_hierarchy.pyx

# cython: boundscheck=False, wraparound=False, cdivision=True
from __future__ import absolute_import

import numpy as np
cimport numpy as np
from libc.math cimport sqrt
from libc.string cimport memset
from cpython.mem cimport PyMem_Malloc, PyMem_Free

cdef extern from "numpy/npy_math.h":
    cdef enum:
        NPY_INFINITYF

ctypedef unsigned char uchar


# _hierarchy_distance_update.pxi includes the definition of linkage_distance_update
# and the distance update functions for the supported linkage methods.
include "_hierarchy_distance_update.pxi"
cdef linkage_distance_update *linkage_methods = [
    _single, _complete, _average, _centroid, _median, _ward, _weighted]
include "_structures.pxi"

cdef inline np.npy_int64 condensed_index(np.npy_int64 n, np.npy_int64 i,
                                         np.npy_int64 j):
    """
    Calculate the condensed index of element (i, j) in an n x n condensed
    matrix.
    """
    if i < j:
        return n * i - (i * (i + 1) / 2) + (j - i - 1)
    elif i > j:
        return n * j - (j * (j + 1) / 2) + (i - j - 1)


cdef inline int is_visited(uchar *bitset, int i):
    """
    Check if node i was visited.
    """
    return bitset[i >> 3] & (1 << (i & 7))


cdef inline void set_visited(uchar *bitset, int i):
    """
    Mark node i as visited.
    """
    bitset[i >> 3] |= 1 << (i & 7)


cpdef void calculate_cluster_sizes(double[:, :] Z, double[:] cs, int n):
    """
    Calculate the size of each cluster. The result is the fourth column of
    the linkage matrix.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix. The fourth column can be empty.
    cs : ndarray
        The array to store the sizes.
    n : ndarray
        The number of observations.
    """
    cdef int i, child_l, child_r

    for i in range(n - 1):
        child_l = <int>Z[i, 0]
        child_r = <int>Z[i, 1]

        if child_l >= n:
            cs[i] += cs[child_l - n]
        else:
            cs[i] += 1

        if child_r >= n:
            cs[i] += cs[child_r - n]
        else:
            cs[i] += 1


def cluster_dist(double[:, :] Z, int[:] T, double cutoff, int n):
    """
    Form flat clusters by distance criterion.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    T : ndarray
        The array to store the cluster numbers. The i'th observation belongs to
        cluster `T[i]`.
    cutoff : double
        Clusters are formed when distances are less than or equal to `cutoff`.
    n : int
        The number of observations.
    """
    cdef double[:] max_dists = np.ndarray(n, dtype=np.double)
    get_max_dist_for_each_cluster(Z, max_dists, n)
    cluster_monocrit(Z, max_dists, T, cutoff, n)


def cluster_in(double[:, :] Z, double[:, :] R, int[:] T, double cutoff, int n):
    """
    Form flat clusters by inconsistent criterion.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    R : ndarray
        The inconsistent matrix.
    T : ndarray
        The array to store the cluster numbers. The i'th observation belongs to
        cluster `T[i]`.
    cutoff : double
        Clusters are formed when the inconsistent values are less than or
        or equal to `cutoff`.
    n : int
        The number of observations.
    """
    cdef double[:] max_inconsists = np.ndarray(n, dtype=np.double)
    get_max_Rfield_for_each_cluster(Z, R, max_inconsists, n, 3)
    cluster_monocrit(Z, max_inconsists, T, cutoff, n)


def cluster_maxclust_dist(double[:, :] Z, int[:] T, int n, int mc):
    """
    Form flat clusters by maxclust criterion.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    T : ndarray
        The array to store the cluster numbers. The i'th observation belongs to
        cluster `T[i]`.
    n : int
        The number of observations.
    mc : int
        The maximum number of clusters.
    """
    cdef double[:] max_dists = np.ndarray(n, dtype=np.double)
    get_max_dist_for_each_cluster(Z, max_dists, n)
    # should use an O(n) algorithm
    cluster_maxclust_monocrit(Z, max_dists, T, n, mc)


