fastcluster_dm.cpp.inc

Go to the documentation of this file.

/*
  fastcluster: Fast hierarchical clustering routines for R and Python
 
  Copyright © 2011 Daniel Müllner
  <http://danifold.net>
 
  This library implements various fast algorithms for hierarchical,
  agglomerative clustering methods:
 
  (1) Algorithms for the "stored matrix approach": the input is the array of
      pairwise dissimilarities.
 
      MST_linkage_core: single linkage clustering with the "minimum spanning
      tree algorithm (Rohlfs)
 
      NN_chain_core: nearest-neighbor-chain algorithm, suitable for single,
      complete, average, weighted and Ward linkage (Murtagh)
 
      generic_linkage: generic algorithm, suitable for all distance update
      formulas (Müllner)
 
  (2) Algorithms for the "stored data approach": the input are points in a
      vector space.
 
      MST_linkage_core_vector: single linkage clustering for vector data
 
      generic_linkage_vector: generic algorithm for vector data, suitable for
      the Ward, centroid and median methods.
 
      generic_linkage_vector_alternative: alternative scheme for updating the
      nearest neighbors. This method seems faster than "generic_linkage_vector"
      for the centroid and median methods but slower for the Ward method.
 
  All these implementation treat infinity values correctly. They throw an
  exception if a NaN distance value occurs.
*/
 
// Older versions of Microsoft Visual Studio do not have the fenv header.
#ifdef _MSC_VER
#if (_MSC_VER == 1500 || _MSC_VER == 1600)
#define NO_INCLUDE_FENV
#endif
#endif
// NaN detection via fenv might not work on systems with software
// floating-point emulation (bug report for Debian armel).
#ifdef __SOFTFP__
#define NO_INCLUDE_FENV
#endif
#ifdef NO_INCLUDE_FENV
// #pragma message("Do not use fenv header.")
#else
// #pragma message("Use fenv header. If there is a warning about unknown #pragma STDC FENV_ACCESS, this can be ignored.")
// TODO
//#pragma STDC FENV_ACCESS ON
#include <fenv.h>
#endif
 
#include <cmath> // for std::pow, std::sqrt
#include <cstddef> // for std::ptrdiff_t
#include <limits> // for std::numeric_limits<...>::infinity()
#include <algorithm> // for std::fill_n
#include <stdexcept> // for std::runtime_error
#include <string> // for std::string
 
#include <cfloat> // also for DBL_MAX, DBL_MIN
#ifndef DBL_MANT_DIG
#error The constant DBL_MANT_DIG could not be defined.
#endif
#define T_FLOAT_MANT_DIG DBL_MANT_DIG
 
#ifndef LONG_MAX
#include <climits>
#endif
#ifndef LONG_MAX
#error The constant LONG_MAX could not be defined.
#endif
#ifndef INT_MAX
#error The constant INT_MAX could not be defined.
#endif
 
#ifndef INT32_MAX
#ifdef _MSC_VER
#if _MSC_VER >= 1600
#define __STDC_LIMIT_MACROS
#include <stdint.h>
#else
typedef __int32 int_fast32_t;
typedef __int64 int64_t;
#endif
#else
#define __STDC_LIMIT_MACROS
#include <stdint.h>
#endif
#endif
 
#define FILL_N std::fill_n
#ifdef _MSC_VER
#if _MSC_VER < 1600
#undef FILL_N
#define FILL_N stdext::unchecked_fill_n
#endif
#endif
 
// Suppress warnings about (potentially) uninitialized variables.
#ifdef _MSC_VER
    #pragma warning (disable:4700)
#endif
 
#ifndef HAVE_DIAGNOSTIC
#if __GNUC__ > 4 || (__GNUC__ == 4 && (__GNUC_MINOR__ >= 6))
#define HAVE_DIAGNOSTIC 1
#endif
#endif
 
#ifndef HAVE_VISIBILITY
#if __GNUC__ >= 4
#define HAVE_VISIBILITY 1
#endif
#endif
 
/* Since the public interface is given by the Python respectively R interface,
 * we do not want other symbols than the interface initalization routines to be
 * visible in the shared object file. The "visibility" switch is a GCC concept.
 * Hiding symbols keeps the relocation table small and decreases startup time.
 * See http://gcc.gnu.org/wiki/Visibility
 */
#if HAVE_VISIBILITY
#pragma GCC visibility push(hidden)
#endif
 
typedef int_fast32_t t_index;
#ifndef INT32_MAX
#define MAX_INDEX 0x7fffffffL
#else
#define MAX_INDEX INT32_MAX
#endif
#if (LONG_MAX < MAX_INDEX)
#error The integer format "t_index" must not have a greater range than "long int".
#endif
#if (INT_MAX > MAX_INDEX)
#error The integer format "int" must not have a greater range than "t_index".
#endif
typedef double t_float;
 
/* Method codes.
 
   These codes must agree with the METHODS array in fastcluster.R and the
   dictionary mthidx in fastcluster.py.
*/
enum method_codes {
  // non-Euclidean methods
  METHOD_METR_SINGLE           = 0,
  METHOD_METR_COMPLETE         = 1,
  METHOD_METR_AVERAGE          = 2,
  METHOD_METR_WEIGHTED         = 3,
  METHOD_METR_WARD             = 4,
  METHOD_METR_WARD_D           = METHOD_METR_WARD,
  METHOD_METR_CENTROID         = 5,
  METHOD_METR_MEDIAN           = 6,
  METHOD_METR_WARD_D2          = 7,
 
  MIN_METHOD_CODE              = 0,
  MAX_METHOD_CODE              = 7
};
 
enum method_codes_vector {
  // Euclidean methods
  METHOD_VECTOR_SINGLE         = 0,
  METHOD_VECTOR_WARD           = 1,
  METHOD_VECTOR_CENTROID       = 2,
  METHOD_VECTOR_MEDIAN         = 3,
 
  MIN_METHOD_VECTOR_CODE       = 0,
  MAX_METHOD_VECTOR_CODE       = 3
};
 
