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# Source code for networkx.algorithms.threshold

    """
    Threshold Graphs - Creation, manipulation and identification.
    """

    from math import sqrt

    import networkx as nx
    from networkx.utils import py_random_state

    __all__ = ["is_threshold_graph", "find_threshold_graph"]

    @nx._dispatchable
    def is_threshold_graph(G):
        """
        Returns `True` if `G` is a threshold graph.

        Parameters
        ----------
        G : NetworkX graph instance
            An instance of `Graph`, `DiGraph`, `MultiGraph` or `MultiDiGraph`

        Returns
        -------
        bool
            `True` if `G` is a threshold graph, `False` otherwise.

        Examples
        --------
        >>> from networkx.algorithms.threshold import is_threshold_graph
        >>> G = nx.path_graph(3)
        >>> is_threshold_graph(G)
        True
        >>> G = nx.barbell_graph(3, 3)
        >>> is_threshold_graph(G)
        False

        References
        ----------
        .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
        """
        return is_threshold_sequence([d for n, d in G.degree()])

    def is_threshold_sequence(degree_sequence):
        """
        Returns True if the sequence is a threshold degree sequence.

        Uses the property that a threshold graph must be constructed by
        adding either dominating or isolated nodes. Thus, it can be
        deconstructed iteratively by removing a node of degree zero or a
        node that connects to the remaining nodes.  If this deconstruction
        fails then the sequence is not a threshold sequence.
        """
        ds = degree_sequence[:]  # get a copy so we don't destroy original
        ds.sort()
        while ds:
            if ds[0] == 0:  # if isolated node
                ds.pop(0)  # remove it
                continue
            if ds[-1] != len(ds) - 1:  # is the largest degree node dominating?
                return False  # no, not a threshold degree sequence
            ds.pop()  # yes, largest is the dominating node
            ds = [d - 1 for d in ds]  # remove it and decrement all degrees
        return True

    def creation_sequence(degree_sequence, with_labels=False, compact=False):
        """
        Determines the creation sequence for the given threshold degree sequence.

        The creation sequence is a list of single characters 'd'
        or 'i': 'd' for dominating or 'i' for isolated vertices.
        Dominating vertices are connected to all vertices present when it
        is added.  The first node added is by convention 'd'.
        This list can be converted to a string if desired using "".join(cs)

        If with_labels==True:
        Returns a list of 2-tuples containing the vertex number
        and a character 'd' or 'i' which describes the type of vertex.

        If compact==True:
        Returns the creation sequence in a compact form that is the number
        of 'i's and 'd's alternating.
        Examples:
        [1,2,2,3] represents d,i,i,d,d,i,i,i
        [3,1,2] represents d,d,d,i,d,d

        Notice that the first number is the first vertex to be used for
        construction and so is always 'd'.

        with_labels and compact cannot both be True.

        Returns None if the sequence is not a threshold sequence
        """
        if with_labels and compact:
            raise ValueError("compact sequences cannot be labeled")

        # make an indexed copy
        if isinstance(degree_sequence, dict):  # labeled degree sequence
            ds = [[degree, label] for (label, degree) in degree_sequence.items()]
        else:
            ds = [[d, i] for i, d in enumerate(degree_sequence)]
        ds.sort()
        cs = []  # creation sequence
        while ds:
            if ds[0][0] == 0:  # isolated node
                (d, v) = ds.pop(0)
                if len(ds) > 0:  # make sure we start with a d
                    cs.insert(0, (v, "i"))
                else:
                    cs.insert(0, (v, "d"))
                continue
            if ds[-1][0] != len(ds) - 1:  # Not dominating node
                return None  # not a threshold degree sequence
            (d, v) = ds.pop()
            cs.insert(0, (v, "d"))
            ds = [[d[0] - 1, d[1]] for d in ds]  # decrement due to removing node

        if with_labels:
            return cs
        if compact:
            return make_compact(cs)
        return [v[1] for v in cs]  # not labeled

    def make_compact(creation_sequence):
        """
        Returns the creation sequence in a compact form
        that is the number of 'i's and 'd's alternating.

        Examples
        --------
        >>> from networkx.algorithms.threshold import make_compact
        >>> make_compact(["d", "i", "i", "d", "d", "i", "i", "i"])
        [1, 2, 2, 3]
        >>> make_compact(["d", "d", "d", "i", "d", "d"])
        [3, 1, 2]

        Notice that the first number is the first vertex
        to be used for construction and so is always 'd'.