cpdef void cluster_maxclust_monocrit(double[:, :] Z, double[:] MC, int[:] T,
                                     int n, int max_nc):
    """
    Form flat clusters by maxclust_monocrit criterion.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    MC : ndarray
        The monotonic criterion array.
    T : ndarray
        The array to store the cluster numbers. The i'th observation belongs to
        cluster `T[i]`.
    n : int
        The number of observations.
    max_nc : int
        The maximum number of clusters.
    """
    cdef int i, k, i_lc, i_rc, root, nc, lower_idx, upper_idx
    cdef double thresh
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError

    lower_idx = 0
    upper_idx = n - 1
    while upper_idx - lower_idx > 1:
        i = (lower_idx + upper_idx) >> 1
        thresh = MC[i]

        memset(visited, 0, visited_size)
        nc = 0
        k = 0
        curr_node[0] = 2 * n - 2

        while k >= 0:
            root = curr_node[k] - n
            i_lc = <int>Z[root, 0]
            i_rc = <int>Z[root, 1]

            if MC[root] <= thresh:  # this subtree forms a cluster
                nc += 1
                if nc > max_nc:  # illegal
                    break
                k -= 1
                set_visited(visited, i_lc)
                set_visited(visited, i_rc)
                continue

            if not is_visited(visited, i_lc):
                set_visited(visited, i_lc)
                if i_lc >= n:
                    k += 1
                    curr_node[k] = i_lc
                    continue
                else:  # singleton cluster
                    nc += 1
                    if nc > max_nc:
                        break

            if not is_visited(visited, i_rc):
                set_visited(visited, i_rc)
                if i_rc >= n:
                    k += 1
                    curr_node[k] = i_rc
                    continue
                else:  # singleton cluster
                    nc += 1
                    if nc > max_nc:
                        break

            k -= 1

        if nc > max_nc:
            lower_idx = i
        else:
            upper_idx = i

    PyMem_Free(visited)
    cluster_monocrit(Z, MC, T, MC[upper_idx], n)


cpdef void cluster_monocrit(double[:, :] Z, double[:] MC, int[:] T,
                            double cutoff, int n):
    """
    Form flat clusters by monocrit criterion.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    MC : ndarray
        The monotonic criterion array.
    T : ndarray
        The array to store the cluster numbers. The i'th observation belongs to
        cluster `T[i]`.
    cutoff : double
        Clusters are formed when the MC values are less than or equal to
        `cutoff`.
    n : int
        The number of observations.
    """
    cdef int k, i_lc, i_rc, root, n_cluster = 0, cluster_leader = -1
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    k = 0
    curr_node[0] = 2 * n - 2
    while k >= 0:
        root = curr_node[k] - n
        i_lc = <int>Z[root, 0]
        i_rc = <int>Z[root, 1]

        if cluster_leader == -1 and MC[root] <= cutoff:  # found a cluster
            cluster_leader = root
            n_cluster += 1

        if i_lc >= n and not is_visited(visited, i_lc):
            set_visited(visited, i_lc)
            k += 1
            curr_node[k] = i_lc
            continue

        if i_rc >= n and not is_visited(visited, i_rc):
            set_visited(visited, i_rc)
            k += 1
            curr_node[k] = i_rc
            continue

        if i_lc < n:
            if cluster_leader == -1:  # singleton cluster
                n_cluster += 1
            T[i_lc] = n_cluster

        if i_rc < n:
            if cluster_leader == -1:  # singleton cluster
                n_cluster += 1
            T[i_rc] = n_cluster

        if cluster_leader == root:  # back to the leader
            cluster_leader = -1
        k -= 1

    PyMem_Free(visited)


def cophenetic_distances(double[:, :] Z, double[:] d, int n):
    """
    Calculate the cophenetic distances between each observation

