from collections import defaultdict
import networkx as nx
__all__ = ["combinatorial_embedding_to_pos"]
[docs]
def combinatorial_embedding_to_pos(embedding, fully_triangulate=False):
"""Assigns every node a (x, y) position based on the given embedding
The algorithm iteratively inserts nodes of the input graph in a certain
order and rearranges previously inserted nodes so that the planar drawing
stays valid. This is done efficiently by only maintaining relative
positions during the node placements and calculating the absolute positions
at the end. For more information see [1]_.
Parameters
----------
embedding : nx.PlanarEmbedding
This defines the order of the edges
fully_triangulate : bool
If set to True the algorithm adds edges to a copy of the input
embedding and makes it chordal.
Returns
-------
pos : dict
Maps each node to a tuple that defines the (x, y) position
References
----------
.. [1] M. Chrobak and T.H. Payne:
A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677
"""
if len(embedding.nodes()) < 4:
# Position the node in any triangle
default_positions = [(0, 0), (2, 0), (1, 1)]
pos = {}
for i, v in enumerate(embedding.nodes()):
pos[v] = default_positions[i]
return pos
embedding, outer_face = triangulate_embedding(embedding, fully_triangulate)
# The following dicts map a node to another node
# If a node is not in the key set it means that the node is not yet in G_k
# If a node maps to None then the corresponding subtree does not exist
left_t_child = {}
right_t_child = {}
# The following dicts map a node to an integer
delta_x = {}
y_coordinate = {}
node_list = get_canonical_ordering(embedding, outer_face)
# 1. Phase: Compute relative positions
# Initialization
v1, v2, v3 = node_list[0][0], node_list[1][0], node_list[2][0]
delta_x[v1] = 0
y_coordinate[v1] = 0
right_t_child[v1] = v3
left_t_child[v1] = None
delta_x[v2] = 1
y_coordinate[v2] = 0
right_t_child[v2] = None
left_t_child[v2] = None
delta_x[v3] = 1
y_coordinate[v3] = 1
right_t_child[v3] = v2
left_t_child[v3] = None
for k in range(3, len(node_list)):
vk, contour_nbrs = node_list[k]
wp = contour_nbrs[0]
wp1 = contour_nbrs[1]
wq = contour_nbrs[-1]
wq1 = contour_nbrs[-2]
adds_mult_tri = len(contour_nbrs) > 2
# Stretch gaps:
delta_x[wp1] += 1
delta_x[wq] += 1
delta_x_wp_wq = sum(delta_x[x] for x in contour_nbrs[1:])
# Adjust offsets
delta_x[vk] = (-y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2
y_coordinate[vk] = (y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2
delta_x[wq] = delta_x_wp_wq - delta_x[vk]
if adds_mult_tri:
delta_x[wp1] -= delta_x[vk]
# Install v_k:
right_t_child[wp] = vk
right_t_child[vk] = wq
if adds_mult_tri:
left_t_child[vk] = wp1
right_t_child[wq1] = None
else:
left_t_child[vk] = None
# 2. Phase: Set absolute positions
pos = {}
pos[v1] = (0, y_coordinate[v1])
remaining_nodes = [v1]
while remaining_nodes:
parent_node = remaining_nodes.pop()
# Calculate position for left child
set_position(
parent_node, left_t_child, remaining_nodes, delta_x, y_coordinate, pos
)
# Calculate position for right child
set_position(
parent_node, right_t_child, remaining_nodes, delta_x, y_coordinate, pos
)
return pos
def set_position(parent, tree, remaining_nodes, delta_x, y_coordinate, pos):
"""Helper method to calculate the absolute position of nodes."""
child = tree[parent]
parent_node_x = pos[parent][0]
if child is not None:
# Calculate pos of child
child_x = parent_node_x + delta_x[child]
pos[child] = (child_x, y_coordinate[child])
# Remember to calculate pos of its children
remaining_nodes.append(child)
def get_canonical_ordering(embedding, outer_face):
"""Returns a canonical ordering of the nodes
The canonical ordering of nodes (v1, ..., vn) must fulfill the following
conditions:
(See Lemma 1 in [2]_)
- For the subgraph G_k of the input graph induced by v1, ..., vk it holds:
- 2-connected
- internally triangulated
- the edge (v1, v2) is part of the outer face
- For a node v(k+1) the following holds:
- The node v(k+1) is part of the outer face of G_k
- It has at least two neighbors in G_k
- All neighbors of v(k+1) in G_k lie consecutively on the outer face of
G_k (excluding the edge (v1, v2)).
The algorithm used here starts with G_n (containing all nodes). It first
selects the nodes v1 and v2. And then tries to find the order of the other
nodes by checking which node can be removed in order to fulfill the
conditions mentioned above. This is done by calculating the number of
chords of nodes on the outer face. For more information see [1]_.
