linear-assignment.c

   1 /*
   2  * Based on: Jonker, R., & Volgenant, A. (1987). <i>A shortest augmenting path
   3  * algorithm for dense and sparse linear assignment problems</i>. Computing,
   4  * 38(4), 325-340.
   5  */
   6 #include "cache.h"
   7 #include "linear-assignment.h"
   8
   9 #define COST(column, row) cost[(column) + column_count * (row)]
  10
  11 /*
  12  * The parameter `cost` is the cost matrix: the cost to assign column j to row
  13  * i is `cost[j + column_count * i].
  14  */
  15 void compute_assignment(int column_count, int row_count, int *cost,
  16                         int *column2row, int *row2column)
  17 {
  18         int *v, *d;
  19         int *free_row, free_count = 0, saved_free_count, *pred, *col;
  20         int i, j, phase;
  21
  22         memset(column2row, -1, sizeof(int) * column_count);
  23         memset(row2column, -1, sizeof(int) * row_count);
  24         ALLOC_ARRAY(v, column_count);
  25
  26         /* column reduction */
  27         for (j = column_count - 1; j >= 0; j--) {
  28                 int i1 = 0;
  29
  30                 for (i = 1; i < row_count; i++)
  31                         if (COST(j, i1) > COST(j, i))
  32                                 i1 = i;
  33                 v[j] = COST(j, i1);
  34                 if (row2column[i1] == -1) {
  35                         /* row i1 unassigned */
  36                         row2column[i1] = j;
  37                         column2row[j] = i1;
  38                 } else {
  39                         if (row2column[i1] >= 0)
  40                                 row2column[i1] = -2 - row2column[i1];
  41                         column2row[j] = -1;
  42                 }
  43         }
  44
  45         /* reduction transfer */
  46         ALLOC_ARRAY(free_row, row_count);
  47         for (i = 0; i < row_count; i++) {
  48                 int j1 = row2column[i];
  49                 if (j1 == -1)
  50                         free_row[free_count++] = i;
  51                 else if (j1 < -1)
  52                         row2column[i] = -2 - j1;
  53                 else {
  54                         int min = COST(!j1, i) - v[!j1];
  55                         for (j = 1; j < column_count; j++)
  56                                 if (j != j1 && min > COST(j, i) - v[j])
  57                                         min = COST(j, i) - v[j];
  58                         v[j1] -= min;
  59                 }
  60         }
  61
  62         if (free_count ==
  63             (column_count < row_count ? row_count - column_count : 0)) {
  64                 free(v);
  65                 free(free_row);
  66                 return;
  67         }
  68
  69         /* augmenting row reduction */
  70         for (phase = 0; phase < 2; phase++) {
  71                 int k = 0;
  72
  73                 saved_free_count = free_count;
  74                 free_count = 0;
  75                 while (k < saved_free_count) {
  76                         int u1, u2;
  77                         int j1 = 0, j2, i0;
  78
  79                         i = free_row[k++];
  80                         u1 = COST(j1, i) - v[j1];
  81                         j2 = -1;
  82                         u2 = INT_MAX;
  83                         for (j = 1; j < column_count; j++) {
  84                                 int c = COST(j, i) - v[j];
  85                                 if (u2 > c) {
  86                                         if (u1 < c) {
  87                                                 u2 = c;
  88                                                 j2 = j;
  89                                         } else {
  90                                                 u2 = u1;
  91                                                 u1 = c;
  92                                                 j2 = j1;
  93                                                 j1 = j;
  94                                         }
  95                                 }
  96                         }
  97                         if (j2 < 0) {
  98                                 j2 = j1;
  99                                 u2 = u1;
 100                         }
 101
 102                         i0 = column2row[j1];
 103                         if (u1 < u2)
 104                                 v[j1] -= u2 - u1;
 105                         else if (i0 >= 0) {
 106                                 j1 = j2;
 107                                 i0 = column2row[j1];
 108                         }
 109
 110                         if (i0 >= 0) {
 111                                 if (u1 < u2)
 112                                         free_row[--k] = i0;
 113                                 else
 114                                         free_row[free_count++] = i0;
 115                         }
 116                         row2column[i] = j1;
 117                         column2row[j1] = i;
 118                 }
 119         }
 120
 121         /* augmentation */
 122         saved_free_count = free_count;
 123         ALLOC_ARRAY(d, column_count);
 124         ALLOC_ARRAY(pred, column_count);
 125         ALLOC_ARRAY(col, column_count);
 126         for (free_count = 0; free_count < saved_free_count; free_count++) {
 127                 int i1 = free_row[free_count], low = 0, up = 0, last, k;
 128                 int min, c, u1;
 129
 130                 for (j = 0; j < column_count; j++) {
 131                         d[j] = COST(j, i1) - v[j];
 132                         pred[j] = i1;
 133                         col[j] = j;
 134                 }
 135
 136                 j = -1;
 137                 do {
 138                         last = low;
 139                         min = d[col[up++]];
 140                         for (k = up; k < column_count; k++) {
 141                                 j = col[k];
 142                                 c = d[j];
 143                                 if (c <= min) {
 144                                         if (c < min) {
 145                                                 up = low;
 146                                                 min = c;
 147                                         }
 148                                         col[k] = col[up];
 149                                         col[up++] = j;
 150                                 }
 151                         }
 152                         for (k = low; k < up; k++)
 153                                 if (column2row[col[k]] == -1)
 154                                         goto update;
 155
 156                         /* scan a row */
 157                         do {
 158                                 int j1 = col[low++];
 159
 160                                 i = column2row[j1];
 161                                 u1 = COST(j1, i) - v[j1] - min;
 162                                 for (k = up; k < column_count; k++) {
 163                                         j = col[k];
 164                                         c = COST(j, i) - v[j] - u1;
 165                                         if (c < d[j]) {
 166                                                 d[j] = c;
 167                                                 pred[j] = i;
 168                                                 if (c == min) {
 169                                                         if (column2row[j] == -1)
 170                                                                 goto update;
 171                                                         col[k] = col[up];
 172                                                         col[up++] = j;
 173                                                 }
 174                                         }
 175                                 }
 176                         } while (low != up);
 177                 } while (low == up);
 178
 179 update:
 180                 /* updating of the column pieces */
 181                 for (k = 0; k < last; k++) {
 182                         int j1 = col[k];
 183                         v[j1] += d[j1] - min;
 184                 }
 185
 186                 /* augmentation */
 187                 do {
 188                         if (j < 0)
 189                                 BUG("negative j: %d", j);
 190                         i = pred[j];
 191                         column2row[j] = i;
 192                         SWAP(j, row2column[i]);
 193                 } while (i1 != i);
 194         }
 195
 196         free(col);
 197         free(pred);
 198         free(d);
 199         free(v);
 200         free(free_row);
 201 }