## Comparison of Wang tiling solvers

12 décembre 2018 | Catégories: sage, slabbe spkg, math | View Comments

During the last year, I have written a Python module to deal with Wang tiles containing about 4K lines of code including doctests and documentation.

It can be installed like this:

sage -pip install slabbe

It can be used like this:

sage: from slabbe import WangTileSet sage: tiles = [(2,4,2,1), (2,2,2,0), (1,1,3,1), (1,2,3,2), (3,1,3,3), ....: (0,1,3,1), (0,0,0,1), (3,1,0,2), (0,2,1,2), (1,2,1,4), (3,3,1,2)] sage: T0 = WangTileSet([map(str,t) for t in tiles]) sage: T0.tikz(ncolumns=11).pdf()

The module on wang tiles contains a class `WangTileSolver` which contains
three reductions of the Wang tiling problem the first using MILP solvers,
the second using SAT solvers and the third using Knuth's dancing links.

Here is one example of a tiling found using the dancing links reduction:

sage: %time tiling = T0.solver(10,10).solve(solver='dancing_links') CPU times: user 36 ms, sys: 12 ms, total: 48 ms Wall time: 65.5 ms sage: tiling.tikz().pdf()

All these reductions now allow me to compare the efficiency of various types of solvers restricted to the Wang tiling type of problems. Here is the list of solvers that I often use.

Solver | Description |
---|---|

'Gurobi' |
MILP solver |

'GLPK' |
MILP solver |

'PPL' |
MILP solver |

'LP' |
a SAT solver using a reduction to LP |

'cryptominisat' |
SAT solver |

'picosat' |
SAT solver |

'glucose' |
SAT solver |

'dancing_links' |
Knuth's algorihm |

In this recent work on the substitutive structure of Jeandel-Rao tilings, I introduced various Wang tile sets \(T_i\) for \(i\in\{0,1,\dots,12\}\). In this blog post, we will concentrate on the 11 Wang tile set \(T_0\) introduced by Jeandel and Rao as well as \(T_2\) containing 20 tiles and \(T_3\) containing 24 tiles.

**Tiling a n x n square**

The most natural question to ask is to find valid Wang tilings of \(n\times n\) square with given Wang tiles. Below is the time spent by each mentionned solvers to find a valid tiling of a \(n\times n\) square in less than 10 seconds for each of the three wang tile sets \(T_0\), \(T_2\) and \(T_3\).

We remark that MILP solvers are slower. Dancing links can solve 20x20 squares with Jeandel Rao tiles \(T_0\) and SAT solvers are performing very well with Glucose being the best as it can find a 55x55 tiling with Jeandel-Rao tiles \(T_0\) in less than 10 seconds.

**Finding all dominoes allowing a surrounding of given radius**

One thing that is often needed in my research is to enumerate all horizontal and vertical dominoes that allow a given surrounding radius. This is a difficult question in general as deciding if a given tile set admits a tiling of the infinite plane is undecidable. But in some cases, the information we get from the dominoes admitting a surrounding of radius 1, 2, 3 or 4 is enough to conclude that the tiling can be desubstituted for instance. This is why we need to answer this question as fast as possible.

Below is the comparison in the time taken by each solver to compute all vertical and horizontal dominoes allowing a surrounding of radius 1, 2 and 3 (in less than 1000 seconds for each execution).

What is surprising at first is that the solvers that performed well in the first \(n\times n\) square experience are not the best in the second experiment computing valid dominoes. Dancing links and the MILP solver Gurobi are now the best algorithms to compute all dominoes. They are followed by picosat and cryptominisat and then glucose.

**The source code of the above comparisons**

The source code of the above comparison can be found in this Jupyter notebook. Note that it depends on the use of Glucose as a Sage optional package (#26361) and on the most recent development version of slabbe optional Sage Package.

## Wooden laser-cut Jeandel-Rao tiles

07 septembre 2018 | Catégories: sage, slabbe spkg, math | View Comments

I have been working on Jeandel-Rao tiles lately.