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    d : ndarray
        The condensed matrix to store the cophenetic distances.
    n : int
        The number of observations.
    """
    cdef int i, j, k, root, i_lc, i_rc, n_lc, n_rc, right_start
    cdef double dist
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)
    cdef int[:] members = np.ndarray(n, dtype=np.intc)
    cdef int[:] left_start = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    k = 0
    curr_node[0] = 2 * n - 2
    left_start[0] = 0
    while k >= 0:
        root = curr_node[k] - n
        i_lc = <int>Z[root, 0]
        i_rc = <int>Z[root, 1]

        if i_lc >= n:  # left child is not a leaf
            n_lc = <int>Z[i_lc - n, 3]

            if not is_visited(visited, i_lc):
                set_visited(visited, i_lc)
                k += 1
                curr_node[k] = i_lc
                left_start[k] = left_start[k - 1]
                continue  # visit the left subtree
        else:
            n_lc = 1
            members[left_start[k]] = i_lc

        if i_rc >= n:  # right child is not a leaf
            n_rc = <int>Z[i_rc - n, 3]

            if not is_visited(visited, i_rc):
                set_visited(visited, i_rc)
                k += 1
                curr_node[k] = i_rc
                left_start[k] = left_start[k - 1] + n_lc
                continue  # visit the right subtree
        else:
            n_rc = 1
            members[left_start[k] + n_lc] = i_rc

        # back to the root of current subtree
        dist = Z[root, 2]
        right_start = left_start[k] + n_lc
        for i in range(left_start[k], right_start):
            for j in range(right_start, right_start + n_rc):
                d[condensed_index(n, members[i], members[j])] = dist

        k -= 1  # back to parent node

    PyMem_Free(visited)


cpdef void get_max_Rfield_for_each_cluster(double[:, :] Z, double[:, :] R,
                                           double[:] max_rfs, int n, int rf):
    """
    Get the maximum statistic for each non-singleton cluster. For the i'th
    non-singleton cluster, max_rfs[i] = max{R[j, rf] j is a descendent of i}.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    R : ndarray
        The R matrix.
    max_rfs : ndarray
        The array to store the result.
    n : int
        The number of observations.
    rf : int
        Indicate which column of `R` is used.
    """
    cdef int k, i_lc, i_rc, root
    cdef double max_rf, max_l, max_r
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    k = 0
    curr_node[0] = 2 * n - 2
    while k >= 0:
        root = curr_node[k] - n
        i_lc = <int>Z[root, 0]
        i_rc = <int>Z[root, 1]

        if i_lc >= n and not is_visited(visited, i_lc):
            set_visited(visited, i_lc)
            k += 1
            curr_node[k] = i_lc
            continue

        if i_rc >= n and not is_visited(visited, i_rc):
            set_visited(visited, i_rc)
            k += 1
            curr_node[k] = i_rc
            continue

        max_rf = R[root, rf]
        if i_lc >= n:
            max_l = max_rfs[i_lc - n]
            if max_l > max_rf:
                max_rf = max_l
        if i_rc >= n:
            max_r = max_rfs[i_rc - n]
            if max_r > max_rf:
                max_rf = max_r
        max_rfs[root] = max_rf

        k -= 1

    PyMem_Free(visited)


cpdef get_max_dist_for_each_cluster(double[:, :] Z, double[:] MD, int n):
    """
    Get the maximum inconsistency coefficient for each non-singleton cluster.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    MD : ndarray
        The array to store the result.
    n : int
        The number of observations.
    """
    cdef int k, i_lc, i_rc, root
    cdef double max_dist, max_l, max_r
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    k = 0
    curr_node[0] = 2 * n - 2
    while k >= 0:
        root = curr_node[k] - n
        i_lc = <int>Z[root, 0]
        i_rc = <int>Z[root, 1]

        if i_lc >= n and not is_visited(visited, i_lc):
            set_visited(visited, i_lc)
            k += 1
            curr_node[k] = i_lc
            continue

        if i_rc >= n and not is_visited(visited, i_rc):
            set_visited(visited, i_rc)
            k += 1
            curr_node[k] = i_rc
            continue

