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# Builtin function that implements the k-Means clustering algorithm
#
# INPUT PARAMETERS:
# ----------------------------------------------------------------------------
# NAME                              TYPE      DEFAULT  MEANING
# ----------------------------------------------------------------------------
# X                                 Double    ---      The input Matrix to do KMeans on.
# k                                 Int       ---      Number of centroids
# runs                              Int       10       Number of runs (with different initial centroids)
# max_iter                          Int       1000     Maximum number of iterations per run
# eps                               Double    0.000001 Tolerance (epsilon) for WCSS change ratio
# is_verbose                        Boolean   FALSE    do not print per-iteration stats
# avg_sample_size_per_centroid      Int       50       Average number of records per centroid in data samples
#
#
# RETURN VALUES
# ----------------------------------------------------------------------------
# NAME     TYPE      DEFAULT  MEANING
# ----------------------------------------------------------------------------
# Y        String    "Y.mtx"  The mapping of records to centroids
# C        String    "C.mtx"  The output matrix with the centroids
# ----------------------------------------------------------------------------


m_kmeans = function(Matrix[Double] X, Integer k = 10, Integer runs = 10, Integer max_iter = 1000,
    Double eps = 0.000001, Boolean is_verbose = FALSE, Integer avg_sample_size_per_centroid = 50,
    Integer seed = -1)
  return (Matrix[Double] C, Matrix[Double] Y)
{
  if( is_verbose )
    print ("BEGIN K-MEANS SCRIPT");
  
  num_records   = nrow (X);
  num_features  = ncol (X);
  num_centroids = k;
  num_runs = runs;

  if(is_verbose)
    print("dim X=" + nrow(X) + "x" + ncol(X))

  sumXsq = sum (X ^ 2);

  # STEP 1: INITIALIZE CENTROIDS FOR ALL RUNS FROM DATA SAMPLES:

  if( is_verbose )
    print ("Taking data samples for initialization...");

  [sample_maps, samples_vs_runs_map, sample_block_size] = get_sample_maps(
    num_records, num_runs, num_centroids * avg_sample_size_per_centroid, seed);

  is_row_in_samples = rowSums (sample_maps);
  X_samples = sample_maps %*% X;
  X_samples_sq_norms = rowSums (X_samples ^ 2);

  if( is_verbose )
    print ("Initializing the centroids for all runs...");
  
  All_Centroids = matrix (0, num_runs * num_centroids, num_features);

  # We select centroids according to the k-Means++ heuristic applied to a sample of X
  # Loop invariant: min_distances ~ sq.distances from X_sample rows to nearest centroids,
  # with the out-of-range X_sample positions in min_distances set to 0.0

  min_distances = is_row_in_samples;  # Pick the 1-st centroids uniformly at random

  seed = ifelse(seed == -1, -1, seed + 1)
  random_row = rand(rows = 1, cols = num_runs, seed = seed);
  for (i in 1 : num_centroids)
  {
    # "Matricize" and prefix-sum to compute the cumulative distribution function:
    min_distances_matrix_form = matrix (min_distances, rows = sample_block_size, cols = num_runs, byrow = FALSE);
    cdf_min_distances = cumsum (min_distances_matrix_form);

    # Select the i-th centroid in each sample as a random sample row id with
    # probability ~ min_distances:
    random_row = random_row + rand(rows = 1, cols = 1, seed = seed + i)
    threshold_matrix = random_row * cdf_min_distances [sample_block_size, ];
    centroid_ids = t(colSums (cdf_min_distances < threshold_matrix)) + 1;

    # Place the selected centroids together, one per run, into a matrix:
    centroid_placer = table (seq (1, num_runs),
    sample_block_size * seq (0, num_runs - 1) + centroid_ids, num_runs, sample_block_size * num_runs);
    centroids = centroid_placer %*% X_samples;

    # Place the selected centroids into their appropriate slots in All_Centroids:
    centroid_placer = table (seq (i, num_centroids * (num_runs - 1) + i, num_centroids),
    seq (1, num_runs, 1), nrow (All_Centroids), num_runs);
    All_Centroids = All_Centroids + centroid_placer %*% centroids;