Before the conference Model Sets and Aperiodic Order held in Durham UK (Sep 3-7 2018), I thought it would be a good idea to bring some real tiles at the conference. So I first decided of some conventions to represent the above tiles as topologically closed disk basically using the representation of integers in base 1:

With these shapes, I created a 33 x 19 patch. With 3cm on each side, the patch takes 99cm x 57cm just within the capacity of the laser cut machine (1m x 60 cm):

With the help of David Renault from LaBRI, we went at Coh@bit, the FabLab of Bordeaux University and we laser cut two 3mm thick plywood for a total of 1282 Wang tiles. This is the result:

One may recreate the 33 x 19 tiling as follows (note that I am using
Cartesian-like coordinates, so the first list `data[0]` actually is the first
column from bottom to top):

sage: data = [[10, 4, 6, 1, 3, 3, 7, 0, 9, 7, 2, 6, 1, 3, 8, 7, 0, 9, 7], ....: [4, 5, 6, 1, 8, 10, 4, 0, 9, 3, 8, 7, 0, 9, 7, 5, 0, 9, 3], ....: [3, 7, 6, 1, 7, 2, 5, 0, 9, 8, 7, 5, 0, 9, 3, 7, 0, 9, 10], ....: [10, 4, 6, 1, 3, 8, 7, 0, 9, 7, 5, 6, 1, 8, 10, 4, 0, 9, 3], ....: [2, 5, 6, 1, 8, 7, 5, 0, 9, 3, 7, 6, 1, 7, 2, 5, 0, 9, 8], ....: [8, 7, 6, 1, 7, 5, 6, 1, 8, 10, 4, 6, 1, 3, 8, 7, 0, 9, 7], ....: [7, 5, 6, 1, 3, 7, 6, 1, 7, 2, 5, 6, 1, 8, 7, 5, 0, 9, 3], ....: [3, 7, 6, 1, 10, 4, 6, 1, 3, 8, 7, 6, 1, 7, 5, 6, 1, 8, 10], ....: [10, 4, 6, 1, 3, 3, 7, 0, 9, 7, 5, 6, 1, 3, 7, 6, 1, 7, 2], ....: [2, 5, 6, 1, 8, 10, 4, 0, 9, 3, 7, 6, 1, 10, 4, 6, 1, 3, 8], ....: [8, 7, 6, 1, 7, 5, 5, 0, 9, 10, 4, 6, 1, 3, 3, 7, 0, 9, 7], ....: [7, 5, 6, 1, 3, 7, 6, 1, 10, 4, 5, 6, 1, 8, 10, 4, 0, 9, 3], ....: [3, 7, 6, 1, 10, 4, 6, 1, 3, 3, 7, 6, 1, 7, 2, 5, 0, 9, 8], ....: [10, 4, 6, 1, 3, 3, 7, 0, 9, 10, 4, 6, 1, 3, 8, 7, 0, 9, 7], ....: [4, 5, 6, 1, 8, 10, 4, 0, 9, 3, 3, 7, 0, 9, 7, 5, 0, 9, 3], ....: [3, 7, 6, 1, 7, 2, 5, 0, 9, 8, 10, 4, 0, 9, 3, 7, 0, 9, 10], ....: [10, 4, 6, 1, 3, 8, 7, 0, 9, 7, 5, 5, 0, 9, 10, 4, 0, 9, 3], ....: [2, 5, 6, 1, 8, 7, 5, 0, 9, 3, 7, 6, 1, 10, 4, 5, 0, 9, 8], ....: [8, 7, 6, 1, 7, 5, 6, 1, 8, 10, 4, 6, 1, 3, 3, 7, 0, 9, 7], ....: [7, 5, 6, 1, 3, 7, 6, 1, 7, 2, 5, 6, 1, 8, 10, 4, 0, 9, 3], ....: [3, 7, 6, 1, 10, 4, 6, 1, 3, 8, 7, 6, 1, 7, 2, 5, 0, 9, 8], ....: [10, 4, 6, 1, 3, 3, 7, 0, 9, 7, 2, 6, 1, 3, 8, 7, 0, 9, 7], ....: [4, 5, 6, 1, 8, 10, 4, 0, 9, 3, 8, 7, 0, 9, 7, 5, 0, 9, 3], ....: [3, 7, 6, 1, 7, 2, 5, 0, 9, 8, 7, 5, 0, 9, 3, 7, 0, 9, 10], ....: [10, 4, 6, 1, 3, 8, 7, 0, 9, 7, 5, 6, 1, 8, 10, 4, 0, 9, 3], ....: [3, 3, 7, 0, 9, 7, 5, 0, 9, 3, 7, 6, 1, 7, 2, 5, 0, 9, 8], ....: [8, 10, 4, 0, 9, 3, 7, 0, 9, 10, 4, 6, 1, 3, 8, 7, 0, 9, 7], ....: [7, 5, 5, 0, 9, 10, 4, 0, 9, 3, 3, 7, 0, 9, 7, 5, 0, 9, 3], ....: [3, 7, 6, 1, 10, 4, 5, 0, 9, 8, 10, 4, 0, 9, 3, 7, 0, 9, 10], ....: [10, 4, 6, 1, 3, 3, 7, 0, 9, 7, 5, 5, 0, 9, 10, 4, 0, 9, 3], ....: [2, 5, 6, 1, 8, 10, 4, 0, 9, 3, 7, 6, 1, 10, 4, 5, 0, 9, 8], ....: [8, 7, 6, 1, 7, 5, 5, 0, 9, 10, 4, 6, 1, 3, 3, 7, 0, 9, 7], ....: [7, 5, 6, 1, 3, 7, 6, 1, 10, 4, 5, 6, 1, 8, 10, 4, 0, 9, 3]]