Parameters
----------
embedding : nx.PlanarEmbedding
The embedding must be triangulated
outer_face : list
The nodes on the outer face of the graph
Returns
-------
ordering : list
A list of tuples `(vk, wp_wq)`. Here `vk` is the node at this position
in the canonical ordering. The element `wp_wq` is a list of nodes that
make up the outer face of G_k.
References
----------
.. [1] Steven Chaplick.
Canonical Orders of Planar Graphs and (some of) Their Applications 2015
https://wuecampus2.uni-wuerzburg.de/moodle/pluginfile.php/545727/mod_resource/content/0/vg-ss15-vl03-canonical-orders-druckversion.pdf
.. [2] M. Chrobak and T.H. Payne:
A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677
"""
v1 = outer_face[0]
v2 = outer_face[1]
chords = defaultdict(int) # Maps nodes to the number of their chords
marked_nodes = set()
ready_to_pick = set(outer_face)
# Initialize outer_face_ccw_nbr (do not include v1 -> v2)
outer_face_ccw_nbr = {}
prev_nbr = v2
for idx in range(2, len(outer_face)):
outer_face_ccw_nbr[prev_nbr] = outer_face[idx]
prev_nbr = outer_face[idx]
outer_face_ccw_nbr[prev_nbr] = v1
# Initialize outer_face_cw_nbr (do not include v2 -> v1)
outer_face_cw_nbr = {}
prev_nbr = v1
for idx in range(len(outer_face) - 1, 0, -1):
outer_face_cw_nbr[prev_nbr] = outer_face[idx]
prev_nbr = outer_face[idx]
def is_outer_face_nbr(x, y):
if x not in outer_face_ccw_nbr:
return outer_face_cw_nbr[x] == y
if x not in outer_face_cw_nbr:
return outer_face_ccw_nbr[x] == y
return outer_face_ccw_nbr[x] == y or outer_face_cw_nbr[x] == y
def is_on_outer_face(x):
return x not in marked_nodes and (x in outer_face_ccw_nbr or x == v1)
# Initialize number of chords
for v in outer_face:
for nbr in embedding.neighbors_cw_order(v):
if is_on_outer_face(nbr) and not is_outer_face_nbr(v, nbr):
chords[v] += 1
ready_to_pick.discard(v)
# Initialize canonical_ordering
canonical_ordering = [None] * len(embedding.nodes())
canonical_ordering[0] = (v1, [])
canonical_ordering[1] = (v2, [])
ready_to_pick.discard(v1)
ready_to_pick.discard(v2)
for k in range(len(embedding.nodes()) - 1, 1, -1):
# 1. Pick v from ready_to_pick
v = ready_to_pick.pop()
marked_nodes.add(v)
# v has exactly two neighbors on the outer face (wp and wq)
wp = None
wq = None
# Iterate over neighbors of v to find wp and wq
nbr_iterator = iter(embedding.neighbors_cw_order(v))
while True:
nbr = next(nbr_iterator)
if nbr in marked_nodes:
# Only consider nodes that are not yet removed
continue
if is_on_outer_face(nbr):
# nbr is either wp or wq
if nbr == v1:
wp = v1
elif nbr == v2:
wq = v2
else:
if outer_face_cw_nbr[nbr] == v:
# nbr is wp
wp = nbr
else:
# nbr is wq
wq = nbr
if wp is not None and wq is not None:
# We don't need to iterate any further
break
# Obtain new nodes on outer face (neighbors of v from wp to wq)
wp_wq = [wp]
nbr = wp
while nbr != wq:
# Get next neighbor (clockwise on the outer face)
next_nbr = embedding[v][nbr]["ccw"]
wp_wq.append(next_nbr)
# Update outer face
outer_face_cw_nbr[nbr] = next_nbr
outer_face_ccw_nbr[next_nbr] = nbr
# Move to next neighbor of v
nbr = next_nbr
if len(wp_wq) == 2:
# There was a chord between wp and wq, decrease number of chords
chords[wp] -= 1
if chords[wp] == 0:
ready_to_pick.add(wp)
chords[wq] -= 1
if chords[wq] == 0:
ready_to_pick.add(wq)
else:
# Update all chords involving w_(p+1) to w_(q-1)
new_face_nodes = set(wp_wq[1:-1])
for w in new_face_nodes:
# If we do not find a chord for w later we can pick it next
ready_to_pick.add(w)
for nbr in embedding.neighbors_cw_order(w):
if is_on_outer_face(nbr) and not is_outer_face_nbr(w, nbr):
# There is a chord involving w
chords[w] += 1
ready_to_pick.discard(w)
if nbr not in new_face_nodes:
# Also increase chord for the neighbor
# We only iterator over new_face_nodes
chords[nbr] += 1
ready_to_pick.discard(nbr)
# Set the canonical ordering node and the list of contour neighbors
canonical_ordering[k] = (v, wp_wq)
return canonical_ordering
def triangulate_face(embedding, v1, v2):
"""Triangulates the face given by half edge (v, w)
Parameters
----------
embedding : nx.PlanarEmbedding
v1 : node
The half-edge (v1, v2) belongs to the face that gets triangulated
v2 : node
"""
_, v3 = embedding.next_face_half_edge(v1, v2)
_, v4 = embedding.next_face_half_edge(v2, v3)
if v1 in (v2, v3):
# The component has less than 3 nodes
return
while v1 != v4:
# Add edge if not already present on other side
if embedding.has_edge(v1, v3):
# Cannot triangulate at this position
v1, v2, v3 = v2, v3, v4
else:
# Add edge for triangulation
embedding.add_half_edge(v1, v3, ccw=v2)
embedding.add_half_edge(v3, v1, cw=v2)
v1, v2, v3 = v1, v3, v4
# Get next node
_, v4 = embedding.next_face_half_edge(v2, v3)
def triangulate_embedding(embedding, fully_triangulate=True):
"""Triangulates the embedding.