// self-destructing array pointer
template <typename type>
class auto_array_ptr{
private:
  type * ptr;
  auto_array_ptr(auto_array_ptr const &); // non construction-copyable
  auto_array_ptr& operator=(auto_array_ptr const &); // non copyable
public:
  auto_array_ptr()
    : ptr(NULL)
  { }
  template <typename index>
  auto_array_ptr(index const size)
    : ptr(new type[size])
  { }
  template <typename index, typename value>
  auto_array_ptr(index const size, value const val)
    : ptr(new type[size])
  {
    FILL_N(ptr, size, val);
  }
  ~auto_array_ptr() {
    delete [] ptr; }
  void free() {
    delete [] ptr;
    ptr = NULL;
  }
  template <typename index>
  void init(index const size) {
    ptr = new type [size];
  }
  template <typename index, typename value>
  void init(index const size, value const val) {
    init(size);
    FILL_N(ptr, size, val);
  }
  inline operator type *() const { return ptr; }
};
 
struct node {
  t_index node1, node2;
  t_float dist;
};
 
inline bool operator< (const node a, const node b) {
  return (a.dist < b.dist);
}
 
class cluster_result {
private:
  auto_array_ptr<node> Z;
  t_index pos;
 
public:
  cluster_result(const t_index size)
    : Z(size)
    , pos(0)
  {}
 
  void append(const t_index node1, const t_index node2, const t_float dist) {
    Z[pos].node1 = node1;
    Z[pos].node2 = node2;
    Z[pos].dist  = dist;
    ++pos;
  }
 
  node * operator[] (const t_index idx) const { return Z + idx; }
 
  /* Define several methods to postprocess the distances. All these functions
     are monotone, so they do not change the sorted order of distances. */
 
  void sqrt() const {
    for (node * ZZ=Z; ZZ!=Z+pos; ++ZZ) {
      ZZ->dist = std::sqrt(ZZ->dist);
    }
  }
 
  void sqrt(const t_float) const { // ignore the argument
    sqrt();
  }
 
  void sqrtdouble(const t_float) const { // ignore the argument
    for (node * ZZ=Z; ZZ!=Z+pos; ++ZZ) {
      ZZ->dist = std::sqrt(2*ZZ->dist);
    }
  }
 
  #ifdef R_pow
  #define my_pow R_pow
  #else
  #define my_pow std::pow
  #endif
 
  void power(const t_float p) const {
    t_float const q = 1/p;
    for (node * ZZ=Z; ZZ!=Z+pos; ++ZZ) {
      ZZ->dist = my_pow(ZZ->dist,q);
    }
  }
 
  void plusone(const t_float) const { // ignore the argument
    for (node * ZZ=Z; ZZ!=Z+pos; ++ZZ) {
      ZZ->dist += 1;
    }
  }
 
  void divide(const t_float denom) const {
    for (node * ZZ=Z; ZZ!=Z+pos; ++ZZ) {
      ZZ->dist /= denom;
    }
  }
};
 
class doubly_linked_list {
  /*
    Class for a doubly linked list. Initially, the list is the integer range
    [0, size]. We provide a forward iterator and a method to delete an index
    from the list.
 
    Typical use: for (i=L.start; L<size; i=L.succ[I])
    or
    for (i=somevalue; L<size; i=L.succ[I])
  */
public:
  t_index start;
  auto_array_ptr<t_index> succ;
 
private:
  auto_array_ptr<t_index> pred;
  // Not necessarily private, we just do not need it in this instance.
 
public:
  doubly_linked_list(const t_index size)
    // Initialize to the given size.
    : start(0)
    , succ(size+1)
    , pred(size+1)
  {
    for (t_index i=0; i<size; ++i) {
      pred[i+1] = i;
      succ[i] = i+1;
    }
    // pred[0] is never accessed!
    //succ[size] is never accessed!
  }
 
  ~doubly_linked_list() {}
 
  void remove(const t_index idx) {
    // Remove an index from the list.
    if (idx==start) {
      start = succ[idx];
    }
    else {
      succ[pred[idx]] = succ[idx];
      pred[succ[idx]] = pred[idx];
    }
    succ[idx] = 0; // Mark as inactive
  }
 
  bool is_inactive(t_index idx) const {
    return (succ[idx]==0);
  }
};
 
// Indexing functions
// D is the upper triangular part of a symmetric (NxN)-matrix
// We require r_ < c_ !
#define D_(r_,c_) ( D[(static_cast<std::ptrdiff_t>(2*N-3-(r_))*(r_)>>1)+(c_)-1] )
// Z is an ((N-1)x4)-array
#define Z_(_r, _c) (Z[(_r)*4 + (_c)])
 
/*
  Lookup function for a union-find data structure.
 
  The function finds the root of idx by going iteratively through all
  parent elements until a root is found. An element i is a root if
  nodes[i] is zero. To make subsequent searches faster, the entry for
  idx and all its parents is updated with the root element.
 */
class union_find {
private:
  auto_array_ptr<t_index> parent;
  t_index nextparent;
 
public:
  union_find(const t_index size)
    : parent(size>0 ? 2*size-1 : 0, 0)
    , nextparent(size)
  { }
 
  t_index Find (t_index idx) const {
    if (parent[idx] != 0 ) { // a → b
      t_index p = idx;
      idx = parent[idx];
      if (parent[idx] != 0 ) { // a → b → c
        do {
          idx = parent[idx];
        } while (parent[idx] != 0);
        do {
          t_index tmp = parent[p];
          parent[p] = idx;
          p = tmp;
        } while (parent[p] != idx);
      }
    }
    return idx;
  }
 
  void Union (const t_index node1, const t_index node2) {
    parent[node1] = parent[node2] = nextparent++;
  }
};
 
class nan_error{};
#ifdef FE_INVALID
class fenv_error{};
#endif
 
static void MST_linkage_core(const t_index N, const t_float * const D,
                             cluster_result & Z2) {
/*
    N: integer, number of data points
    D: condensed distance matrix N*(N-1)/2
    Z2: output data structure
 
    The basis of this algorithm is an algorithm by Rohlf:
 
    F. James Rohlf, Hierarchical clustering using the minimum spanning tree,
    The Computer Journal, vol. 16, 1973, p. 93–95.
*/
  t_index i;
  t_index idx2;
  doubly_linked_list active_nodes(N);
  auto_array_ptr<t_float> d(N);
 
  t_index prev_node;
  t_float min;
 
  // first iteration
  idx2 = 1;
  min = std::numeric_limits<t_float>::infinity();
  for (i=1; i<N; ++i) {
    d[i] = D[i-1];
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
    if (d[i] < min) {
      min = d[i];
      idx2 = i;
    }
    else if (fc_isnan(d[i]))
      assert(false && "fastcluster: nan error");
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  }
  Z2.append(0, idx2, min);
 
  for (t_index j=1; j<N-1; ++j) {
    prev_node = idx2;
    active_nodes.remove(prev_node);
 
    idx2 = active_nodes.succ[0];
    min = d[idx2];
    for (i=idx2; i<prev_node; i=active_nodes.succ[i]) {
      t_float tmp = D_(i, prev_node);
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
      if (tmp < d[i])
        d[i] = tmp;
      else if (fc_isnan(tmp))
        assert(false && "fastcluster: nan error");
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
      if (d[i] < min) {
        min = d[i];
        idx2 = i;
      }
    }
    for (; i<N; i=active_nodes.succ[i]) {
      t_float tmp = D_(prev_node, i);
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
      if (d[i] > tmp)
        d[i] = tmp;
      else if (fc_isnan(tmp))
        assert(false && "fastcluster: nan error");
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
      if (d[i] < min) {
        min = d[i];
        idx2 = i;
      }
    }
    Z2.append(prev_node, idx2, min);
  }
}
 