        Labeled creation sequences lose their labels in the
        compact representation.

        >>> make_compact([3, 1, 2])
        [3, 1, 2]
        """
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            cs = creation_sequence[:]
        elif isinstance(first, tuple):  # labeled creation sequence
            cs = [s[1] for s in creation_sequence]
        elif isinstance(first, int):  # compact creation sequence
            return creation_sequence
        else:
            raise TypeError("Not a valid creation sequence type")

        ccs = []
        count = 1  # count the run lengths of d's or i's.
        for i in range(1, len(cs)):
            if cs[i] == cs[i - 1]:
                count += 1
            else:
                ccs.append(count)
                count = 1
        ccs.append(count)  # don't forget the last one
        return ccs

    def uncompact(creation_sequence):
        """
        Converts a compact creation sequence for a threshold
        graph to a standard creation sequence (unlabeled).
        If the creation_sequence is already standard, return it.
        See creation_sequence.
        """
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            return creation_sequence
        elif isinstance(first, tuple):  # labeled creation sequence
            return creation_sequence
        elif isinstance(first, int):  # compact creation sequence
            ccscopy = creation_sequence[:]
        else:
            raise TypeError("Not a valid creation sequence type")
        cs = []
        while ccscopy:
            cs.extend(ccscopy.pop(0) * ["d"])
            if ccscopy:
                cs.extend(ccscopy.pop(0) * ["i"])
        return cs

    def creation_sequence_to_weights(creation_sequence):
        """
        Returns a list of node weights which create the threshold
        graph designated by the creation sequence.  The weights
        are scaled so that the threshold is 1.0.  The order of the
        nodes is the same as that in the creation sequence.
        """
        # Turn input sequence into a labeled creation sequence
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            if isinstance(creation_sequence, list):
                wseq = creation_sequence[:]
            else:
                wseq = list(creation_sequence)  # string like 'ddidid'
        elif isinstance(first, tuple):  # labeled creation sequence
            wseq = [v[1] for v in creation_sequence]
        elif isinstance(first, int):  # compact creation sequence
            wseq = uncompact(creation_sequence)
        else:
            raise TypeError("Not a valid creation sequence type")
        # pass through twice--first backwards
        wseq.reverse()
        w = 0
        prev = "i"
        for j, s in enumerate(wseq):
            if s == "i":
                wseq[j] = w
                prev = s
            elif prev == "i":
                prev = s
                w += 1
        wseq.reverse()  # now pass through forwards
        for j, s in enumerate(wseq):
            if s == "d":
                wseq[j] = w
                prev = s
            elif prev == "d":
                prev = s
                w += 1
        # Now scale weights
        if prev == "d":
            w += 1
        wscale = 1 / w
        return [ww * wscale for ww in wseq]
        # return wseq

    def weights_to_creation_sequence(
        weights, threshold=1, with_labels=False, compact=False
    ):
        """
        Returns a creation sequence for a threshold graph
        determined by the weights and threshold given as input.
        If the sum of two node weights is greater than the
        threshold value, an edge is created between these nodes.

        The creation sequence is a list of single characters 'd'
        or 'i': 'd' for dominating or 'i' for isolated vertices.
        Dominating vertices are connected to all vertices present
        when it is added.  The first node added is by convention 'd'.

        If with_labels==True:
        Returns a list of 2-tuples containing the vertex number
        and a character 'd' or 'i' which describes the type of vertex.

        If compact==True:
        Returns the creation sequence in a compact form that is the number
        of 'i's and 'd's alternating.
        Examples:
        [1,2,2,3] represents d,i,i,d,d,i,i,i
        [3,1,2] represents d,d,d,i,d,d

        Notice that the first number is the first vertex to be used for
        construction and so is always 'd'.

        with_labels and compact cannot both be True.
        """
        if with_labels and compact:
            raise ValueError("compact sequences cannot be labeled")