        max_dist = Z[root, 2]
        if i_lc >= n:
            max_l = MD[i_lc - n]
            if max_l > max_dist:
                max_dist = max_l
        if i_rc >= n:
            max_r = MD[i_rc - n]
            if max_r > max_dist:
                max_dist = max_r
        MD[root] = max_dist

        k -= 1

    PyMem_Free(visited)


def inconsistent(double[:, :] Z, double[:, :] R, int n, int d):
    """
    Calculate the inconsistency statistics.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    R : ndarray
        A (n - 1) x 5 matrix to store the result. The inconsistency statistics
        `R[i]` are calculated over `d` levels below cluster i. `R[i, 0]` is the
        mean of distances. `R[i, 1]` is the standard deviation of distances.
        `R[i, 2]` is the number of clusters included. `R[i, 3]` is the
        inconsistency coefficient.

        .. math:: \\frac{\\mathtt{Z[i,2]}-\\mathtt{R[i,0]}} {R[i,1]}

    n : int
        The number of observations.
    d : int
        The number of levels included in calculation below a node.
    """
    cdef int i, k, i_lc, i_rc, root, level_count
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)
    cdef double level_sum, level_std_sum, level_std, dist

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError

    for i in range(n - 1):
        k = 0
        level_count = 0
        level_sum = 0
        level_std_sum = 0
        memset(visited, 0, visited_size)
        curr_node[0] = i

        while k >= 0:
            root = curr_node[k]

            if k < d - 1:
                i_lc = <int>Z[root, 0]
                if i_lc >= n and not is_visited(visited, i_lc):
                    set_visited(visited, i_lc)
                    k += 1
                    curr_node[k] = i_lc - n
                    continue

                i_rc = <int>Z[root, 1]
                if i_rc >= n and not is_visited(visited, i_rc):
                    set_visited(visited, i_rc)
                    k += 1
                    curr_node[k] = i_rc - n
                    continue

            dist = Z[root, 2]
            level_count += 1
            level_sum += dist
            level_std_sum += dist * dist
            k -= 1

        R[i, 0] = level_sum / level_count
        R[i, 2] = level_count
        if level_count < 2:
            level_std = (level_std_sum - (level_sum * level_sum)) / level_count
        else:
            level_std = ((level_std_sum -
                         ((level_sum * level_sum) / level_count)) /
                         (level_count - 1))
        if level_std > 0:
            level_std = sqrt(level_std)
            R[i, 1] = level_std
            R[i, 3] = (Z[i, 2] - R[i, 0]) / level_std
        else:
            R[i, 1] = 0

    PyMem_Free(visited)


def leaders(double[:, :] Z, int[:] T, int[:] L, int[:] M, int nc, int n):
    """
    Find the leader (root) of each flat cluster.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    T : ndarray
        The flat clusters assignment returned by `fcluster` or `fclusterdata`.
    L : ndarray
        `L` and `M` store the result. The leader of flat cluster `L[i]` is
        node `M[i]`.
    M : ndarray
        `L` and `M` store the result. The leader of flat cluster `L[i]` is
        node `M[i]`.
    nc : int
        The number of flat clusters.
    n : int
        The number of observations.

    Returns
    -------
    err_node : int
        Found that `T` is invalid when examining node `err_node`.
        `-1` indicates success.
    """
    cdef int k, i_lc, i_rc, root, cid_lc, cid_rc, leader_idx, result = -1
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)
    cdef int[:] cluster_ids = np.ndarray(n * 2 - 1, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    cluster_ids[:n] = T[:]
    cluster_ids[n:] = -1
    k = 0
    curr_node[0] = 2 * n - 2
    leader_idx = 0
    while k >= 0:
        root = curr_node[k] - n
        i_lc = <int>Z[root, 0]
        i_rc = <int>Z[root, 1]

        if i_lc >= n and not is_visited(visited, i_lc):
            set_visited(visited, i_lc)
            k += 1
            curr_node[k] = i_lc
            continue

