# Source code for networkx.generators.lattice

"""Functions for generating grid graphs and lattices

The :func:grid_2d_graph, :func:triangular_lattice_graph, and
:func:hexagonal_lattice_graph functions correspond to the three
regular tilings of the plane_, the square, triangular, and hexagonal
tilings, respectively. :func:grid_graph and :func:hypercube_graph
are similar for arbitrary dimensions. Useful relevant discussion can
be found about Triangular Tiling_, and Square, Hex and Triangle Grids_

.. _regular tilings of the plane: https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds#Euclidean_tilings
.. _Square, Hex and Triangle Grids: http://www-cs-students.stanford.edu/~amitp/game-programming/grids/
.. _Triangular Tiling: https://en.wikipedia.org/wiki/Triangular_tiling

"""

from itertools import repeat
from math import sqrt

from networkx.classes import set_node_attributes
from networkx.exception import NetworkXError
from networkx.generators.classic import cycle_graph, empty_graph, path_graph
from networkx.relabel import relabel_nodes
from networkx.utils import flatten, nodes_or_number, pairwise

__all__ = [
"grid_2d_graph",
"grid_graph",
"hypercube_graph",
"triangular_lattice_graph",
"hexagonal_lattice_graph",
]

[docs]@nodes_or_number([0, 1]) def grid_2d_graph(m, n, periodic=False, create_using=None): """Returns the two-dimensional grid graph. The grid graph has each node connected to its four nearest neighbors. Parameters ---------- m, n : int or iterable container of nodes If an integer, nodes are from range(n). If a container, elements become the coordinate of the nodes. periodic : bool or iterable If periodic is True, both dimensions are periodic. If False, none are periodic. If periodic is iterable, it should yield 2 bool values indicating whether the 1st and 2nd axes, respectively, are periodic. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- NetworkX graph The (possibly periodic) grid graph of the specified dimensions. """ G = empty_graph(0, create_using) row_name, rows = m col_name, cols = n G.add_nodes_from((i, j) for i in rows for j in cols) G.add_edges_from(((i, j), (pi, j)) for pi, i in pairwise(rows) for j in cols) G.add_edges_from(((i, j), (i, pj)) for i in rows for pj, j in pairwise(cols)) try: periodic_r, periodic_c = periodic except TypeError: periodic_r = periodic_c = periodic if periodic_r and len(rows) > 2: first = rows[0] last = rows[-1] G.add_edges_from(((first, j), (last, j)) for j in cols) if periodic_c and len(cols) > 2: first = cols[0] last = cols[-1] G.add_edges_from(((i, first), (i, last)) for i in rows) # both directions for directed if G.is_directed(): G.add_edges_from((v, u) for u, v in G.edges()) return G
[docs]def grid_graph(dim, periodic=False): """Returns the *n*-dimensional grid graph. The dimension *n* is the length of the list dim and the size in each dimension is the value of the corresponding list element. Parameters ---------- dim : list or tuple of numbers or iterables of nodes 'dim' is a tuple or list with, for each dimension, either a number that is the size of that dimension or an iterable of nodes for that dimension. The dimension of the grid_graph is the length of dim. periodic : bool or iterable If periodic is True, all dimensions are periodic. If False all dimensions are not periodic. If periodic is iterable, it should yield dim bool values each of which indicates whether the corresponding axis is periodic. Returns ------- NetworkX graph The (possibly periodic) grid graph of the specified dimensions. Examples -------- To produce a 2 by 3 by 4 grid graph, a graph on 24 nodes: >>> from networkx import grid_graph >>> G = grid_graph(dim=(2, 3, 4)) >>> len(G) 24 >>> G = grid_graph(dim=(range(7, 9), range(3, 6))) >>> len(G) 6 """ from networkx.algorithms.operators.product import cartesian_product if not dim: return empty_graph(0) try: func = (cycle_graph if p else path_graph for p in periodic) except TypeError: func = repeat(cycle_graph if periodic else path_graph) G = next(func)(dim[0]) for current_dim in dim[1:]: Gnew = next(func)(current_dim) G = cartesian_product(Gnew, G) # graph G is done but has labels of the form (1, (2, (3, 1))) so relabel H = relabel_nodes(G, flatten) return H
