-cliques are stream will then change, but it will not be uniform at random. - Once you have formed the DAG using the node ordering (core ordering or any other ordering), you do not need to process the nodes in that same order. You can use another random ordering for that. It will add some randomness in the stream, but the order will still not be uniform at random. Please let me know if you have any better ideas.
> Another extension: can we guarantee a given order on the output stream? that it is uniformly random, for instance? One possible way to do is using a buffer, however, it remains non uniform. A buffer of size n/100 can be filled at first using the first n/100 output. Afterwards, one K-clique is randomly selected to be « outputted » and replaced with a new k-clique. The larger the buffer, the closer the ouput will be to a uniformly random output.
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