        if i_rc >= n and not is_visited(visited, i_rc):
            set_visited(visited, i_rc)
            k += 1
            curr_node[k] = i_rc
            continue

        cid_lc = cluster_ids[i_lc]
        cid_rc = cluster_ids[i_rc]

        if cid_lc == cid_rc:  # left and right children in the same cluster
            cluster_ids[root + n] = cid_lc
        else:  # left and right children are both leaders
            if cid_lc != -1:
                if leader_idx >= nc:
                    result = root + n
                    break
                L[leader_idx] = i_lc
                M[leader_idx] = cid_lc
                leader_idx += 1

            if cid_rc != -1:
                if leader_idx >= nc:
                    result = root + n
                    break
                L[leader_idx] = i_rc
                M[leader_idx] = cid_rc
                leader_idx += 1

            cluster_ids[root + n] = -1

        k -= 1

    if result == -1:
        i_lc = <int>Z[n - 2, 0]
        i_rc = <int>Z[n - 2, 1]
        cid_lc = cluster_ids[i_lc]
        cid_rc = cluster_ids[i_rc]
        if cid_lc == cid_rc and cid_lc != -1:
            if leader_idx >= nc:
                result = 2 * n - 2
            else:
                L[leader_idx] = 2 * n - 2
                M[leader_idx] = cid_lc

    PyMem_Free(visited)
    return result  # -1 means success here


def linkage(double[:] dists, np.npy_int64 n, int method):
    """
    Perform hierarchy clustering.

    Parameters
    ----------
    dists : ndarray
        A condensed matrix stores the pairwise distances of the observations.
    n : int
        The number of observations.
    method : int
        The linkage method. 0: single 1: complete 2: average 3: centroid
        4: median 5: ward 6: weighted

    Returns
    -------
    Z : ndarray, shape (n - 1, 4)
        Computed linkage matrix.
    """
    Z_arr = np.empty((n - 1, 4))
    cdef double[:, :] Z = Z_arr

    cdef int i, j, k, x, y, nx, ny, ni, id_x, id_y, id_i
    cdef np.npy_int64 i_start
    cdef double current_min
    # inter-cluster dists
    cdef double[:] D = np.ndarray(n * (n - 1) / 2, dtype=np.double)
    # map the indices to node ids
    cdef int[:] id_map = np.ndarray(n, dtype=np.intc)
    cdef linkage_distance_update new_dist

    new_dist = linkage_methods[method]

    D[:] = dists
    for i in range(n):
        id_map[i] = i

    for k in range(n - 1):
        # find two closest clusters x, y (x < y)
        current_min = NPY_INFINITYF
        for i in range(n - 1):
            if id_map[i] == -1:
                continue

            i_start = condensed_index(n, i, i + 1)
            for j in range(n - i - 1):
                if D[i_start + j] < current_min:
                    current_min = D[i_start + j]
                    x = i
                    y = i + j + 1

        id_x = id_map[x]
        id_y = id_map[y]

        # get the original numbers of points in clusters x and y
        nx = 1 if id_x < n else <int>Z[id_x - n, 3]
        ny = 1 if id_y < n else <int>Z[id_y - n, 3]

        # record the new node
        Z[k, 0] = min(id_x, id_y)
        Z[k, 1] = max(id_y, id_x)
        Z[k, 2] = current_min
        Z[k, 3] = nx + ny
        id_map[x] = -1  # cluster x will be dropped
        id_map[y] = n + k  # cluster y will be replaced with the new cluster

        # update the distance matrix
        for i in range(n):
            id_i = id_map[i]
            if id_i == -1 or id_i == n + k:
                continue

            ni = 1 if id_i < n else <int>Z[id_i - n, 3]
            D[condensed_index(n, i, y)] = new_dist(
                D[condensed_index(n, i, x)],
                D[condensed_index(n, i, y)],
                current_min, nx, ny, ni)
            if i < x:
                D[condensed_index(n, i, x)] = NPY_INFINITYF
    return Z_arr


cdef Pair find_min_dist(int n, double[:] D, int[:] size, int x):
    cdef double current_min = NPY_INFINITYF
    cdef int y = -1
    cdef int i
    cdef double dist

    for i in range(x + 1, n):
        if size[i] == 0:
            continue

        dist = D[condensed_index(n, x, i)]
        if dist < current_min:
            current_min = dist
            y = i

    return Pair(y, current_min)


def fast_linkage(double[:] dists, int n, int method):
    """Perform hierarchy clustering.