Traverses faces of the embedding and adds edges to a copy of the
embedding to triangulate it.
The method also ensures that the resulting graph is 2-connected by adding
edges if the same vertex is contained twice on a path around a face.
Parameters
----------
embedding : nx.PlanarEmbedding
The input graph must contain at least 3 nodes.
fully_triangulate : bool
If set to False the face with the most nodes is chooses as outer face.
This outer face does not get triangulated.
Returns
-------
(embedding, outer_face) : (nx.PlanarEmbedding, list) tuple
The element `embedding` is a new embedding containing all edges from
the input embedding and the additional edges to triangulate the graph.
The element `outer_face` is a list of nodes that lie on the outer face.
If the graph is fully triangulated these are three arbitrary connected
nodes.
"""
if len(embedding.nodes) <= 1:
return embedding, list(embedding.nodes)
embedding = nx.PlanarEmbedding(embedding)
# Get a list with a node for each connected component
component_nodes = [next(iter(x)) for x in nx.connected_components(embedding)]
# 1. Make graph a single component (add edge between components)
for i in range(len(component_nodes) - 1):
v1 = component_nodes[i]
v2 = component_nodes[i + 1]
embedding.connect_components(v1, v2)
# 2. Calculate faces, ensure 2-connectedness and determine outer face
outer_face = [] # A face with the most number of nodes
face_list = []
edges_visited = set() # Used to keep track of already visited faces
for v in embedding.nodes():
for w in embedding.neighbors_cw_order(v):
new_face = make_bi_connected(embedding, v, w, edges_visited)
if new_face:
# Found a new face
face_list.append(new_face)
if len(new_face) > len(outer_face):
# The face is a candidate to be the outer face
outer_face = new_face
# 3. Triangulate (internal) faces
for face in face_list:
if face is not outer_face or fully_triangulate:
# Triangulate this face
triangulate_face(embedding, face[0], face[1])
if fully_triangulate:
v1 = outer_face[0]
v2 = outer_face[1]
v3 = embedding[v2][v1]["ccw"]
outer_face = [v1, v2, v3]
return embedding, outer_face
def make_bi_connected(embedding, starting_node, outgoing_node, edges_counted):
"""Triangulate a face and make it 2-connected
This method also adds all edges on the face to `edges_counted`.
Parameters
----------
embedding: nx.PlanarEmbedding
The embedding that defines the faces
starting_node : node
A node on the face
outgoing_node : node
A node such that the half edge (starting_node, outgoing_node) belongs
to the face
edges_counted: set
Set of all half-edges that belong to a face that have been visited
Returns
-------
face_nodes: list
A list of all nodes at the border of this face
"""
# Check if the face has already been calculated
if (starting_node, outgoing_node) in edges_counted:
# This face was already counted
return []
edges_counted.add((starting_node, outgoing_node))
# Add all edges to edges_counted which have this face to their left
v1 = starting_node
v2 = outgoing_node
face_list = [starting_node] # List of nodes around the face
face_set = set(face_list) # Set for faster queries
_, v3 = embedding.next_face_half_edge(v1, v2)
# Move the nodes v1, v2, v3 around the face:
while v2 != starting_node or v3 != outgoing_node:
if v1 == v2:
raise nx.NetworkXException("Invalid half-edge")
# cycle is not completed yet
if v2 in face_set:
# v2 encountered twice: Add edge to ensure 2-connectedness
embedding.add_half_edge(v1, v3, ccw=v2)
embedding.add_half_edge(v3, v1, cw=v2)
edges_counted.add((v2, v3))
edges_counted.add((v3, v1))
v2 = v1
else:
face_set.add(v2)
face_list.append(v2)
# set next edge
v1 = v2
v2, v3 = embedding.next_face_half_edge(v2, v3)
# remember that this edge has been counted
edges_counted.add((v1, v2))
return face_list