[docs]def hypercube_graph(n): """Returns the *n*-dimensional hypercube graph. The nodes are the integers between 0 and 2 ** n - 1, inclusive. For more information on the hypercube graph, see the Wikipedia article Hypercube graph_. .. _Hypercube graph: https://en.wikipedia.org/wiki/Hypercube_graph Parameters ---------- n : int The dimension of the hypercube. The number of nodes in the graph will be 2 ** n. Returns ------- NetworkX graph The hypercube graph of dimension *n*. """ dim = n * [2] G = grid_graph(dim) return G
[docs]def triangular_lattice_graph( m, n, periodic=False, with_positions=True, create_using=None ): r"""Returns the $m$ by $n$ triangular lattice graph. The triangular lattice graph_ is a two-dimensional grid graph_ in which each square unit has a diagonal edge (each grid unit has a chord). The returned graph has $m$ rows and $n$ columns of triangles. Rows and columns include both triangles pointing up and down. Rows form a strip of constant height. Columns form a series of diamond shapes, staggered with the columns on either side. Another way to state the size is that the nodes form a grid of m+1 rows and (n + 1) // 2 columns. The odd row nodes are shifted horizontally relative to the even rows. Directed graph types have edges pointed up or right. Positions of nodes are computed by default or with_positions is True. The position of each node (embedded in a euclidean plane) is stored in the graph using equilateral triangles with sidelength 1. The height between rows of nodes is thus $\sqrt(3)/2$. Nodes lie in the first quadrant with the node $(0, 0)$ at the origin. .. _triangular lattice graph: http://mathworld.wolfram.com/TriangularGrid.html .. _grid graph: http://www-cs-students.stanford.edu/~amitp/game-programming/grids/ .. _Triangular Tiling: https://en.wikipedia.org/wiki/Triangular_tiling Parameters ---------- m : int The number of rows in the lattice. n : int The number of columns in the lattice. periodic : bool (default: False) If True, join the boundary vertices of the grid using periodic boundary conditions. The join between boundaries is the final row and column of triangles. This means there is one row and one column fewer nodes for the periodic lattice. Periodic lattices require m >= 3, n >= 5 and are allowed but misaligned if m or n are odd with_positions : bool (default: True) Store the coordinates of each node in the graph node attribute 'pos'. The coordinates provide a lattice with equilateral triangles. Periodic positions shift the nodes vertically in a nonlinear way so the edges don't overlap so much. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- NetworkX graph The *m* by *n* triangular lattice graph. """ H = empty_graph(0, create_using) if n == 0 or m == 0: return H if periodic: if n < 5 or m < 3: msg = f"m > 2 and n > 4 required for periodic. m={m}, n={n}" raise NetworkXError(msg) N = (n + 1) // 2 # number of nodes in row rows = range(m + 1) cols = range(N + 1) # Make grid H.add_edges_from(((i, j), (i + 1, j)) for j in rows for i in cols[:N]) H.add_edges_from(((i, j), (i, j + 1)) for j in rows[:m] for i in cols) # add diagonals H.add_edges_from(((i, j), (i + 1, j + 1)) for j in rows[1:m:2] for i in cols[:N]) H.add_edges_from(((i + 1, j), (i, j + 1)) for j in rows[:m:2] for i in cols[:N]) # identify boundary nodes if periodic from networkx.algorithms.minors