# AssignmentProblemDense.java

Below is the syntax highlighted version of AssignmentProblemDense.java from §6.5 Reductions.

```/******************************************************************************
*  Compilation:  javac AssignmentProblemDense.java
*  Execution:    java AssignmentProblemDense n
*
*  Solve an n-by-n assignment problem using successive shortest path
*  algorithm in n^3 time.
*
*  Assumes n-by-n cost matrix is nonnegative. If not, can add positive
*  constant to all entries.
*
******************************************************************************/

public class AssignmentProblemDense {
private static final double FLOATING_POINT_EPSILON = 1E-14;
private static final int UNMATCHED = -1;

private int n;              // number of rows and columns
private double[][] weight;  // the n-by-n cost matrix
private double[] px;        // px[i] = dual variable for row i
private double[] py;        // py[j] = dual variable for col j
private int[] xy;           // xy[i] = j means i-j is a match
private int[] yx;           // yx[j] = i means i-j is a match

public AssignmentProblemDense(double[][] weight) {
this.weight = weight.clone();
n = weight.length;

// dual variables
px = new double[n];
py = new double[n];

// initial matching is empty
xy = new int[n];
yx = new int[n];
for (int i = 0; i < n; i++)
xy[i] = UNMATCHED;
for (int j = 0; j < n; j++)
yx[j] = UNMATCHED;

// add N edges to matching
for (int k = 0; k < n; k++) {
StdOut.println(k);
assert isDualFeasible();
assert isComplementarySlack();
augment();
}
assert check();
}

// find shortest augmenting path and upate
private void augment() {

// build residual graph
int s = 2*n, t = 2*n+1;
for (int i = 0; i < n; i++) {
if (xy[i] == UNMATCHED)
}
for (int j = 0; j < n; j++) {
if (yx[j] == UNMATCHED)
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (xy[i] == j) G.addEdge(new DirectedEdge(n+j, i, 0.0));
else            G.addEdge(new DirectedEdge(i, n+j, reducedCost(i, j)));
}
}

// compute shortest path from s to every other vertex
DenseDijkstraSP spt = new DenseDijkstraSP(G, s);

// augment along alternating path
for (DirectedEdge e : spt.pathTo(t)) {
int v = e.from(), w = e.to() - n;
if (v < n) {
xy[v] = w;
yx[w] = v;
}
}

// update dual variables
for (int i = 0; i < n; i++)
px[i] += spt.distTo(i);
for (int j = 0; j < n; j++)
py[j] += spt.distTo(n+j);
}

// reduced cost of i-j
// (subtracting off minWeight reweights all weights to be non-negative)
private double reducedCost(int i, int j) {
double reducedCost = weight[i][j] + px[i] - py[j];

// to avoid issues with floating-point precision
double magnitude = Math.abs(weight[i][j]) + Math.abs(px[i]) + Math.abs(py[j]);
if (Math.abs(reducedCost) <= FLOATING_POINT_EPSILON * magnitude) return 0.0;

assert reducedCost >= 0.0;
return reducedCost;
}

// total weight of min weight perfect matching
public double weight() {
double total = 0.0;
for (int i = 0; i < n; i++) {
if (xy[i] != UNMATCHED)
total += weight[i][xy[i]];
}
}

public int sol(int i) {
return xy[i];
}

// check that dual variables are feasible
private boolean isDualFeasible() {
// check that all edges have >= 0 reduced cost
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (reducedCost(i, j) < 0) {
StdOut.println("Dual variables are not feasible");
return false;
}
}
}
return true;
}

// check that primal and dual variables are complementary slack
private boolean isComplementarySlack() {

// check that all matched edges have 0-reduced cost
for (int i = 0; i < n; i++) {
if ((xy[i] != UNMATCHED) && (reducedCost(i, xy[i]) != 0)) {
StdOut.println("Primal and dual variables are not complementary slack");
return false;
}
}
return true;
}

// check that primal variables are a perfect matching
private boolean isPerfectMatching() {

// check that xy[] is a perfect matching
boolean[] perm = new boolean[n];
for (int i = 0; i < n; i++) {
if (perm[xy[i]]) {
StdOut.println("Not a perfect matching");
return false;
}
perm[xy[i]] = true;
}

// check that xy[] and yx[] are inverses
for (int j = 0; j < n; j++) {
if (xy[yx[j]] != j) {
StdOut.println("xy[] and yx[] are not inverses");
return false;
}
}
for (int i = 0; i < n; i++) {
if (yx[xy[i]] != i) {
StdOut.println("xy[] and yx[] are not inverses");
return false;
}
}

return true;
}

// check optimality conditions
private boolean check() {
return isPerfectMatching() && isDualFeasible() && isComplementarySlack();
}

public static void main(String[] args) {

int n = Integer.parseInt(args[0]);
double[][] weight = new double[n][n];
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
weight[i][j] = StdRandom.uniform();

AssignmentProblemDense assignment = new AssignmentProblemDense(weight);
StdOut.println("weight = " + assignment.weight());
for (int i = 0; i < n; i++)
StdOut.println(i + "-" + assignment.sol(i));
}

}
```