/* Functions for the update of the dissimilarity array */
 
inline static void f_single( t_float * const b, const t_float a ) {
  if (*b > a) *b = a;
}
inline static void f_complete( t_float * const b, const t_float a ) {
  if (*b < a) *b = a;
}
inline static void f_average( t_float * const b, const t_float a, const t_float s, const t_float t) {
  *b = s*a + t*(*b);
  #ifndef FE_INVALID
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
  if (fc_isnan(*b)) {
    assert(false && "fastcluster: nan error");
  }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  #endif
}
inline static void f_weighted( t_float * const b, const t_float a) {
  *b = (a+*b)*.5;
  #ifndef FE_INVALID
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
  if (fc_isnan(*b)) {
    assert(false && "fastcluster: nan error");
  }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  #endif
}
inline static void f_ward( t_float * const b, const t_float a, const t_float c, const t_float s, const t_float t, const t_float v) {
  *b = ( (v+s)*a - v*c + (v+t)*(*b) ) / (s+t+v);
  //*b = a+(*b)-(t*a+s*(*b)+v*c)/(s+t+v);
  #ifndef FE_INVALID
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
  if (fc_isnan(*b)) {
    assert(false && "fastcluster: nan error");
  }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  #endif
}
inline static void f_centroid( t_float * const b, const t_float a, const t_float stc, const t_float s, const t_float t) {
  *b = s*a - stc + t*(*b);
  #ifndef FE_INVALID
  if (fc_isnan(*b)) {
    assert(false && "fastcluster: nan error");
  }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  #endif
}
inline static void f_median( t_float * const b, const t_float a, const t_float c_4) {
  *b = (a+(*b))*.5 - c_4;
  #ifndef FE_INVALID
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
  if (fc_isnan(*b)) {
    assert(false && "fastcluster: nan error");
  }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  #endif
}
 
template <method_codes method, typename t_members>
static void NN_chain_core(const t_index N, t_float * const D, t_members * const members, cluster_result & Z2) {
/*
    N: integer
    D: condensed distance matrix N*(N-1)/2
    Z2: output data structure
 
    This is the NN-chain algorithm, described on page 86 in the following book:
 
    Fionn Murtagh, Multidimensional Clustering Algorithms,
    Vienna, Würzburg: Physica-Verlag, 1985.
*/
  t_index i;
 
  auto_array_ptr<t_index> NN_chain(N);
  t_index NN_chain_tip = 0;
 
  t_index idx1, idx2;
 
  t_float size1, size2;
  doubly_linked_list active_nodes(N);
 
  t_float min;
 
  for (t_float const * DD=D; DD!=D+(static_cast<std::ptrdiff_t>(N)*(N-1)>>1);
       ++DD) {
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
    if (fc_isnan(*DD)) {
      assert(false && "fastcluster: nan error");
    }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  }
 
  #ifdef FE_INVALID
  if (feclearexcept(FE_INVALID)) assert(false && "fastcluster: fenv error");
  #endif
 
  for (t_index j=0; j<N-1; ++j) {
    if (NN_chain_tip <= 3) {
      NN_chain[0] = idx1 = active_nodes.start;
      NN_chain_tip = 1;
 
      idx2 = active_nodes.succ[idx1];
      min = D_(idx1,idx2);
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i]) {
        if (D_(idx1,i) < min) {
          min = D_(idx1,i);
          idx2 = i;
        }
      }
    }  // a: idx1   b: idx2
    else {
      NN_chain_tip -= 3;
      idx1 = NN_chain[NN_chain_tip-1];
      idx2 = NN_chain[NN_chain_tip];
      min = idx1<idx2 ? D_(idx1,idx2) : D_(idx2,idx1);
    }  // a: idx1   b: idx2
 
    do {
      NN_chain[NN_chain_tip] = idx2;
 
      for (i=active_nodes.start; i<idx2; i=active_nodes.succ[i]) {
        if (D_(i,idx2) < min) {
          min = D_(i,idx2);
          idx1 = i;
        }
      }
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i]) {
        if (D_(idx2,i) < min) {
          min = D_(idx2,i);
          idx1 = i;
        }
      }
 
      idx2 = idx1;
      idx1 = NN_chain[NN_chain_tip++];
 
    } while (idx2 != NN_chain[NN_chain_tip-2]);
 
    Z2.append(idx1, idx2, min);
 
    if (idx1>idx2) {
      t_index tmp = idx1;
      idx1 = idx2;
      idx2 = tmp;
    }
 
    if (method==METHOD_METR_AVERAGE ||
        method==METHOD_METR_WARD) {
      size1 = static_cast<t_float>(members[idx1]);
      size2 = static_cast<t_float>(members[idx2]);
      members[idx2] += members[idx1];
    }
 
    // Remove the smaller index from the valid indices (active_nodes).
    active_nodes.remove(idx1);
 
    switch (method) {
    case METHOD_METR_SINGLE:
      /*
      Single linkage.
 
      Characteristic: new distances are never longer than the old distances.
      */
      // Update the distance matrix in the range [start, idx1).
      for (i=active_nodes.start; i<idx1; i=active_nodes.succ[i])
        f_single(&D_(i, idx2), D_(i, idx1) );
      // Update the distance matrix in the range (idx1, idx2).
      for (; i<idx2; i=active_nodes.succ[i])
        f_single(&D_(i, idx2), D_(idx1, i) );
      // Update the distance matrix in the range (idx2, N).
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i])
        f_single(&D_(idx2, i), D_(idx1, i) );
      break;
 
    case METHOD_METR_COMPLETE:
      /*
      Complete linkage.
 
      Characteristic: new distances are never shorter than the old distances.
      */
      // Update the distance matrix in the range [start, idx1).
      for (i=active_nodes.start; i<idx1; i=active_nodes.succ[i])
        f_complete(&D_(i, idx2), D_(i, idx1) );
      // Update the distance matrix in the range (idx1, idx2).
      for (; i<idx2; i=active_nodes.succ[i])
        f_complete(&D_(i, idx2), D_(idx1, i) );
      // Update the distance matrix in the range (idx2, N).
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i])
        f_complete(&D_(idx2, i), D_(idx1, i) );
      break;
 
    case METHOD_METR_AVERAGE: {
      /*
      Average linkage.
 