The above patch have been chosen among 1000 other randomly generated as the closest to the asymptotic frequencies of the tiles in Jeandel-Rao tilings (or at least in the minimal subshift that I describe in the preprint):

sage: from collections import Counter sage: c = Counter(flatten(data)) sage: tile_count = [c[i] for i in range(11)]

The asymptotic frequencies:

sage: phi = golden_ratio.n() sage: Linv = [2*phi + 6, 2*phi + 6, 18*phi + 10, 2*phi + 6, 8*phi + 2, ....: 5*phi + 4, 2*phi + 6, 12/5*phi + 14/5, 8*phi + 2, ....: 2*phi + 6, 8*phi + 2] sage: perfect_proportions = vector([1/a for a in Linv])

Comparison of the number of tiles of each type with the expected frequency:

sage: header_row = ['tile id', 'Asymptotic frequency', 'Expected nb of copies', ....: 'Nb copies in the 33x19 patch'] sage: columns = [range(11), perfect_proportions, vector(perfect_proportions)*33*19, tile_count] sage: table(columns=columns, header_row=header_row) tile id Asymptotic frequency Expected nb of copies Nb copies in the 33x19 patch +---------+----------------------+-----------------------+------------------------------+ 0 0.108271182329550 67.8860313206280 67 1 0.108271182329550 67.8860313206280 65 2 0.0255593590340479 16.0257181143480 16 3 0.108271182329550 67.8860313206280 71 4 0.0669152706817991 41.9558747174880 42 5 0.0827118232955023 51.8603132062800 51 6 0.108271182329550 67.8860313206280 65 7 0.149627093977301 93.8161879237680 95 8 0.0669152706817991 41.9558747174880 44 9 0.108271182329550 67.8860313206280 67 10 0.0669152706817991 41.9558747174880 44

I brought the \(33\times19=641\) tiles at the conference and offered to the first 7 persons to find a \(7\times 7\) tiling the opportunity to keep the 49 tiles they used. 49 is a good number since the frequency of the lowest tile (with id 2) is about 2% which allows to have at least one copy of each tile in a subset of 49 tiles allowing a solution.