        # make an indexed copy
        if isinstance(weights, dict):  # labeled weights
            wseq = [[w, label] for (label, w) in weights.items()]
        else:
            wseq = [[w, i] for i, w in enumerate(weights)]
        wseq.sort()
        cs = []  # creation sequence
        cutoff = threshold - wseq[-1][0]
        while wseq:
            if wseq[0][0] < cutoff:  # isolated node
                (w, label) = wseq.pop(0)
                cs.append((label, "i"))
            else:
                (w, label) = wseq.pop()
                cs.append((label, "d"))
                cutoff = threshold - wseq[-1][0]
            if len(wseq) == 1:  # make sure we start with a d
                (w, label) = wseq.pop()
                cs.append((label, "d"))
        # put in correct order
        cs.reverse()

        if with_labels:
            return cs
        if compact:
            return make_compact(cs)
        return [v[1] for v in cs]  # not labeled

    # Manipulating NetworkX.Graphs in context of threshold graphs
    @nx._dispatchable(graphs=None, returns_graph=True)
    def threshold_graph(creation_sequence, create_using=None):
        """
        Create a threshold graph from the creation sequence or compact
        creation_sequence.

        The input sequence can be a

        creation sequence (e.g. ['d','i','d','d','d','i'])
        labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])
        compact creation sequence (e.g. [2,1,1,2,0])

        Use cs=creation_sequence(degree_sequence,labeled=True)
        to convert a degree sequence to a creation sequence.

        Returns None if the sequence is not valid
        """
        # Turn input sequence into a labeled creation sequence
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            ci = list(enumerate(creation_sequence))
        elif isinstance(first, tuple):  # labeled creation sequence
            ci = creation_sequence[:]
        elif isinstance(first, int):  # compact creation sequence
            cs = uncompact(creation_sequence)
            ci = list(enumerate(cs))
        else:
            print("not a valid creation sequence type")
            return None

        G = nx.empty_graph(0, create_using)
        if G.is_directed():
            raise nx.NetworkXError("Directed Graph not supported")

        G.name = "Threshold Graph"

        # add nodes and edges
        # if type is 'i' just add nodea
        # if type is a d connect to everything previous
        while ci:
            (v, node_type) = ci.pop(0)
            if node_type == "d":  # dominating type, connect to all existing nodes
                # We use `for u in list(G):` instead of
                # `for u in G:` because we edit the graph `G` in
                # the loop. Hence using an iterator will result in
                # `RuntimeError: dictionary changed size during iteration`
                for u in list(G):
                    G.add_edge(v, u)
            G.add_node(v)
        return G

    @nx._dispatchable
    def find_alternating_4_cycle(G):
        """
        Returns False if there aren't any alternating 4 cycles.
        Otherwise returns the cycle as [a,b,c,d] where (a,b)
        and (c,d) are edges and (a,c) and (b,d) are not.
        """
        for u, v in G.edges():
            for w in G.nodes():
                if not G.has_edge(u, w) and u != w:
                    for x in G.neighbors(w):
                        if not G.has_edge(v, x) and v != x:
                            return [u, v, w, x]
        return False

    @nx._dispatchable(returns_graph=True)
    def find_threshold_graph(G, create_using=None):
        """
        Returns a threshold subgraph that is close to largest in `G`.

        The threshold graph will contain the largest degree node in G.

        Parameters
        ----------
        G : NetworkX graph instance
            An instance of `Graph`, or `MultiDiGraph`
        create_using : NetworkX graph class or `None` (default), optional
            Type of graph to use when constructing the threshold graph.
            If `None`, infer the appropriate graph type from the input.

        Returns
        -------
        graph :
            A graph instance representing the threshold graph

        Examples
        --------
        >>> from networkx.algorithms.threshold import find_threshold_graph
        >>> G = nx.barbell_graph(3, 3)
        >>> T = find_threshold_graph(G)
        >>> T.nodes  # may vary
        NodeView((7, 8, 5, 6))

        References
        ----------
        .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
        """
        return threshold_graph(find_creation_sequence(G), create_using)