      Shorter and longer distances can occur.
      */
      // Update the distance matrix in the range [start, idx1).
      t_float s = size1/(size1+size2);
      t_float t = size2/(size1+size2);
      for (i=active_nodes.start; i<idx1; i=active_nodes.succ[i])
        f_average(&D_(i, idx2), D_(i, idx1), s, t );
      // Update the distance matrix in the range (idx1, idx2).
      for (; i<idx2; i=active_nodes.succ[i])
        f_average(&D_(i, idx2), D_(idx1, i), s, t );
      // Update the distance matrix in the range (idx2, N).
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i])
        f_average(&D_(idx2, i), D_(idx1, i), s, t );
      break;
    }
 
    case METHOD_METR_WEIGHTED:
      /*
      Weighted linkage.
 
      Shorter and longer distances can occur.
      */
      // Update the distance matrix in the range [start, idx1).
      for (i=active_nodes.start; i<idx1; i=active_nodes.succ[i])
        f_weighted(&D_(i, idx2), D_(i, idx1) );
      // Update the distance matrix in the range (idx1, idx2).
      for (; i<idx2; i=active_nodes.succ[i])
        f_weighted(&D_(i, idx2), D_(idx1, i) );
      // Update the distance matrix in the range (idx2, N).
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i])
        f_weighted(&D_(idx2, i), D_(idx1, i) );
      break;
 
    case METHOD_METR_WARD:
      /*
      Ward linkage.
 
      Shorter and longer distances can occur, not smaller than min(d1,d2)
      but maybe bigger than max(d1,d2).
      */
      // Update the distance matrix in the range [start, idx1).
      //t_float v = static_cast<t_float>(members[i]);
      for (i=active_nodes.start; i<idx1; i=active_nodes.succ[i])
        f_ward(&D_(i, idx2), D_(i, idx1), min,
               size1, size2, static_cast<t_float>(members[i]) );
      // Update the distance matrix in the range (idx1, idx2).
      for (; i<idx2; i=active_nodes.succ[i])
        f_ward(&D_(i, idx2), D_(idx1, i), min,
               size1, size2, static_cast<t_float>(members[i]) );
      // Update the distance matrix in the range (idx2, N).
      for (i=active_nodes.succ[idx2]; i<N; i=active_nodes.succ[i])
        f_ward(&D_(idx2, i), D_(idx1, i), min,
               size1, size2, static_cast<t_float>(members[i]) );
      break;
 
    default:
      assert(false && "fastcluster: invalid method");
    }
  }
  #ifdef FE_INVALID
  if (fetestexcept(FE_INVALID)) assert(false && "fastcluster: fenv error");
  #endif
}
 
class binary_min_heap {
  /*
  Class for a binary min-heap. The data resides in an array A. The elements of
  A are not changed but two lists I and R of indices are generated which point
  to elements of A and backwards.
 
  The heap tree structure is
 
     H[2*i+1]     H[2*i+2]
         \            /
          \          /
           ≤        ≤
            \      /
             \    /
              H[i]
 
  where the children must be less or equal than their parent. Thus, H[0]
  contains the minimum. The lists I and R are made such that H[i] = A[I[i]]
  and R[I[i]] = i.
 
  This implementation is not designed to handle NaN values.
  */
private:
  t_float * const A;
  t_index size;
  auto_array_ptr<t_index> I;
  auto_array_ptr<t_index> R;
 
  // no default constructor
  binary_min_heap();
  // noncopyable
  binary_min_heap(binary_min_heap const &);
  binary_min_heap & operator=(binary_min_heap const &);
 
public:
  binary_min_heap(t_float * const A_, const t_index size_)
    : A(A_), size(size_), I(size), R(size)
  { // Allocate memory and initialize the lists I and R to the identity. This
    // does not make it a heap. Call heapify afterwards!
    for (t_index i=0; i<size; ++i)
      R[i] = I[i] = i;
  }
 
  binary_min_heap(t_float * const A_, const t_index size1, const t_index size2,
                  const t_index start)
    : A(A_), size(size1), I(size1), R(size2)
  { // Allocate memory and initialize the lists I and R to the identity. This
    // does not make it a heap. Call heapify afterwards!
    for (t_index i=0; i<size; ++i) {
      R[i+start] = i;
      I[i] = i + start;
    }
  }
 
  ~binary_min_heap() {}
 
  void heapify() {
    // Arrange the indices I and R so that H[i] := A[I[i]] satisfies the heap
    // condition H[i] < H[2*i+1] and H[i] < H[2*i+2] for each i.
    //
    // Complexity: Θ(size)
    // Reference: Cormen, Leiserson, Rivest, Stein, Introduction to Algorithms,
    // 3rd ed., 2009, Section 6.3 “Building a heap”
    t_index idx;
    for (idx=(size>>1); idx>0; ) {
      --idx;
      update_geq_(idx);
    }
  }
 
  inline t_index argmin() const {
    // Return the minimal element.
    return I[0];
  }
 
  void heap_pop() {
    // Remove the minimal element from the heap.
    --size;
    I[0] = I[size];
    R[I[0]] = 0;
    update_geq_(0);
  }
 
  void remove(t_index idx) {
    // Remove an element from the heap.
    --size;
    R[I[size]] = R[idx];
    I[R[idx]] = I[size];
    if ( H(size)<=A[idx] ) {
      update_leq_(R[idx]);
    }
    else {
      update_geq_(R[idx]);
    }
  }
 
  void replace ( const t_index idxold, const t_index idxnew,
                 const t_float val) {
    R[idxnew] = R[idxold];
    I[R[idxnew]] = idxnew;
    if (val<=A[idxold])
      update_leq(idxnew, val);
    else
      update_geq(idxnew, val);
  }
 
  void update ( const t_index idx, const t_float val ) const {
    // Update the element A[i] with val and re-arrange the indices to preserve
    // the heap condition.
    if (val<=A[idx])
      update_leq(idx, val);
    else
      update_geq(idx, val);
  }
 
  void update_leq ( const t_index idx, const t_float val ) const {
    // Use this when the new value is not more than the old value.
    A[idx] = val;
    update_leq_(R[idx]);
  }
 
  void update_geq ( const t_index idx, const t_float val ) const {
    // Use this when the new value is not less than the old value.
    A[idx] = val;
    update_geq_(R[idx]);
  }
 