A natural question to ask is how many such \(7\times 7\) tilings does there
exist? With ticket #25125 that was merged in Sage 8.3 this Spring, it is
possible to enumerate and count solutions in parallel with Knuth dancing links
algorithm. After the installation of the Sage Optional package slabbe (`sage
-pip install slabbe`), one may compute that there are 152244 solutions.

sage: from slabbe import WangTileSet sage: tiles = [(2,4,2,1), (2,2,2,0), (1,1,3,1), (1,2,3,2), (3,1,3,3), ....: (0,1,3,1), (0,0,0,1), (3,1,0,2), (0,2,1,2), (1,2,1,4), (3,3,1,2)] sage: T0 = WangTileSet(tiles) sage: T0_solver = T0.solver(7,7) sage: %time T0_solver.number_of_solutions(ncpus=8) CPU times: user 16 ms, sys: 82.3 ms, total: 98.3 ms Wall time: 388 ms 152244

One may also get the list of all solutions and print one of them:

sage: %time L = T0_solver.all_solutions(); print(len(L)) 152244 CPU times: user 6.46 s, sys: 344 ms, total: 6.8 s Wall time: 6.82 s sage: L[0] A wang tiling of a 7 x 7 rectangle sage: L[0].table() # warning: the output is in Cartesian-like coordinates [[1, 8, 10, 4, 5, 0, 9], [1, 7, 2, 5, 6, 1, 8], [1, 3, 8, 7, 6, 1, 7], [0, 9, 7, 5, 6, 1, 3], [0, 9, 3, 7, 6, 1, 8], [1, 8, 10, 4, 6, 1, 7], [1, 7, 2, 2, 6, 1, 3]]

This is the number of distinct sets of 49 tiles which admits a 7x7 solution:

sage: from collections import Counter sage: def count_tiles(tiling): ....: C = Counter(flatten(tiling.table())) ....: return tuple(C.get(a,0) for a in range(11)) sage: Lfreq = map(count_tiles, L) sage: Lfreq_count = Counter(Lfreq) sage: len(Lfreq_count) 83258

Number of other solutions with the same set of 49 tiles:

sage: Counter(Lfreq_count.values()) Counter({1: 49076, 2: 19849, 3: 6313, 4: 3664, 6: 1410, 5: 1341, 7: 705, 8: 293, 9: 159, 14: 116, 10: 104, 12: 97, 18: 44, 11: 26, 15: 24, 13: 10, 17: 8, 22: 6, 32: 6, 16: 3, 28: 2, 19: 1, 21: 1})

How the number of \(k\times k\)-solutions grows for k from 0 to 9:

sage: [T0.solver(k,k).number_of_solutions() for k in range(10)] [0, 11, 85, 444, 1723, 9172, 50638, 152244, 262019, 1641695]

Unfortunately, most of those \(k\times k\)-solutions are not extendable to a tiling of the whole plane. Indeed the number of \(k\times k\) patches in the language of the minimal aperiodic subshift that I am able to describe and which is a proper subset of Jeandel-Rao tilings seems, according to some heuristic, to be something like:

[1, 11, 49, 108, 184, 268, 367, 483]

I do not share my (ugly) code for this computation yet, as I will rather share clean code soon when times come. So among the 152244 about only 483 (0.32%) of them are prolongable into a uniformly recurrent tiling of the plane.

## A time evolution picture of packages built in parallel by Sage

16 décembre 2016 | Catégories: sage | View Comments

Compiling sage takes a while and does a lot of stuff. Each time I am wondering
which components takes so much time and which are fast. I wrote a module in my
slabbe version `0.3b2` package available on PyPI to figure this out.

This is after compiling 7.5.beta6 after an upgrade from 7.5.beta4:

sage: from slabbe.analyze_sage_build import draw_sage_build sage: draw_sage_build().pdf()

From scratch from a fresh git clone of 7.5.beta6, after running `MAKE='make
-j4' make ptestlong`, I get:

sage: from slabbe.analyze_sage_build import draw_sage_build sage: draw_sage_build().pdf()

The picture does not include the start and ptestlong because there was an error compiling the documentation.