    It implements "Generic Clustering Algorithm" from [1]. The worst case
    time complexity is O(N^3), but the best case time complexity is O(N^2) and
    it usually works quite close to the best case.

    Parameters
    ----------
    dists : ndarray
        A condensed matrix stores the pairwise distances of the observations.
    n : int
        The number of observations.
    method : int
        The linkage method. 0: single 1: complete 2: average 3: centroid
        4: median 5: ward 6: weighted

    Returns
    -------
    Z : ndarray, shape (n - 1, 4)
        Computed linkage matrix.

    References
    ----------
    .. [1] Daniel Mullner, "Modern hierarchical, agglomerative clustering
       algorithms", :arXiv:`1109.2378v1`.
    """
    cdef double[:, :] Z = np.empty((n - 1, 4))

    cdef double[:] D = dists.copy()  # Distances between clusters.
    cdef int[:] size = np.ones(n, dtype=np.intc)  # Sizes of clusters.
    # ID of a cluster to put into linkage matrix.
    cdef int[:] cluster_id = np.arange(n, dtype=np.intc)

    # Nearest neighbor candidate and lower bound of the distance to the
    # true nearest neighbor for each cluster among clusters with higher
    # indices (thus size is n - 1).
    cdef int[:] neighbor = np.empty(n - 1, dtype=np.intc)
    cdef double[:] min_dist = np.empty(n - 1)

    cdef linkage_distance_update new_dist = linkage_methods[method]

    cdef int i, k
    cdef int x, y, z
    cdef int nx, ny, nz
    cdef int id_x, id_y
    cdef double dist
    cdef Pair pair

    for x in range(n - 1):
        pair = find_min_dist(n, D, size, x)
        neighbor[x] = pair.key
        min_dist[x] = pair.value
    cdef Heap min_dist_heap = Heap(min_dist)

    for k in range(n - 1):
        # Theoretically speaking, this can be implemented as "while True", but
        # having a fixed size loop when floating point computations involved
        # looks more reliable. The idea that we should find the two closest
        # clusters in no more that n - k (1 for the last iteration) distance
        # updates.
        for i in range(n - k):
            pair = min_dist_heap.get_min()
            x, dist = pair.key, pair.value
            y = neighbor[x]

            if dist == D[condensed_index(n, x, y)]:
                break

            pair = find_min_dist(n, D, size, x)
            y, dist = pair.key, pair.value
            neighbor[x] = y
            min_dist[x] = dist
            min_dist_heap.change_value(x, dist)
        min_dist_heap.remove_min()

        id_x = cluster_id[x]
        id_y = cluster_id[y]
        nx = size[x]
        ny = size[y]

        if id_x > id_y:
            id_x, id_y = id_y, id_x

        Z[k, 0] = id_x
        Z[k, 1] = id_y
        Z[k, 2] = dist
        Z[k, 3] = nx + ny

        size[x] = 0  # Cluster x will be dropped.
        size[y] = nx + ny  # Cluster y will be replaced with the new cluster.
        cluster_id[y] = n + k  # Update ID of y.