    @nx._dispatchable
    def find_creation_sequence(G):
        """
        Find a threshold subgraph that is close to largest in G.
        Returns the labeled creation sequence of that threshold graph.
        """
        cs = []
        # get a local pointer to the working part of the graph
        H = G
        while H.order() > 0:
            # get new degree sequence on subgraph
            dsdict = dict(H.degree())
            ds = [(d, v) for v, d in dsdict.items()]
            ds.sort()
            # Update threshold graph nodes
            if ds[-1][0] == 0:  # all are isolated
                cs.extend(zip(dsdict, ["i"] * (len(ds) - 1) + ["d"]))
                break  # Done!
            # pull off isolated nodes
            while ds[0][0] == 0:
                (d, iso) = ds.pop(0)
                cs.append((iso, "i"))
            # find new biggest node
            (d, bigv) = ds.pop()
            # add edges of star to t_g
            cs.append((bigv, "d"))
            # form subgraph of neighbors of big node
            H = H.subgraph(H.neighbors(bigv))
        cs.reverse()
        return cs

    # Properties of Threshold Graphs
    def triangles(creation_sequence):
        """
        Compute number of triangles in the threshold graph with the
        given creation sequence.
        """
        # shortcut algorithm that doesn't require computing number
        # of triangles at each node.
        cs = creation_sequence  # alias
        dr = cs.count("d")  # number of d's in sequence
        ntri = dr * (dr - 1) * (dr - 2) / 6  # number of triangles in clique of nd d's
        # now add dr choose 2 triangles for every 'i' in sequence where
        # dr is the number of d's to the right of the current i
        for i, typ in enumerate(cs):
            if typ == "i":
                ntri += dr * (dr - 1) / 2
            else:
                dr -= 1
        return ntri

    def triangle_sequence(creation_sequence):
        """
        Return triangle sequence for the given threshold graph creation sequence.

        """
        cs = creation_sequence
        seq = []
        dr = cs.count("d")  # number of d's to the right of the current pos
        dcur = (dr - 1) * (dr - 2) // 2  # number of triangles through a node of clique dr
        irun = 0  # number of i's in the last run
        drun = 0  # number of d's in the last run
        for i, sym in enumerate(cs):
            if sym == "d":
                drun += 1
                tri = dcur + (dr - 1) * irun  # new triangles at this d
            else:  # cs[i]="i":
                if prevsym == "d":  # new string of i's
                    dcur += (dr - 1) * irun  # accumulate shared shortest paths
                    irun = 0  # reset i run counter
                    dr -= drun  # reduce number of d's to right
                    drun = 0  # reset d run counter
                irun += 1
                tri = dr * (dr - 1) // 2  # new triangles at this i
            seq.append(tri)
            prevsym = sym
        return seq

    def cluster_sequence(creation_sequence):
        """
        Return cluster sequence for the given threshold graph creation sequence.
        """
        triseq = triangle_sequence(creation_sequence)
        degseq = degree_sequence(creation_sequence)
        cseq = []
        for i, deg in enumerate(degseq):
            tri = triseq[i]
            if deg <= 1:  # isolated vertex or single pair gets cc 0
                cseq.append(0)
                continue
            max_size = (deg * (deg - 1)) // 2
            cseq.append(tri / max_size)
        return cseq

    def degree_sequence(creation_sequence):
        """
        Return degree sequence for the threshold graph with the given
        creation sequence
        """
        cs = creation_sequence  # alias
        seq = []
        rd = cs.count("d")  # number of d to the right
        for i, sym in enumerate(cs):
            if sym == "d":
                rd -= 1
                seq.append(rd + i)
            else:
                seq.append(rd)
        return seq

    def density(creation_sequence):
        """
        Return the density of the graph with this creation_sequence.
        The density is the fraction of possible edges present.
        """
        N = len(creation_sequence)
        two_size = sum(degree_sequence(creation_sequence))
        two_possible = N * (N - 1)
        den = two_size / two_possible
        return den

    def degree_correlation(creation_sequence):
        """
        Return the degree-degree correlation over all edges.
        """
        cs = creation_sequence
        s1 = 0  # deg_i*deg_j
        s2 = 0  # deg_i^2+deg_j^2
        s3 = 0  # deg_i+deg_j
        m = 0  # number of edges
        rd = cs.count("d")  # number of d nodes to the right
        rdi = [i for i, sym in enumerate(cs) if sym == "d"]  # index of "d"s
        ds = degree_sequence(cs)
        for i, sym in enumerate(cs):
            if sym == "d":
                if i != rdi[0]:
                    print("Logic error in degree_correlation", i, rdi)
                    raise ValueError
                rdi.pop(0)
            degi = ds[i]
            for dj in rdi:
                degj = ds[dj]
                s1 += degj * degi
                s2 += degi**2 + degj**2
                s3 += degi + degj
                m += 1
        denom = 2 * m * s2 - s3 * s3
        numer = 4 * m * s1 - s3 * s3
        if denom == 0:
            if numer == 0:
                return 1
            raise ValueError(f"Zero Denominator but Numerator is {numer}")
        return numer / denom

    def shortest_path(creation_sequence, u, v):
        """
        Find the shortest path between u and v in a
        threshold graph G with the given creation_sequence.