By default, `draw_sage_build` considers all of the logs files in
`logs/pkgs` but options are available to consider only log files created in a
given interval of time. See `draw_sage_build?` for more info.

## unsupported operand parent for *, Matrix over number field, vector over symbolic ring

18 février 2016 | Catégories: sage | View Comments

Yesterday I received this email (in french):

Salut, avec Thomas on a une question bête: K.<x>=NumberField(x*x-x-1) J'aimerais multiplier une matrice avec des coefficients en x par un vecteur contenant des variables a et b. Il dit "unsupported operand parent for *, Matrix over number field, vector over symbolic ring" Est ce grave ?

Here is my answer. Indeed, in Sage, symbolic variables can't multiply with elements in an Number Field in x:

sage: x = var('x') sage: K.<x> = NumberField(x*x-x-1) sage: a = var('a') sage: a*x Traceback (most recent call last) ... TypeError: unsupported operand parent(s) for '*': 'Symbolic Ring' and 'Number Field in x with defining polynomial x^2 - x - 1'

But, we can define a polynomial ring with variables in a,b and coefficients in
the NumberField. Then, we are able to multiply `a` with `x`:

sage: x = var('x') sage: K.<x> = NumberField(x*x-x-1) sage: K Number Field in x with defining polynomial x^2 - x - 1 sage: R.<a,b> = K['a','b'] sage: R Multivariate Polynomial Ring in a, b over Number Field in x with defining polynomial x^2 - x - 1 sage: a*x (x)*a

With two square brackets, we obtain powers series:

sage: R.<a,b> = K[['a','b']] sage: R Multivariate Power Series Ring in a, b over Number Field in x with defining polynomial x^2 - x - 1 sage: a*x*b (x)*a*b

It works with matrices:

sage: MS = MatrixSpace(R,2,2) sage: MS Full MatrixSpace of 2 by 2 dense matrices over Multivariate Power Series Ring in a, b over Number Field in x with defining polynomial x^2 - x - 1 sage: MS([0,a,b,x]) [ 0 a] [ b (x)] sage: m1 = MS([0,a,b,x]) sage: m2 = MS([0,a+x,b*b+x,x*x]) sage: m1 + m2 * m1 [ (x)*b + a*b (x + 1) + (x + 1)*a] [ (x + 2)*b (3*x + 1) + (x)*a + a*b^2]

## slabbe-0.2.spkg released

30 novembre 2015 | Catégories: sage, slabbe spkg | View Comments

These is a summary of the functionalities present in slabbe-0.2.spkg optional
Sage package. It works on version 6.8 of Sage but will work best with sage-6.10
(it is using the new code for `cartesian_product` merged the the betas of
sage-6.10). It contains 7 new modules:

finite_word.pylanguage.pylyapunov.pymatrix_cocycle.pymult_cont_frac.pyxranking_scale.pytikz_picture.py

**Cheat Sheets**

The best way to have a quick look at what can be computed with the optional
Sage package `slabbe-0.2.spkg` is to look at the 3-dimensional Continued
Fraction Algorithms Cheat Sheets available on the arXiv since today. It
gathers a handful of informations on different 3-dimensional Continued Fraction
Algorithms including well-known and old ones (Poincaré, Brun, Selmer, Fully
Subtractive) and new ones (Arnoux-Rauzy-Poincaré, Reverse, Cassaigne).