private:
  void update_leq_ (t_index i) const {
    t_index j;
    for ( ; (i>0) && ( H(i)<H(j=(i-1)>>1) ); i=j)
      heap_swap(i,j);
  }
 
  void update_geq_ (t_index i) const {
    t_index j;
    for ( ; (j=2*i+1)<size; i=j) {
      if ( H(j)>=H(i) ) {
        ++j;
        if ( j>=size || H(j)>=H(i) ) break;
      }
      else if ( j+1<size && H(j+1)<H(j) ) ++j;
      heap_swap(i, j);
    }
  }
 
  void heap_swap(const t_index i, const t_index j) const {
    // Swap two indices.
    t_index tmp = I[i];
    I[i] = I[j];
    I[j] = tmp;
    R[I[i]] = i;
    R[I[j]] = j;
  }
 
  inline t_float H(const t_index i) const {
    return A[I[i]];
  }
 
};
 
template <method_codes method, typename t_members>
static void generic_linkage(const t_index N, t_float * const D, t_members * const members, cluster_result & Z2) {
  /*
    N: integer, number of data points
    D: condensed distance matrix N*(N-1)/2
    Z2: output data structure
  */
 
  const t_index N_1 = N-1;
  t_index i, j; // loop variables
  t_index idx1, idx2; // row and column indices
 
  auto_array_ptr<t_index> n_nghbr(N_1); // array of nearest neighbors
  auto_array_ptr<t_float> mindist(N_1); // distances to the nearest neighbors
  auto_array_ptr<t_index> row_repr(N); // row_repr[i]: node number that the
                                       // i-th row represents
  doubly_linked_list active_nodes(N);
  binary_min_heap nn_distances(&*mindist, N_1); // minimum heap structure for
                        // the distance to the nearest neighbor of each point
  t_index node1, node2; // node numbers in the output
  t_float size1, size2; // and their cardinalities
 
  t_float min; // minimum and row index for nearest-neighbor search
  t_index idx;
 
  for (i=0; i<N; ++i)
    // Build a list of row ↔ node label assignments.
    // Initially i ↦ i
    row_repr[i] = i;
 
  // Initialize the minimal distances:
  // Find the nearest neighbor of each point.
  // n_nghbr[i] = argmin_{j>i} D(i,j) for i in range(N-1)
  t_float const * DD = D;
  for (i=0; i<N_1; ++i) {
    min = std::numeric_limits<t_float>::infinity();
    for (idx=j=i+1; j<N; ++j, ++DD) {
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
      if (*DD<min) {
        min = *DD;
        idx = j;
      }
      else if (fc_isnan(*DD))
        assert(false && "fastcluster: nan error");
    }
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
    mindist[i] = min;
    n_nghbr[i] = idx;
  }
 
  // Put the minimal distances into a heap structure to make the repeated
  // global minimum searches fast.
  nn_distances.heapify();
 
  #ifdef FE_INVALID
  if (feclearexcept(FE_INVALID)) assert(false && "fastcluster: fenv error");
  #endif
 
  // Main loop: We have N-1 merging steps.
  for (i=0; i<N_1; ++i) {
    /*
      Here is a special feature that allows fast bookkeeping and updates of the
      minimal distances.
 
      mindist[i] stores a lower bound on the minimum distance of the point i to
      all points of higher index:
 
          mindist[i] ≥ min_{j>i} D(i,j)
 
      Normally, we have equality. However, this minimum may become invalid due
      to the updates in the distance matrix. The rules are:
 
      1) If mindist[i] is equal to D(i, n_nghbr[i]), this is the correct
         minimum and n_nghbr[i] is a nearest neighbor.
 
      2) If mindist[i] is smaller than D(i, n_nghbr[i]), this might not be the
         correct minimum. The minimum needs to be recomputed.
 
      3) mindist[i] is never bigger than the true minimum. Hence, we never
         miss the true minimum if we take the smallest mindist entry,
         re-compute the value if necessary (thus maybe increasing it) and
         looking for the now smallest mindist entry until a valid minimal
         entry is found. This step is done in the lines below.
 
      The update process for D below takes care that these rules are
      fulfilled. This makes sure that the minima in the rows D(i,i+1:)of D are
      re-calculated when necessary but re-calculation is avoided whenever
      possible.
 
      The re-calculation of the minima makes the worst-case runtime of this
      algorithm cubic in N. We avoid this whenever possible, and in most cases
      the runtime appears to be quadratic.
    */
    idx1 = nn_distances.argmin();
    if (method != METHOD_METR_SINGLE) {
      while ( mindist[idx1] < D_(idx1, n_nghbr[idx1]) ) {
        // Recompute the minimum mindist[idx1] and n_nghbr[idx1].
        n_nghbr[idx1] = j = active_nodes.succ[idx1]; // exists, maximally N-1
        min = D_(idx1,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          if (D_(idx1,j)<min) {
            min = D_(idx1,j);
            n_nghbr[idx1] = j;
          }
        }
        /* Update the heap with the new true minimum and search for the
           (possibly different) minimal entry. */
        nn_distances.update_geq(idx1, min);
        idx1 = nn_distances.argmin();
      }
    }
 
    nn_distances.heap_pop(); // Remove the current minimum from the heap.
    idx2 = n_nghbr[idx1];
 
    // Write the newly found minimal pair of nodes to the output array.
    node1 = row_repr[idx1];
    node2 = row_repr[idx2];
 
    if (method==METHOD_METR_AVERAGE ||
        method==METHOD_METR_WARD ||
        method==METHOD_METR_CENTROID) {
      size1 = static_cast<t_float>(members[idx1]);
      size2 = static_cast<t_float>(members[idx2]);
      members[idx2] += members[idx1];
    }
    Z2.append(node1, node2, mindist[idx1]);
 
    // Remove idx1 from the list of active indices (active_nodes).
    active_nodes.remove(idx1);
    // Index idx2 now represents the new (merged) node with label N+i.
    row_repr[idx2] = N+i;
 
    // Update the distance matrix
    switch (method) {
    case METHOD_METR_SINGLE:
      /*
        Single linkage.
 
        Characteristic: new distances are never longer than the old distances.
      */
      // Update the distance matrix in the range [start, idx1).
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_single(&D_(j, idx2), D_(j, idx1));
        if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_single(&D_(j, idx2), D_(idx1, j));
        // If the new value is below the old minimum in a row, update
        // the mindist and n_nghbr arrays.
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      // Recompute the minimum mindist[idx2] and n_nghbr[idx2].
      if (idx2<N_1) {
        min = mindist[idx2];
        for (j=active_nodes.succ[idx2]; j<N; j=active_nodes.succ[j]) {
          f_single(&D_(idx2, j), D_(idx1, j) );
          if (D_(idx2, j) < min) {
            n_nghbr[idx2] = j;
            min = D_(idx2, j);
          }
        }
        nn_distances.update_leq(idx2, min);
      }
      break;
 
    case METHOD_METR_COMPLETE:
      /*
        Complete linkage.
 
