"""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 math import sqrt
from networkx.classes import set_node_attributes
from networkx.algorithms.minors import contracted_nodes
from networkx.algorithms.operators.product import cartesian_product
from networkx.exception import NetworkXError
from networkx.relabel import relabel_nodes
from networkx.utils import flatten, nodes_or_number, pairwise, iterable
from networkx.generators.classic import cycle_graph
from networkx.generators.classic import empty_graph
from networkx.generators.classic import path_graph
from itertools import repeat
__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))
if iterable(periodic):
periodic_r, periodic_c = periodic
else:
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
"""
if not dim:
return empty_graph(0)
if iterable(periodic):
func = (cycle_graph if p else path_graph for p in periodic)
else:
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
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 odd 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
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