``` 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 ``` ```package algs44; // section 6.5 import stdlib.*; public class XAssignmentProblemDense { private static final int UNMATCHED = -1; private final int N; // number of rows and columns private final double[][] weight; // the N-by-N cost matrix private final double[] px; // px[i] = dual variable for row i private final double[] py; // py[j] = dual variable for col j private final int[] xy; // xy[i] = j means i-j is a match private final int[] yx; // yx[j] = i means i-j is a match public XAssignmentProblemDense(double[][] weight) { this.weight = weight; 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 EdgeWeightedDigraph G = new EdgeWeightedDigraph(2*N+2); int s = 2*N, t = 2*N+1; for (int i = 0; i < N; i++) { if (xy[i] == UNMATCHED) G.addEdge(new DirectedEdge(s, i, 0.0)); } for (int j = 0; j < N; j++) { if (yx[j] == UNMATCHED) G.addEdge(new DirectedEdge(N+j, t, py[j])); } 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, reduced(i, j))); } } // compute shortest path from s to every other vertex // DenseDijkstraSP spt = new DenseDijkstraSP(G, s); DijkstraSP spt = new DijkstraSP(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 private double reduced(int i, int j) { return weight[i][j] + px[i] - py[j]; } // 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]]; } return total; } 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 (reduced(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) && (reduced(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.random(); XAssignmentProblemDense assignment = new XAssignmentProblemDense(weight); StdOut.println("weight = " + assignment.weight()); for (int i = 0; i < N; i++) StdOut.println(i + "-" + assignment.sol(i)); } } ```