        For an unlabeled creation_sequence, the vertices
        u and v must be integers in (0,len(sequence)) referring
        to the position of the desired vertices in the sequence.

        For a labeled creation_sequence, u and v are labels of vertices.

        Use cs=creation_sequence(degree_sequence,with_labels=True)
        to convert a degree sequence to a creation sequence.

        Returns a list of vertices from u to v.
        Example: if they are neighbors, it returns [u,v]
        """
        # Turn input sequence into a labeled creation sequence
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            cs = [(i, creation_sequence[i]) for i in range(len(creation_sequence))]
        elif isinstance(first, tuple):  # labeled creation sequence
            cs = creation_sequence[:]
        elif isinstance(first, int):  # compact creation sequence
            ci = uncompact(creation_sequence)
            cs = [(i, ci[i]) for i in range(len(ci))]
        else:
            raise TypeError("Not a valid creation sequence type")

        verts = [s[0] for s in cs]
        if v not in verts:
            raise ValueError(f"Vertex {v} not in graph from creation_sequence")
        if u not in verts:
            raise ValueError(f"Vertex {u} not in graph from creation_sequence")
        # Done checking
        if u == v:
            return [u]

        uindex = verts.index(u)
        vindex = verts.index(v)
        bigind = max(uindex, vindex)
        if cs[bigind][1] == "d":
            return [u, v]
        # must be that cs[bigind][1]=='i'
        cs = cs[bigind:]
        while cs:
            vert = cs.pop()
            if vert[1] == "d":
                return [u, vert[0], v]
        # All after u are type 'i' so no connection
        return -1

    def shortest_path_length(creation_sequence, i):
        """
        Return the shortest path length from indicated node to
        every other node for the threshold graph with the given
        creation sequence.
        Node is indicated by index i in creation_sequence unless
        creation_sequence is labeled in which case, i is taken to
        be the label of the node.

        Paths lengths in threshold graphs are at most 2.
        Length to unreachable nodes is set to -1.
        """
        # Turn input sequence into a labeled creation sequence
        first = creation_sequence[0]
        if isinstance(first, str):  # creation sequence
            if isinstance(creation_sequence, list):
                cs = creation_sequence[:]
            else:
                cs = list(creation_sequence)
        elif isinstance(first, tuple):  # labeled creation sequence
            cs = [v[1] for v in creation_sequence]
            i = [v[0] for v in creation_sequence].index(i)
        elif isinstance(first, int):  # compact creation sequence
            cs = uncompact(creation_sequence)
        else:
            raise TypeError("Not a valid creation sequence type")

        # Compute
        N = len(cs)
        spl = [2] * N  # length 2 to every node
        spl[i] = 0  # except self which is 0
        # 1 for all d's to the right
        for j in range(i + 1, N):
            if cs[j] == "d":
                spl[j] = 1
        if cs[i] == "d":  # 1 for all nodes to the left
            for j in range(i):
                spl[j] = 1
        # and -1 for any trailing i to indicate unreachable
        for j in range(N - 1, 0, -1):
            if cs[j] == "d":
                break
            spl[j] = -1
        return spl

    def betweenness_sequence(creation_sequence, normalized=True):
        """
        Return betweenness for the threshold graph with the given creation
        sequence.  The result is unscaled.  To scale the values
        to the interval [0,1] divide by (n-1)*(n-2).
        """
        cs = creation_sequence
        seq = []  # betweenness
        lastchar = "d"  # first node is always a 'd'
        dr = float(cs.count("d"))  # number of d's to the right of current pos
        irun = 0  # number of i's in the last run
        drun = 0  # number of d's in the last run
        dlast = 0.0  # betweenness of last d
        for i, c in enumerate(cs):
            if c == "d":  # cs[i]=="d":
                # betweenness = amt shared with earlier d's and i's
                #             + new isolated nodes covered
                #             + new paths to all previous nodes
                b = dlast + (irun - 1) * irun / dr + 2 * irun * (i - drun - irun) / dr
                drun += 1  # update counter
            else:  # cs[i]="i":
                if lastchar == "d":  # if this is a new run of i's
                    dlast = b  # accumulate betweenness
                    dr -= drun  # update number of d's to the right
                    drun = 0  # reset d counter
                    irun = 0  # reset i counter
                b = 0  # isolated nodes have zero betweenness
                irun += 1  # add another i to the run
            seq.append(float(b))
            lastchar = c