import contracted_nodes if periodic is True: for i in cols: H = contracted_nodes(H, (i, 0), (i, m)) for j in rows[:m]: H = contracted_nodes(H, (0, j), (N, j)) elif n % 2: # remove extra nodes H.remove_nodes_from((N, j) for j in rows[1::2]) # Add position node attributes if with_positions: ii = (i for i in cols for j in rows) jj = (j for i in cols for j in rows) xx = (0.5 * (j % 2) + i for i in cols for j in rows) h = sqrt(3) / 2 if periodic: yy = (h * j + 0.01 * i * i for i in cols for j in rows) else: yy = (h * j for i in cols for j in rows) pos = {(i, j): (x, y) for i, j, x, y in zip(ii, jj, xx, yy) if (i, j) in H} set_node_attributes(H, pos, "pos") return H
[docs]def hexagonal_lattice_graph( m, n, periodic=False, with_positions=True, create_using=None ): """Returns an m by n hexagonal lattice graph. The *hexagonal lattice graph* is a graph whose nodes and edges are the hexagonal tiling_ of the plane. The returned graph will have m rows and n columns of hexagons. Odd numbered columns_ are shifted up relative to even numbered columns. Positions of nodes are computed by default or with_positions is True. Node positions creating the standard embedding in the plane with sidelength 1 and are stored in the node attribute 'pos'. pos = nx.get_node_attributes(G, 'pos') creates a dict ready for drawing. .. _hexagonal tiling: https://en.wikipedia.org/wiki/Hexagonal_tiling .. _Odd numbered columns: http://www-cs-students.stanford.edu/~amitp/game-programming/grids/ Parameters ---------- m : int The number of rows of hexagons in the lattice. n : int The number of columns of hexagons in the lattice. periodic : bool Whether to make a periodic grid by joining the boundary vertices. For this to work n must be even and both n > 1 and m > 1. The periodic connections create another row and column of hexagons so these graphs have fewer nodes as boundary nodes are identified. with_positions : bool (default: True) Store the coordinates of each node in the graph node attribute 'pos'. The coordinates provide a lattice with vertical columns of hexagons offset to interleave and cover the plane. Periodic positions shift the nodes vertically in a nonlinear way so the edges don't overlap so much. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. If graph is directed, edges will point up or right. Returns ------- NetworkX graph The *m* by *n* hexagonal lattice graph. """ G = empty_graph(0, create_using) if m == 0 or n == 0: return G if periodic and (n % 2 == 1 or m < 2 or n < 2): msg = "periodic hexagonal lattice needs m > 1, n > 1 and even n" raise NetworkXError(msg) M = 2 * m # twice as many nodes as hexagons vertically rows = range(M + 2) cols = range(n + 1) # make lattice col_edges = (((i, j), (i, j + 1)) for i in cols for j in rows[: M + 1]) row_edges = (((i, j), (i + 1, j)) for i in cols[:n] for j in rows if i % 2 == j % 2) G.add_edges_from(col_edges) G.add_edges_from(row_edges) # Remove corner nodes with one edge G.remove_node((0, M + 1)) G.remove_node((n, (M + 1) * (n % 2))) # identify boundary nodes if periodic from networkx.algorithms.minors import contracted_nodes if periodic: for i in cols[:n]: G = contracted_nodes(G, (i, 0), (i, M)) for i in cols[1:]: G = contracted_nodes(G, (i, 1), (i, M + 1)) for j in rows[1:M]: G = contracted_nodes(G, (0, j), (n, j)) G.remove_node((n, M)) # calc position in embedded space ii = (i for i in cols for j in rows) jj = (j for i in cols for j in rows) xx = (0.5 + i + i // 2 + (j % 2) * ((i % 2) - 0.5) for i in cols for j in rows) h = sqrt(3) / 2 if periodic: yy = (h * j + 0.01 * i * i for i in cols for j in rows) else: yy = (h * j for i in cols for j in rows) # exclude nodes not in G pos = {(i, j): (x, y) for i, j, x, y in zip(ii, jj, xx, yy) if (i, j) in G} set_node_attributes(G, pos, "pos") return G