        Characteristic: new distances are never shorter than the old distances.
      */
      // Update the distance matrix in the range [start, idx1).
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_complete(&D_(j, idx2), D_(j, idx1) );
        if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j])
        f_complete(&D_(j, idx2), D_(idx1, j) );
      // Update the distance matrix in the range (idx2, N).
      for (j=active_nodes.succ[idx2]; j<N; j=active_nodes.succ[j])
        f_complete(&D_(idx2, j), D_(idx1, j) );
      break;
 
    case METHOD_METR_AVERAGE: {
      /*
        Average linkage.
 
        Shorter and longer distances can occur.
      */
      // Update the distance matrix in the range [start, idx1).
      t_float s = size1/(size1+size2);
      t_float t = size2/(size1+size2);
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_average(&D_(j, idx2), D_(j, idx1), s, t);
        if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_average(&D_(j, idx2), D_(idx1, j), s, t);
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        f_average(&D_(idx2, j), D_(idx1, j), s, t);
        min = D_(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          f_average(&D_(idx2, j), D_(idx1, j), s, t);
          if (D_(idx2,j) < min) {
            min = D_(idx2,j);
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
    }
 
    case METHOD_METR_WEIGHTED:
      /*
        Weighted linkage.
 
        Shorter and longer distances can occur.
      */
      // Update the distance matrix in the range [start, idx1).
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_weighted(&D_(j, idx2), D_(j, idx1) );
        if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_weighted(&D_(j, idx2), D_(idx1, j) );
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        f_weighted(&D_(idx2, j), D_(idx1, j) );
        min = D_(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          f_weighted(&D_(idx2, j), D_(idx1, j) );
          if (D_(idx2,j) < min) {
            min = D_(idx2,j);
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
 
    case METHOD_METR_WARD:
      /*
        Ward linkage.
 
        Shorter and longer distances can occur, not smaller than min(d1,d2)
        but maybe bigger than max(d1,d2).
      */
      // Update the distance matrix in the range [start, idx1).
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_ward(&D_(j, idx2), D_(j, idx1), mindist[idx1],
               size1, size2, static_cast<t_float>(members[j]) );
        if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_ward(&D_(j, idx2), D_(idx1, j), mindist[idx1], size1, size2,
               static_cast<t_float>(members[j]) );
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        f_ward(&D_(idx2, j), D_(idx1, j), mindist[idx1],
               size1, size2, static_cast<t_float>(members[j]) );
        min = D_(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          f_ward(&D_(idx2, j), D_(idx1, j), mindist[idx1],
                 size1, size2, static_cast<t_float>(members[j]) );
          if (D_(idx2,j) < min) {
            min = D_(idx2,j);
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
 
    case METHOD_METR_CENTROID: {
      /*
        Centroid linkage.
 
        Shorter and longer distances can occur, not bigger than max(d1,d2)
        but maybe smaller than min(d1,d2).
      */
      // Update the distance matrix in the range [start, idx1).
      t_float s = size1/(size1+size2);
      t_float t = size2/(size1+size2);
      t_float stc = s*t*mindist[idx1];
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_centroid(&D_(j, idx2), D_(j, idx1), stc, s, t);
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
        else if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_centroid(&D_(j, idx2), D_(idx1, j), stc, s, t);
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        f_centroid(&D_(idx2, j), D_(idx1, j), stc, s, t);
        min = D_(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          f_centroid(&D_(idx2, j), D_(idx1, j), stc, s, t);
          if (D_(idx2,j) < min) {
            min = D_(idx2,j);
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
    }
 
    case METHOD_METR_MEDIAN: {
      /*
        Median linkage.
 
        Shorter and longer distances can occur, not bigger than max(d1,d2)
        but maybe smaller than min(d1,d2).
      */
      // Update the distance matrix in the range [start, idx1).
      t_float c_4 = mindist[idx1]*.25;
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        f_median(&D_(j, idx2), D_(j, idx1), c_4 );
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
        else if (n_nghbr[j] == idx1)
          n_nghbr[j] = idx2;
      }
      // Update the distance matrix in the range (idx1, idx2).
      for (; j<idx2; j=active_nodes.succ[j]) {
        f_median(&D_(j, idx2), D_(idx1, j), c_4 );
        if (D_(j, idx2) < mindist[j]) {
          nn_distances.update_leq(j, D_(j, idx2));
          n_nghbr[j] = idx2;
        }
      }
      // Update the distance matrix in the range (idx2, N).
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        f_median(&D_(idx2, j), D_(idx1, j), c_4 );
        min = D_(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          f_median(&D_(idx2, j), D_(idx1, j), c_4 );
          if (D_(idx2,j) < min) {
            min = D_(idx2,j);
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
    }
 
    default:
      assert(false && "fastcluster: invalid method");
    }
  }
  #ifdef FE_INVALID
  if (fetestexcept(FE_INVALID)) assert(false && "fastcluster: fenv error");
  #endif
}
 
/*
  Clustering methods for vector data
*/
 
template <typename t_dissimilarity>
static void MST_linkage_core_vector(const t_index N,
                                    t_dissimilarity & dist,
                                    cluster_result & Z2) {
/*
    N: integer, number of data points
    dist: function pointer to the metric
    Z2: output data structure
 
    The basis of this algorithm is an algorithm by Rohlf:
 
    F. James Rohlf, Hierarchical clustering using the minimum spanning tree,
    The Computer Journal, vol. 16, 1973, p. 93–95.
*/
  t_index i;
  t_index idx2;
  doubly_linked_list active_nodes(N);
  auto_array_ptr<t_float> d(N);
 
  t_index prev_node;
  t_float min;
 
  // first iteration
  idx2 = 1;
  min = std::numeric_limits<t_float>::infinity();
  for (i=1; i<N; ++i) {
    d[i] = dist(0,i);
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
    if (d[i] < min) {
      min = d[i];
      idx2 = i;
    }
    else if (fc_isnan(d[i]))
      assert(false && "fastcluster: nan error");
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
  }
 