    # Update min_distances to preserve the loop invariant:
    distances = X_samples_sq_norms + samples_vs_runs_map %*% rowSums (centroids ^ 2)
                - 2 * rowSums (X_samples * (samples_vs_runs_map %*% centroids));
    min_distances = ifelse(i==1, is_row_in_samples*distances, min(min_distances,distances));
  }

  # STEP 2: PERFORM K-MEANS ITERATIONS FOR ALL RUNS:

  termination_code = matrix (0, rows = num_runs, cols = 1);
  final_wcss = matrix (0, rows = num_runs, cols = 1);
  num_iterations = matrix (0, rows = num_runs, cols = 1);

  if( is_verbose )
    print ("Performing k-means iterations for all runs...");

  parfor (run_index in 1 : num_runs, check = 0)
  {
    C = All_Centroids [(num_centroids * (run_index - 1) + 1) : (num_centroids * run_index), ];
    C_old = C;
    iter_count = 0;
    term_code = 0;
    wcss = Inf

    while (term_code == 0)
    {
      # Compute Euclidean squared distances from records (X rows) to centroids (C rows)
      # without the C-independent term, then take the minimum for each record
      D = -2 * (X %*% t(C)) + t(rowSums (C ^ 2));
      minD = rowMins (D);
      # Compute the current centroid-based within-cluster sum of squares (WCSS)
      wcss_old = wcss;
      wcss = sumXsq + sum (minD);
      if( is_verbose ) {
        if (iter_count == 0)
          print ("Run " + run_index + ", At Start-Up:  Centroid WCSS = " + wcss);
        else
          print ("Run " + run_index + ", Iteration " + iter_count + ":  Centroid WCSS = " + wcss
          + ";  Centroid change (avg.sq.dist.) = " + (sum ((C - C_old) ^ 2) / num_centroids));
      }

      # Find the closest centroid for each record
      P = D <= minD;
      # If some records belong to multiple centroids, share them equally
      P = P / rowSums (P);
      # Compute the column normalization factor for P
      P_denom = colSums (P);
      # Compute new centroids as weighted averages over the records
      C_new = (t(P) %*% X) / t(P_denom);

      # Check if convergence or maximum iteration has been reached
      iter_count = iter_count + 1
      if(wcss_old - wcss < eps * wcss)
        term_code = 1; # Convergence reached
      else if(iter_count >= max_iter)
        term_code = 2; # Max iteration reached
      else if(sum (P_denom <= 0) > 0)
        term_code = 3; # "Runaway" centroid (0.0 denom)
      else
        C_old = C; C = C_new;
    }

    if(is_verbose)
      print ("Run " + run_index + ", Iteration " + iter_count + ":  Terminated with code = "
        + term_code + ",  Centroid WCSS = " + wcss);

    All_Centroids [(num_centroids * (run_index - 1) + 1) : (num_centroids * run_index), ] = C;
    final_wcss [run_index, 1] = wcss;
    termination_code [run_index, 1] = term_code;
    num_iterations [run_index, 1] = iter_count;
  }

  # STEP 3: SELECT THE RUN WITH BEST CENTROID-WCSS AND OUTPUT ITS CENTROIDS:

  termination_bitmap = matrix (0, num_runs, 3);
  termination_bitmap_raw = table (seq (1, num_runs, 1), termination_code);
  termination_bitmap [, 1 : ncol(termination_bitmap_raw)] = termination_bitmap_raw;
  termination_stats = colSums (termination_bitmap);

  if(is_verbose){

    print ("Number of successful runs = " + as.integer (as.scalar (termination_stats [1, 1])));
    print ("Number of incomplete runs = " + as.integer (as.scalar (termination_stats [1, 2])));
    print ("Number of failed runs (with lost centroids) = " + as.integer (as.scalar (termination_stats [1, 3])));
  }

  num_successful_runs = as.scalar (termination_stats [1, 1]);

  if (num_successful_runs > 0)
  {
    final_wcss_successful = replace(target = final_wcss, pattern = Inf, replacement = 0)* termination_bitmap [, 1];
    worst_wcss = max (final_wcss_successful);
    best_wcss = min (final_wcss_successful + (10 * worst_wcss + 10) * (1 - termination_bitmap [, 1]));
    avg_wcss = sum (final_wcss_successful) / num_successful_runs;
    best_index_vector = (final_wcss_successful == best_wcss);
    aggr_best_index_vector = cumsum (best_index_vector);
    best_index = as.integer (sum (aggr_best_index_vector == 0) + 1);

    if(is_verbose)
      print ("Successful runs:  Best run is " + best_index + " with Centroid WCSS = " + best_wcss
             + ";  Avg WCSS = " + avg_wcss + ";  Worst WCSS = " + worst_wcss);

