linear-assignment: a function to solve least-cost assignment problems

The problem solved by the code introduced in this commit goes like this:
given two sets of items, and a cost matrix which says how much it
"costs" to assign any given item of the first set to any given item of
the second, assign all items (except when the sets have different size)
in the cheapest way.

We use the Jonker-Volgenant algorithm to solve the assignment problem to
answer questions such as: given two different versions of a topic branch
(or iterations of a patch series), what is the best pairing of
commits/patches between the different versions?

Signed-off-by: Johannes Schindelin <johannes.schindelin@gmx.de>
Signed-off-by: Junio C Hamano <gitster@pobox.com>

1 files changed, 201 insertions, 0 deletions

diff --git a/linear-assignment.c b/linear-assignment.c
new file mode 100644
index 0000000..9b3e56e
--- /dev/null
+++ b/linear-assignment.c
@@ -0,0 +1,201 @@
+/*
+ * Based on: Jonker, R., & Volgenant, A. (1987). <i>A shortest augmenting path
+ * algorithm for dense and sparse linear assignment problems</i>. Computing,
+ * 38(4), 325-340.
+ */
+#include "cache.h"
+#include "linear-assignment.h"
+
+#define COST(column, row) cost[(column) + column_count * (row)]
+
+/*
+ * The parameter `cost` is the cost matrix: the cost to assign column j to row
+ * i is `cost[j + column_count * i].
+ */
+void compute_assignment(int column_count, int row_count, int *cost,
+			int *column2row, int *row2column)
+{
+	int *v, *d;
+	int *free_row, free_count = 0, saved_free_count, *pred, *col;
+	int i, j, phase;
+
+	memset(column2row, -1, sizeof(int) * column_count);
+	memset(row2column, -1, sizeof(int) * row_count);
+	ALLOC_ARRAY(v, column_count);
+
+	/* column reduction */
+	for (j = column_count - 1; j >= 0; j--) {
+		int i1 = 0;
+
+		for (i = 1; i < row_count; i++)
+			if (COST(j, i1) > COST(j, i))
+				i1 = i;
+		v[j] = COST(j, i1);
+		if (row2column[i1] == -1) {
+			/* row i1 unassigned */
+			row2column[i1] = j;
+			column2row[j] = i1;
+		} else {
+			if (row2column[i1] >= 0)
+				row2column[i1] = -2 - row2column[i1];
+			column2row[j] = -1;
+		}
+	}
+
+	/* reduction transfer */
+	ALLOC_ARRAY(free_row, row_count);
+	for (i = 0; i < row_count; i++) {
+		int j1 = row2column[i];
+		if (j1 == -1)
+			free_row[free_count++] = i;
+		else if (j1 < -1)
+			row2column[i] = -2 - j1;
+		else {
+			int min = COST(!j1, i) - v[!j1];
+			for (j = 1; j < column_count; j++)
+				if (j != j1 && min > COST(j, i) - v[j])
+					min = COST(j, i) - v[j];
+			v[j1] -= min;
+		}
+	}
+
+	if (free_count ==
+	    (column_count < row_count ? row_count - column_count : 0)) {
+		free(v);
+		free(free_row);
+		return;
+	}
+
+	/* augmenting row reduction */
+	for (phase = 0; phase < 2; phase++) {
+		int k = 0;
+
+		saved_free_count = free_count;
+		free_count = 0;
+		while (k < saved_free_count) {
+			int u1, u2;
+			int j1 = 0, j2, i0;
+
+			i = free_row[k++];
+			u1 = COST(j1, i) - v[j1];
+			j2 = -1;
+			u2 = INT_MAX;
+			for (j = 1; j < column_count; j++) {
+				int c = COST(j, i) - v[j];
+				if (u2 > c) {
+					if (u1 < c) {
+						u2 = c;
+						j2 = j;
+					} else {
+						u2 = u1;
+						u1 = c;
+						j2 = j1;
+						j1 = j;
+					}
+				}
+			}
+			if (j2 < 0) {
+				j2 = j1;
+				u2 = u1;
+			}
+
+			i0 = column2row[j1];
+			if (u1 < u2)
+				v[j1] -= u2 - u1;
+			else if (i0 >= 0) {
+				j1 = j2;
+				i0 = column2row[j1];
+			}
+
+			if (i0 >= 0) {
+				if (u1 < u2)
+					free_row[--k] = i0;
+				else
+					free_row[free_count++] = i0;
+			}
+			row2column[i] = j1;
+			column2row[j1] = i;
+		}
+	}
+
+	/* augmentation */
+	saved_free_count = free_count;
+	ALLOC_ARRAY(d, column_count);
+	ALLOC_ARRAY(pred, column_count);
+	ALLOC_ARRAY(col, column_count);
+	for (free_count = 0; free_count < saved_free_count; free_count++) {
+		int i1 = free_row[free_count], low = 0, up = 0, last, k;
+		int min, c, u1;
+
+		for (j = 0; j < column_count; j++) {
+			d[j] = COST(j, i1) - v[j];
+			pred[j] = i1;
+			col[j] = j;
+		}
+
+		j = -1;
+		do {
+			last = low;
+			min = d[col[up++]];
+			for (k = up; k < column_count; k++) {
+				j = col[k];
+				c = d[j];
+				if (c <= min) {
+					if (c < min) {
+						up = low;
+						min = c;
+					}
+					col[k] = col[up];
+					col[up++] = j;
+				}
+			}
+			for (k = low; k < up; k++)
+				if (column2row[col[k]] == -1)
+					goto update;
+
+			/* scan a row */
+			do {
+				int j1 = col[low++];
+
+				i = column2row[j1];
+				u1 = COST(j1, i) - v[j1] - min;
+				for (k = up; k < column_count; k++) {
+					j = col[k];
+					c = COST(j, i) - v[j] - u1;
+					if (c < d[j]) {
+						d[j] = c;
+						pred[j] = i;
+						if (c == min) {
+							if (column2row[j] == -1)
+								goto update;
+							col[k] = col[up];
+							col[up++] = j;
+						}
+					}
+				}
+			} while (low != up);
+		} while (low == up);
+
+update:
+		/* updating of the column pieces */
+		for (k = 0; k < last; k++) {
+			int j1 = col[k];
+			v[j1] += d[j1] - min;
+		}
+
+		/* augmentation */
+		do {
+			if (j < 0)
+				BUG("negative j: %d", j);
+			i = pred[j];
+			column2row[j] = i;
+			SWAP(j, row2column[i]);
+		} while (i1 != i);
+	}
+
+	free(col);
+	free(pred);
+	free(d);
+	free(v);
+	free(free_row);
+}