  Z2.append(0, idx2, min);
 
  for (t_index j=1; j<N-1; ++j) {
    prev_node = idx2;
    active_nodes.remove(prev_node);
 
    idx2 = active_nodes.succ[0];
    min = d[idx2];
 
    for (i=idx2; i<N; i=active_nodes.succ[i]) {
      t_float tmp = dist(i, prev_node);
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
      if (d[i] > tmp)
        d[i] = tmp;
      else if (fc_isnan(tmp))
        assert(false && "fastcluster: nan error");
#if HAVE_DIAGNOSTIC
#pragma GCC diagnostic pop
#endif
      if (d[i] < min) {
        min = d[i];
        idx2 = i;
      }
    }
    Z2.append(prev_node, idx2, min);
  }
}
 
template <method_codes_vector method, typename t_dissimilarity>
static void generic_linkage_vector(const t_index N,
                                   t_dissimilarity & dist,
                                   cluster_result & Z2) {
  /*
    N: integer, number of data points
    dist: function pointer to the metric
    Z2: output data structure
 
    This algorithm is valid for the distance update methods
    "Ward", "centroid" and "median" only!
  */
  const t_index N_1 = N-1;
  t_index i, j; // loop variables
  t_index idx1, idx2; // row and column indices
 
  auto_array_ptr<t_index> n_nghbr(N_1); // array of nearest neighbors
  auto_array_ptr<t_float> mindist(N_1); // distances to the nearest neighbors
  auto_array_ptr<t_index> row_repr(N); // row_repr[i]: node number that the
                                       // i-th row represents
  doubly_linked_list active_nodes(N);
  binary_min_heap nn_distances(&*mindist, N_1); // minimum heap structure for
                        // the distance to the nearest neighbor of each point
  t_index node1, node2;     // node numbers in the output
  t_float min; // minimum and row index for nearest-neighbor search
 
  for (i=0; i<N; ++i)
    // Build a list of row ↔ node label assignments.
    // Initially i ↦ i
    row_repr[i] = i;
 
  // Initialize the minimal distances:
  // Find the nearest neighbor of each point.
  // n_nghbr[i] = argmin_{j>i} D(i,j) for i in range(N-1)
  for (i=0; i<N_1; ++i) {
    min = std::numeric_limits<t_float>::infinity();
    t_index idx;
    for (idx=j=i+1; j<N; ++j) {
      t_float tmp;
      switch (method) {
      case METHOD_VECTOR_WARD:
        tmp = dist.ward_initial(i,j);
        break;
      default:
        tmp = dist.template sqeuclidean<true>(i,j);
      }
      if (tmp<min) {
        min = tmp;
        idx = j;
      }
    }
    switch (method) {
    case METHOD_VECTOR_WARD:
      mindist[i] = t_dissimilarity::ward_initial_conversion(min);
      break;
    default:
      mindist[i] = min;
    }
    n_nghbr[i] = idx;
  }
 
  // Put the minimal distances into a heap structure to make the repeated
  // global minimum searches fast.
  nn_distances.heapify();
 
  // Main loop: We have N-1 merging steps.
  for (i=0; i<N_1; ++i) {
    idx1 = nn_distances.argmin();
 
    while ( active_nodes.is_inactive(n_nghbr[idx1]) ) {
      // Recompute the minimum mindist[idx1] and n_nghbr[idx1].
      n_nghbr[idx1] = j = active_nodes.succ[idx1]; // exists, maximally N-1
      switch (method) {
      case METHOD_VECTOR_WARD:
        min = dist.ward(idx1,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          t_float const tmp = dist.ward(idx1,j);
          if (tmp<min) {
            min = tmp;
            n_nghbr[idx1] = j;
          }
        }
        break;
      default:
        min = dist.template sqeuclidean<true>(idx1,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          t_float const tmp = dist.template sqeuclidean<true>(idx1,j);
          if (tmp<min) {
            min = tmp;
            n_nghbr[idx1] = j;
          }
        }
      }
      /* Update the heap with the new true minimum and search for the (possibly
         different) minimal entry. */
      nn_distances.update_geq(idx1, min);
      idx1 = nn_distances.argmin();
    }
 
    nn_distances.heap_pop(); // Remove the current minimum from the heap.
    idx2 = n_nghbr[idx1];
 
    // Write the newly found minimal pair of nodes to the output array.
    node1 = row_repr[idx1];
    node2 = row_repr[idx2];
 
    Z2.append(node1, node2, mindist[idx1]);
 
    switch (method) {
    case METHOD_VECTOR_WARD:
    case METHOD_VECTOR_CENTROID:
      dist.merge_inplace(idx1, idx2);
      break;
    case METHOD_VECTOR_MEDIAN:
      dist.merge_inplace_weighted(idx1, idx2);
      break;
    default:
      assert(false && "fastcluster: invalid method");
    }
 
    // Index idx2 now represents the new (merged) node with label N+i.
    row_repr[idx2] = N+i;
    // Remove idx1 from the list of active indices (active_nodes).
    active_nodes.remove(idx1);  // TBD later!!!
 
    // Update the distance matrix
    switch (method) {
    case METHOD_VECTOR_WARD:
      /*
        Ward linkage.
 
        Shorter and longer distances can occur, not smaller than min(d1,d2)
        but maybe bigger than max(d1,d2).
      */
      // Update the distance matrix in the range [start, idx1).
      for (j=active_nodes.start; j<idx1; j=active_nodes.succ[j]) {
        if (n_nghbr[j] == idx2) {
          n_nghbr[j] = idx1; // invalidate
        }
      }
      // Update the distance matrix in the range (idx1, idx2).
      for ( ; j<idx2; j=active_nodes.succ[j]) {
        t_float const tmp = dist.ward(j, idx2);
        if (tmp < mindist[j]) {
          nn_distances.update_leq(j, tmp);
          n_nghbr[j] = idx2;
        }
        else if (n_nghbr[j]==idx2) {
          n_nghbr[j] = idx1; // invalidate
        }
      }
      // Find the nearest neighbor for idx2.
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        min = dist.ward(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          t_float const tmp = dist.ward(idx2,j);
          if (tmp < min) {
            min = tmp;
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
      break;
 
    default:
      /*
        Centroid and median linkage.
 