        # Update the distance matrix.
        for z in range(n):
            nz = size[z]
            if nz == 0 or z == y:
                continue

            D[condensed_index(n, z, y)] = new_dist(
                D[condensed_index(n, z, x)], D[condensed_index(n, z, y)],
                dist, nx, ny, nz)

        # Reassign neighbor candidates from x to y.
        # This reassignment is just a (logical) guess.
        for z in range(x):
            if size[z] > 0 and neighbor[z] == x:
                neighbor[z] = y

        # Update lower bounds of distance.
        for z in range(y):
            if size[z] == 0:
                continue

            dist = D[condensed_index(n, z, y)]
            if dist < min_dist[z]:
                neighbor[z] = y
                min_dist[z] = dist
                min_dist_heap.change_value(z, dist)

        # Find nearest neighbor for y.
        if y < n - 1:
            pair = find_min_dist(n, D, size, y)
            z, dist = pair.key, pair.value
            if z != -1:
                neighbor[y] = z
                min_dist[y] = dist
                min_dist_heap.change_value(y, dist)

    return Z.base


def nn_chain(double[:] dists, int n, int method):
    """Perform hierarchy clustering using nearest-neighbor chain algorithm.

    Parameters
    ----------
    dists : ndarray
        A condensed matrix stores the pairwise distances of the observations.
    n : int
        The number of observations.
    method : int
        The linkage method. 0: single 1: complete 2: average 3: centroid
        4: median 5: ward 6: weighted

    Returns
    -------
    Z : ndarray, shape (n - 1, 4)
        Computed linkage matrix.
    """
    Z_arr = np.empty((n - 1, 4))
    cdef double[:, :] Z = Z_arr

    cdef double[:] D = dists.copy()  # Distances between clusters.
    cdef int[:] size = np.ones(n, dtype=np.intc)  # Sizes of clusters.

    cdef linkage_distance_update new_dist = linkage_methods[method]

    # Variables to store neighbors chain.
    cdef int[:] cluster_chain = np.ndarray(n, dtype=np.intc)
    cdef int chain_length = 0

    cdef int i, j, k, x, y, nx, ny, ni
    cdef double dist, current_min

    for k in range(n - 1):
        if chain_length == 0:
            chain_length = 1
            for i in range(n):
                if size[i] > 0:
                    cluster_chain[0] = i
                    break

        # Go through chain of neighbors until two mutual neighbors are found.
        while True:
            x = cluster_chain[chain_length - 1]

            # We want to prefer the previous element in the chain as the
            # minimum, to avoid potentially going in cycles.
            if chain_length > 1:
                y = cluster_chain[chain_length - 2]
                current_min = D[condensed_index(n, x, y)]
            else:
                current_min = NPY_INFINITYF

            for i in range(n):
                if size[i] == 0 or x == i:
                    continue

                dist = D[condensed_index(n, x, i)]
                if dist < current_min:
                    current_min = dist
                    y = i

            if chain_length > 1 and y == cluster_chain[chain_length - 2]:
                break

            cluster_chain[chain_length] = y
            chain_length += 1

        # Merge clusters x and y and pop them from stack.
        chain_length -= 2

        # This is a convention used in fastcluster.
        if x > y:
            x, y = y, x

        # get the original numbers of points in clusters x and y
        nx = size[x]
        ny = size[y]

        # Record the new node.
        Z[k, 0] = x
        Z[k, 1] = y
        Z[k, 2] = current_min
        Z[k, 3] = nx + ny
        size[x] = 0  # Cluster x will be dropped.
        size[y] = nx + ny  # Cluster y will be replaced with the new cluster

        # Update the distance matrix.
        for i in range(n):
            ni = size[i]
            if ni == 0 or i == y:
                continue

            D[condensed_index(n, i, y)] = new_dist(
                D[condensed_index(n, i, x)],
                D[condensed_index(n, i, y)],
                current_min, nx, ny, ni)

    # Sort Z by cluster distances.
    order = np.argsort(Z_arr[:, 2], kind='mergesort')
    Z_arr = Z_arr[order]

    # Find correct cluster labels inplace.
    label(Z_arr, n)

    return Z_arr


def mst_single_linkage(double[:] dists, int n):
    """Perform hierarchy clustering using MST algorithm for single linkage.

    Parameters
    ----------
    dists : ndarray
        A condensed matrix stores the pairwise distances of the observations.
    n : int
        The number of observations.

