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# Source code for networkx.algorithms.planar_drawing

    from collections import defaultdict

    import networkx as nx

    __all__ = ["combinatorial_embedding_to_pos"]

    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