        Shorter and longer distances can occur, not bigger than max(d1,d2)
        but maybe smaller than min(d1,d2).
      */
      for (j=active_nodes.start; j<idx2; j=active_nodes.succ[j]) {
        t_float const tmp = dist.template sqeuclidean<true>(j, idx2);
        if (tmp < mindist[j]) {
          nn_distances.update_leq(j, tmp);
          n_nghbr[j] = idx2;
        }
        else if (n_nghbr[j] == idx2)
          n_nghbr[j] = idx1; // invalidate
      }
      // Find the nearest neighbor for idx2.
      if (idx2<N_1) {
        n_nghbr[idx2] = j = active_nodes.succ[idx2]; // exists, maximally N-1
        min = dist.template sqeuclidean<true>(idx2,j);
        for (j=active_nodes.succ[j]; j<N; j=active_nodes.succ[j]) {
          t_float const tmp = dist.template sqeuclidean<true>(idx2, j);
          if (tmp < min) {
            min = tmp;
            n_nghbr[idx2] = j;
          }
        }
        nn_distances.update(idx2, min);
      }
    }
  }
}
 
template <method_codes_vector method, typename t_dissimilarity>
static void generic_linkage_vector_alternative(const t_index N,
                                               t_dissimilarity & dist,
                                               cluster_result & Z2) {
  /*
    N: integer, number of data points
    dist: function pointer to the metric
    Z2: output data structure
 
    This algorithm is valid for the distance update methods
    "Ward", "centroid" and "median" only!
  */
  const t_index N_1 = N-1;
  t_index i, j=0; // loop variables
  t_index idx1, idx2; // row and column indices
 
  auto_array_ptr<t_index> n_nghbr(2*N-2); // array of nearest neighbors
  auto_array_ptr<t_float> mindist(2*N-2); // distances to the nearest neighbors
 
  doubly_linked_list active_nodes(N+N_1);
  binary_min_heap nn_distances(&*mindist, N_1, 2*N-2, 1); // minimum heap
      // structure for the distance to the nearest neighbor of each point
 
  t_float min; // minimum for nearest-neighbor searches
 
  // Initialize the minimal distances:
  // Find the nearest neighbor of each point.
  // n_nghbr[i] = argmin_{j>i} D(i,j) for i in range(N-1)
  for (i=1; i<N; ++i) {
    min = std::numeric_limits<t_float>::infinity();
    t_index idx;
    for (idx=j=0; j<i; ++j) {
      t_float tmp;
      switch (method) {
      case METHOD_VECTOR_WARD:
        tmp = dist.ward_initial(i,j);
        break;
      default:
        tmp = dist.template sqeuclidean<true>(i,j);
      }
      if (tmp<min) {
        min = tmp;
        idx = j;
      }
    }
    switch (method) {
    case METHOD_VECTOR_WARD:
      mindist[i] = t_dissimilarity::ward_initial_conversion(min);
      break;
    default:
      mindist[i] = min;
    }
    n_nghbr[i] = idx;
  }
 
  // Put the minimal distances into a heap structure to make the repeated
  // global minimum searches fast.
  nn_distances.heapify();
 
  // Main loop: We have N-1 merging steps.
  for (i=N; i<N+N_1; ++i) {
    /*
      The bookkeeping is different from the "stored matrix approach" algorithm
      generic_linkage.
 
      mindist[i] stores a lower bound on the minimum distance of the point i to
      all points of *lower* index:
 
          mindist[i] ≥ min_{j<i} D(i,j)
 
      Moreover, new nodes do not re-use one of the old indices, but they are
      given a new, unique index (SciPy convention: initial nodes are 0,…,N−1,
      new nodes are N,…,2N−2).
 
      Invalid nearest neighbors are not recognized by the fact that the stored
      distance is smaller than the actual distance, but the list active_nodes
      maintains a flag whether a node is inactive. If n_nghbr[i] points to an
      active node, the entries nn_distances[i] and n_nghbr[i] are valid,
      otherwise they must be recomputed.
    */
    idx1 = nn_distances.argmin();
    while ( active_nodes.is_inactive(n_nghbr[idx1]) ) {
      // Recompute the minimum mindist[idx1] and n_nghbr[idx1].
      n_nghbr[idx1] = j = active_nodes.start;
      switch (method) {
      case METHOD_VECTOR_WARD:
        min = dist.ward_extended(idx1,j);
        for (j=active_nodes.succ[j]; j<idx1; j=active_nodes.succ[j]) {
          t_float tmp = dist.ward_extended(idx1,j);
          if (tmp<min) {
            min = tmp;
            n_nghbr[idx1] = j;
          }
        }
        break;
      default:
        min = dist.sqeuclidean_extended(idx1,j);
        for (j=active_nodes.succ[j]; j<idx1; j=active_nodes.succ[j]) {
          t_float const tmp = dist.sqeuclidean_extended(idx1,j);
          if (tmp<min) {
            min = tmp;
            n_nghbr[idx1] = j;
          }
        }
      }
      /* Update the heap with the new true minimum and search for the (possibly
         different) minimal entry. */
      nn_distances.update_geq(idx1, min);
      idx1 = nn_distances.argmin();
    }
 
    idx2 = n_nghbr[idx1];
    active_nodes.remove(idx1);
    active_nodes.remove(idx2);
 
    Z2.append(idx1, idx2, mindist[idx1]);
 
    if (i<2*N_1) {
      switch (method) {
      case METHOD_VECTOR_WARD:
      case METHOD_VECTOR_CENTROID:
        dist.merge(idx1, idx2, i);
        break;
 
      case METHOD_VECTOR_MEDIAN:
        dist.merge_weighted(idx1, idx2, i);
        break;
 
      default:
        assert(false && "fastcluster: invalid method");
      }
 
      n_nghbr[i] = active_nodes.start;
      if (method==METHOD_VECTOR_WARD) {
        /*
          Ward linkage.
 
          Shorter and longer distances can occur, not smaller than min(d1,d2)
          but maybe bigger than max(d1,d2).
        */
        min = dist.ward_extended(active_nodes.start, i);
        for (j=active_nodes.succ[active_nodes.start]; j<i;
             j=active_nodes.succ[j]) {
          t_float tmp = dist.ward_extended(j, i);
          if (tmp < min) {
            min = tmp;
            n_nghbr[i] = j;
          }
        }
      }
      else {
        /*
          Centroid and median linkage.
 
          Shorter and longer distances can occur, not bigger than max(d1,d2)
          but maybe smaller than min(d1,d2).
        */
        min = dist.sqeuclidean_extended(active_nodes.start, i);
        for (j=active_nodes.succ[active_nodes.start]; j<i;
             j=active_nodes.succ[j]) {
          t_float tmp = dist.sqeuclidean_extended(j, i);
          if (tmp < min) {
            min = tmp;
            n_nghbr[i] = j;
          }
        }
      }
      if (idx2<active_nodes.start)  {
        nn_distances.remove(active_nodes.start);
      } else {
        nn_distances.remove(idx2);
      }
      nn_distances.replace(idx1, i, min);
    }
  }
}
 
#if HAVE_VISIBILITY
#pragma GCC visibility pop
#endif