    Returns
    -------
    Z : ndarray, shape (n - 1, 4)
        Computed linkage matrix.
    """
    Z_arr = np.empty((n - 1, 4))
    cdef double[:, :] Z = Z_arr

    # Which nodes were already merged.
    cdef int[:] merged = np.zeros(n, dtype=np.intc)

    cdef double[:] D = np.empty(n)
    D[:] = NPY_INFINITYF

    cdef int i, k, x, y
    cdef double dist, current_min

    x = 0
    for k in range(n - 1):
        current_min = NPY_INFINITYF
        merged[x] = 1
        for i in range(n):
            if merged[i] == 1:
                continue

            dist = dists[condensed_index(n, x, i)]
            if D[i] > dist:
                D[i] = dist

            if D[i] < current_min:
                y = i
                current_min = D[i]

        Z[k, 0] = x
        Z[k, 1] = y
        Z[k, 2] = current_min
        x = y

    # Sort Z by cluster distances.
    order = np.argsort(Z_arr[:, 2], kind='mergesort')
    Z_arr = Z_arr[order]

    # Find correct cluster labels and compute cluster sizes inplace.
    label(Z_arr, n)

    return Z_arr


cdef class LinkageUnionFind:
    """Structure for fast cluster labeling in unsorted dendrogram."""
    cdef int[:] parent
    cdef int[:] size
    cdef int next_label

    def __init__(self, int n):
        self.parent = np.arange(2 * n - 1, dtype=np.intc)
        self.next_label = n
        self.size = np.ones(2 * n - 1, dtype=np.intc)

    cdef int merge(self, int x, int y):
        self.parent[x] = self.next_label
        self.parent[y] = self.next_label
        cdef int size = self.size[x] + self.size[y]
        self.size[self.next_label] = size
        self.next_label += 1
        return size

    cdef find(self, int x):
        cdef int p = x

        while self.parent[x] != x:
            x = self.parent[x]

        while self.parent[p] != x:
            p, self.parent[p] = self.parent[p], x

        return x


cdef label(double[:, :] Z, int n):
    """Correctly label clusters in unsorted dendrogram."""
    cdef LinkageUnionFind uf = LinkageUnionFind(n)
    cdef int i, x, y, x_root, y_root
    for i in range(n - 1):
        x, y = int(Z[i, 0]), int(Z[i, 1])
        x_root, y_root = uf.find(x), uf.find(y)
        if x_root < y_root:
            Z[i, 0], Z[i, 1] = x_root, y_root
        else:
            Z[i, 0], Z[i, 1] = y_root, x_root
        Z[i, 3] = uf.merge(x_root, y_root)


def prelist(double[:, :] Z, int[:] members, int n):
    """
    Perform a pre-order traversal on the linkage tree and get a list of ids
    of the leaves.

    Parameters
    ----------
    Z : ndarray
        The linkage matrix.
    members : ndarray
        The array to store the result.
    n : int
        The number of observations.
    """
    cdef int k, i_lc, i_rc, root, mem_idx
    cdef int[:] curr_node = np.ndarray(n, dtype=np.intc)

    cdef int visited_size = (((n * 2) - 1) >> 3) + 1
    cdef uchar *visited = <uchar *>PyMem_Malloc(visited_size)
    if not visited:
        raise MemoryError
    memset(visited, 0, visited_size)

    mem_idx = 0
    k = 0
    curr_node[0] = 2 * n - 2
    while k >= 0:
        root = curr_node[k] - n

        i_lc = <int>Z[root, 0]
        if not is_visited(visited, i_lc):
            set_visited(visited, i_lc)
            if i_lc >= n:
                k += 1
                curr_node[k] = i_lc
                continue
            else:
                members[mem_idx] = i_lc
                mem_idx += 1

        i_rc = <int>Z[root, 1]
        if not is_visited(visited, i_rc):
            set_visited(visited, i_rc)
            if i_rc >= n:
                k += 1
                curr_node[k] = i_rc
                continue
            else:
                members[mem_idx] = i_rc
                mem_idx += 1

        k -= 1

    PyMem_Free(visited)