**Installation**

sage -i http://www.slabbe.org/Sage/slabbe-0.2.spkg # on sage 6.8 sage -p http://www.slabbe.org/Sage/slabbe-0.2.spkg # on sage 6.9 or beyond

**Examples**

Computing the orbit of Brun algorithm on some input in \(\mathbb{R}^3_+\) including dual coordinates:

sage: from slabbe.mult_cont_frac import Brun sage: algo = Brun() sage: algo.cone_orbit_list((100, 87, 15), 4) [(13.0, 87.0, 15.0, 1.0, 2.0, 1.0, 321), (13.0, 72.0, 15.0, 1.0, 2.0, 3.0, 132), (13.0, 57.0, 15.0, 1.0, 2.0, 5.0, 132), (13.0, 42.0, 15.0, 1.0, 2.0, 7.0, 132)]

Computing the invariant measure:

sage: fig = algo.invariant_measure_wireframe_plot(n_iterations=10^6, ndivs=30) sage: fig.savefig('a.png')

Drawing the cylinders:

sage: cocycle = algo.matrix_cocycle() sage: t = cocycle.tikz_n_cylinders(3, scale=3) sage: t.png()

Computing the Lyapunov exponents of the 3-dimensional Brun algorithm:

sage: from slabbe.lyapunov import lyapunov_table sage: lyapunov_table(algo, n_orbits=30, n_iterations=10^7) 30 succesful orbits min mean max std +-----------------------+---------+---------+---------+---------+ $\theta_1$ 0.3026 0.3045 0.3051 0.00046 $\theta_2$ -0.1125 -0.1122 -0.1115 0.00020 $1-\theta_2/\theta_1$ 1.3680 1.3684 1.3689 0.00024

**Dealing with tikzpictures**

Since I create lots of tikzpictures in my code and also because I was unhappy
at how the `view` command of Sage handles them (a tikzpicture is not a math
expression to put inside dollar signs), I decided to create a class for
tikzpictures. I think this module could be useful in Sage so I will propose
its inclusion soon.

I am using the standalone document class which allows some configurations like the border:

sage: from slabbe import TikzPicture sage: g = graphs.PetersenGraph() sage: s = latex(g) sage: t = TikzPicture(s, standalone_configs=["border=4mm"], packages=['tkz-graph'])

The `repr` method does not print all of the string since it is often very
long. Though it shows how many lines are not printed:

sage: t \documentclass[tikz]{standalone} \standaloneconfig{border=4mm} \usepackage{tkz-graph} \begin{document} \begin{tikzpicture} % \useasboundingbox (0,0) rectangle (5.0cm,5.0cm); % \definecolor{cv0}{rgb}{0.0,0.0,0.0} ... ... 68 lines not printed (3748 characters in total) ... ... \Edge[lw=0.1cm,style={color=cv6v8,},](v6)(v8) \Edge[lw=0.1cm,style={color=cv6v9,},](v6)(v9) \Edge[lw=0.1cm,style={color=cv7v9,},](v7)(v9) % \end{tikzpicture} \end{document}

There is a method to generates a pdf and another for generating a png. Both
opens the file in a viewer by default unless `view=False`:

sage: pathtofile = t.png(density=60, view=False) sage: pathtofile = t.pdf()

Compare this with the output of `view(s, tightpage=True)` which does not
allow to control the border and also creates a second empty page on some
operating system (osx, only one page on ubuntu):

sage: view(s, tightpage=True)

One can also provide the filename where to save the file in which case the file is not open in a viewer:

sage: _ = t.pdf('petersen_graph.pdf')

Another example with polyhedron code taken from this Sage thematic tutorial Draw polytopes in LateX using TikZ:

sage: V = [[1,0,1],[1,0,0],[1,1,0],[0,0,-1],[0,1,0],[-1,0,0],[0,1,1],[0,0,1],[0,-1,0]] sage: P = Polyhedron(vertices=V).polar() sage: s = P.projection().tikz([674,108,-731],112) sage: t = TikzPicture(s) sage: t \documentclass[tikz]{standalone} \begin{document} \begin{tikzpicture}% [x={(0.249656cm, -0.577639cm)}, y={(0.777700cm, -0.358578cm)}, z={(-0.576936cm, -0.733318cm)}, scale=2.000000, ... ... 80 lines not printed (4889 characters in total) ... ... \node[vertex] at (1.00000, 1.00000, -1.00000) {}; \node[vertex] at (1.00000, 1.00000, 1.00000) {}; %% %% \end{tikzpicture} \end{document} sage: _ = t.pdf()

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