        # normalize by the number of possible shortest paths
        if normalized:
            order = len(cs)
            scale = 1.0 / ((order - 1) * (order - 2))
            seq = [s * scale for s in seq]

        return seq

    def eigenvectors(creation_sequence):
        """
        Return a 2-tuple of Laplacian eigenvalues and eigenvectors
        for the threshold network with creation_sequence.
        The first value is a list of eigenvalues.
        The second value is a list of eigenvectors.
        The lists are in the same order so corresponding eigenvectors
        and eigenvalues are in the same position in the two lists.

        Notice that the order of the eigenvalues returned by eigenvalues(cs)
        may not correspond to the order of these eigenvectors.
        """
        ccs = make_compact(creation_sequence)
        N = sum(ccs)
        vec = [0] * N
        val = vec[:]
        # get number of type d nodes to the right (all for first node)
        dr = sum(ccs[::2])

        nn = ccs[0]
        vec[0] = [1.0 / sqrt(N)] * N
        val[0] = 0
        e = dr
        dr -= nn
        type_d = True
        i = 1
        dd = 1
        while dd < nn:
            scale = 1.0 / sqrt(dd * dd + i)
            vec[i] = i * [-scale] + [dd * scale] + [0] * (N - i - 1)
            val[i] = e
            i += 1
            dd += 1
        if len(ccs) == 1:
            return (val, vec)
        for nn in ccs[1:]:
            scale = 1.0 / sqrt(nn * i * (i + nn))
            vec[i] = i * [-nn * scale] + nn * [i * scale] + [0] * (N - i - nn)
            # find eigenvalue
            type_d = not type_d
            if type_d:
                e = i + dr
                dr -= nn
            else:
                e = dr
            val[i] = e
            st = i
            i += 1
            dd = 1
            while dd < nn:
                scale = 1.0 / sqrt(i - st + dd * dd)
                vec[i] = [0] * st + (i - st) * [-scale] + [dd * scale] + [0] * (N - i - 1)
                val[i] = e
                i += 1
                dd += 1
        return (val, vec)

    def spectral_projection(u, eigenpairs):
        """
        Returns the coefficients of each eigenvector
        in a projection of the vector u onto the normalized
        eigenvectors which are contained in eigenpairs.

        eigenpairs should be a list of two objects.  The
        first is a list of eigenvalues and the second a list
        of eigenvectors.  The eigenvectors should be lists.

        There's not a lot of error checking on lengths of
        arrays, etc. so be careful.
        """
        coeff = []
        evect = eigenpairs[1]
        for ev in evect:
            c = sum(evv * uv for (evv, uv) in zip(ev, u))
            coeff.append(c)
        return coeff

    def eigenvalues(creation_sequence):
        """
        Return sequence of eigenvalues of the Laplacian of the threshold
        graph for the given creation_sequence.

        Based on the Ferrer's diagram method.  The spectrum is integral
        and is the conjugate of the degree sequence.

        See::

          @Article{degree-merris-1994,
           author = {Russel Merris},
           title = {Degree maximal graphs are Laplacian integral},
           journal = {Linear Algebra Appl.},
           year = {1994},
           volume = {199},
           pages = {381--389},
          }

        """
        degseq = degree_sequence(creation_sequence)
        degseq.sort()
        eiglist = []  # zero is always one eigenvalue
        eig = 0
        row = len(degseq)
        bigdeg = degseq.pop()
        while row:
            if bigdeg < row:
                eiglist.append(eig)
                row -= 1
            else:
                eig += 1
                if degseq:
                    bigdeg = degseq.pop()
                else:
                    bigdeg = 0
        return eiglist

    # Threshold graph creation routines

    @py_random_state(2)
    def random_threshold_sequence(n, p, seed=None):
        """
        Create a random threshold sequence of size n.
        A creation sequence is built by randomly choosing d's with
        probability p and i's with probability 1-p.

        s=nx.random_threshold_sequence(10,0.5)

        returns a threshold sequence of length 10 with equal
        probably of an i or a d at each position.