    C = All_Centroids [(num_centroids * (best_index - 1) + 1) : (num_centroids * best_index), ];
    D =  -2 * (X %*% t(C)) + t(rowSums (C ^ 2));
    P = (D <= rowMins (D));
    aggr_P = t(cumsum (t(P)));
    Y = rowSums (aggr_P == 0) + 1

    if(is_verbose)
      print("dim C=" + nrow(C) + "x" + ncol(C) + ", dim Y=" + nrow(Y) + "x" + ncol(Y))

  }
  else{
    print ("K-means: No output is produced. Try increasing the number of iterations and/or lower eps.");
    C = matrix(0, num_centroids,  num_records)
    Y = matrix(-1, 1, num_records)
  }
}


get_sample_maps = function (int num_records, int num_samples, int approx_sample_size, int seed)
                  return (Matrix[double] sample_maps, Matrix[double] sample_col_map, int sample_block_size)
{
  if (approx_sample_size < num_records)
  {
    # Input value "approx_sample_size" is the average sample size; increase it by ~10 std.dev's
    # to get the sample block size (to allocate space):
    sample_block_size = as.integer (approx_sample_size + round (10 * sqrt (approx_sample_size)));
    num_rows = sample_block_size * num_samples;

    # Generate all samples in parallel by converting uniform random values into random
    # integer skip-ahead intervals and prefix-summing them:
    sample_rec_ids = Rand (rows = sample_block_size, cols = num_samples, min = 0.0, max = 1.0, seed = seed);
    sample_rec_ids = round (log (sample_rec_ids) / log (1.0 - approx_sample_size / num_records) + 0.5);
    # Prob [k-1 < log(uniform)/log(1-p) < k] = p*(1-p)^(k-1) = Prob [k-1 zeros before a one]
    sample_rec_ids = cumsum (sample_rec_ids);  #  (skip to next one) --> (skip to i-th one)

    # Replace all sample record ids over "num_records" (i.e. out of range) by "num_records + 1":
    is_sample_rec_id_within_range = (sample_rec_ids <= num_records);
    sample_rec_ids = sample_rec_ids * is_sample_rec_id_within_range
      + (num_records + 1) * (1 - is_sample_rec_id_within_range);

    # Rearrange all samples (and their out-of-range indicators) into one column-vector:
    sample_rec_ids = matrix (sample_rec_ids, rows = num_rows, cols = 1, byrow = FALSE);
    is_row_in_samples = matrix (is_sample_rec_id_within_range, rows = num_rows, cols = 1, byrow = FALSE);

    # Use contingency table to create the "sample_maps" matrix that is a vertical concatenation
    # of 0-1-matrices, one per sample, each with 1s at (i, sample_record[i]) and 0s elsewhere:
    sample_maps = table (seq (1, num_rows), sample_rec_ids, num_rows, num_records);

    # Create a 0-1-matrix that maps each sample column ID into all row positions of the
    # corresponding sample; map out-of-sample-range positions to row id = num_rows + 1:
    sample_positions = (num_rows + 1) - is_row_in_samples * seq (num_rows, 1, -1);
    # Column ID positions = 1, 1, ..., 1, 2, 2, ..., 2, . . . , n_c, n_c, ..., n_c:
    col_positions = round (0.5 + seq (0, num_rows - 1, 1) / sample_block_size);
    sample_col_map = table (sample_positions, col_positions);
    # Remove the out-of-sample-range positions by cutting off the last row:
    sample_col_map = sample_col_map [1 : (num_rows), ];
  }
  else {
    one_per_record = matrix (1, num_records, 1);
    sample_block_size = num_records;
    sample_maps    = matrix (0, (num_records * num_samples), num_records);
    sample_col_map = matrix (0, (num_records * num_samples), num_samples);
    for (i in 1:num_samples) {
      sample_maps    [(num_records * (i - 1) + 1) : (num_records * i),  ] = diag (one_per_record);
      sample_col_map [(num_records * (i - 1) + 1) : (num_records * i), i] = one_per_record;
    }
  }
}