        A "random" threshold graph can be built with

        G=nx.threshold_graph(s)

        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        """
        if not (0 <= p <= 1):
            raise ValueError("p must be in [0,1]")

        cs = ["d"]  # threshold sequences always start with a d
        for i in range(1, n):
            if seed.random() < p:
                cs.append("d")
            else:
                cs.append("i")
        return cs

    # maybe *_d_threshold_sequence routines should
    # be (or be called from) a single routine with a more descriptive name
    # and a keyword parameter?
    def right_d_threshold_sequence(n, m):
        """
        Create a skewed threshold graph with a given number
        of vertices (n) and a given number of edges (m).

        The routine returns an unlabeled creation sequence
        for the threshold graph.

        FIXME: describe algorithm

        """
        cs = ["d"] + ["i"] * (n - 1)  # create sequence with n insolated nodes

        #  m <n : not enough edges, make disconnected
        if m < n:
            cs[m] = "d"
            return cs

        # too many edges
        if m > n * (n - 1) / 2:
            raise ValueError("Too many edges for this many nodes.")

        # connected case m >n-1
        ind = n - 1
        sum = n - 1
        while sum < m:
            cs[ind] = "d"
            ind -= 1
            sum += ind
        ind = m - (sum - ind)
        cs[ind] = "d"
        return cs

    def left_d_threshold_sequence(n, m):
        """
        Create a skewed threshold graph with a given number
        of vertices (n) and a given number of edges (m).

        The routine returns an unlabeled creation sequence
        for the threshold graph.

        FIXME: describe algorithm

        """
        cs = ["d"] + ["i"] * (n - 1)  # create sequence with n insolated nodes

        #  m <n : not enough edges, make disconnected
        if m < n:
            cs[m] = "d"
            return cs

        # too many edges
        if m > n * (n - 1) / 2:
            raise ValueError("Too many edges for this many nodes.")

        # Connected case when M>N-1
        cs[n - 1] = "d"
        sum = n - 1
        ind = 1
        while sum < m:
            cs[ind] = "d"
            sum += ind
            ind += 1
        if sum > m:  # be sure not to change the first vertex
            cs[sum - m] = "i"
        return cs

    @py_random_state(3)
    def swap_d(cs, p_split=1.0, p_combine=1.0, seed=None):
        """
        Perform a "swap" operation on a threshold sequence.

        The swap preserves the number of nodes and edges
        in the graph for the given sequence.
        The resulting sequence is still a threshold sequence.

        Perform one split and one combine operation on the
        'd's of a creation sequence for a threshold graph.
        This operation maintains the number of nodes and edges
        in the graph, but shifts the edges from node to node
        maintaining the threshold quality of the graph.

        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        """
        # preprocess the creation sequence
        dlist = [i for (i, node_type) in enumerate(cs[1:-1]) if node_type == "d"]
        # split
        if seed.random() < p_split:
            choice = seed.choice(dlist)
            split_to = seed.choice(range(choice))
            flip_side = choice - split_to
            if split_to != flip_side and cs[split_to] == "i" and cs[flip_side] == "i":
                cs[choice] = "i"
                cs[split_to] = "d"
                cs[flip_side] = "d"
                dlist.remove(choice)
                # don't add or combine may reverse this action
                # dlist.extend([split_to,flip_side])
        #            print >>sys.stderr,"split at %s to %s and %s"%(choice,split_to,flip_side)
        # combine
        if seed.random() < p_combine and dlist:
            first_choice = seed.choice(dlist)
            second_choice = seed.choice(dlist)
            target = first_choice + second_choice
            if target >= len(cs) or cs[target] == "d" or first_choice == second_choice:
                return cs
            # OK to combine
            cs[first_choice] = "i"
            cs[second_choice] = "i"
            cs[target] = "d"
        #        print >>sys.stderr,"combine %s and %s to make %s."%(first_choice,second_choice,target)

        return cs