《算法》第六章部分程序 part 7

▶ 书中第六章部分程序，加上自己补充的代码，包括全局最小切分 Stoer-Wagner 算法，最小权值二分图匹配

● 全局最小切分 Stoer-Wagner 算法

package package01;

import edu.princeton.cs.algs4.In;
import edu.princeton.cs.algs4.StdOut;
import edu.princeton.cs.algs4.EdgeWeightedGraph;
import edu.princeton.cs.algs4.Edge;
import edu.princeton.cs.algs4.UF;
import edu.princeton.cs.algs4.IndexMaxPQ;

public class class01
{
    private static final double FLOATING_POINT_EPSILON = 1E-11;
    private double weight = Double.POSITIVE_INFINITY;           // 输出的最小割值
    private boolean[] cut;                                      // 顶点是否在割除集 T 中
    private int V;

    private class CutPhase                                      // 最小 s-t 割类（cut-of-the-phase）
    {
        private double weight;
        private int s;
        private int t;

        public CutPhase(double inputWeight, int inputS, int inputT)
        {
            weight = inputWeight;
            s = inputS;
            t = inputT;
        }
    }

    public class01(EdgeWeightedGraph G)
    {
        UF uf = new UF(G.V());                              // 用类 union–find 来表示顶点的合并情况
        boolean[] marked = new boolean[G.V()];              // 已合并的顶点集，初始化为空
        cut = new boolean[G.V()];                           // 割除集 T，初始化为空
        CutPhase cp = new CutPhase(0.0, 0, 0);              // 用于首次搜索的割，无意义
        for (int v = G.V(); --v > 0; marked[cp.t] = true)   // 遍历 V-1 次，每次标记被合并的顶点，以后不再遍历该点
        {
            cp = minCutPhase(G, marked, cp);                // 更新最小割
            if (cp.weight < weight)                         // 发现权值更小的割，更新全局割
            {
                weight = cp.weight;
                for (int j = 0; j < G.V(); j++)             // 在最新的图中，与 cp.t 相连的顶点都是 T 的元素
                    cut[j] = uf.connected(j, cp.t);
            }
            G = contractEdge(G, cp.s, cp.t);                // 顶点 t 合并到顶点 s，更新图
            uf.union(cp.s, cp.t);                           // 顶点 t 加入 T 中
        }
    }

    public double weight()
    {
        return weight;
    }

    public boolean cut(int v)
    {
        return cut[v];
    }

    private CutPhase minCutPhase(EdgeWeightedGraph G, boolean[] marked, CutPhase cp)    // 计算 s - t 的最小割，称为 maximum adjacency (cardinality) search
    {
        IndexMaxPQ<Double> pq = new IndexMaxPQ<Double>(G.V());      // 用于挑选权值最大的点用于合并
        pq.insert(cp.s, Double.POSITIVE_INFINITY);                  // 顶点 s 自己的权值为 +∞
        for (int v = 0; v < G.V(); v++)                             // 其他顶点权值初始化为 0
        {
            if (v != cp.s && !marked[v])
                pq.insert(v, 0.0);
        }
        for (; !pq.isEmpty();)
        {
            cp.s = cp.t;                                            // 记录最后取出的两个顶点，每次往前挪一格
            cp.t = pq.delMax();                                     // 取走权值最大的顶点 v
            for (Edge e : G.adj(cp.t))                              // 只要 cp.t 的邻居还在 pq 中没有被取走，就更新邻居的权值
            {
                int w = e.other(cp.t);
                if (pq.contains(w))                                 
                    pq.increaseKey(w, pq.keyOf(w) + e.weight());    // 顶点 w 的权值自增边 e 的权值
            }
        }
        cp.weight = 0.0;                                            // 计算最后加入 T 的顶点的权值，即为最小割值
        for (Edge e : G.adj(cp.t))
            cp.weight += e.weight();
        return cp;
    }

    private EdgeWeightedGraph contractEdge(EdgeWeightedGraph G, int s, int t)   // 把顶点 t 合并到顶点 s，更新其他边的权值
    {
        EdgeWeightedGraph H = new EdgeWeightedGraph(G.V());         // 合并后的图，顶点与原来相同
        for (int v = 0; v < G.V(); v++)
        {
            for (Edge e : G.adj(v))                                 // 依顶点序遍历所有边
            {
                int w = e.other(v);
                if (v == s && w == t || v == t && w == s)           // 边 s-t 自身不要
                    continue;
                if (v < w)                                          // 只考虑后向边，滤掉重复
                {
                    if (w == t)                                     // 远端顶点 w 是被合并的 t，边 v - w(t) 替换成边 v - s
                        H.addEdge(new Edge(v, s, e.weight()));
                    else if (v == t)                                // 近端顶点 v 是被合并的 t，边 v(t) - w 替换成边 s - w
                        H.addEdge(new Edge(w, s, e.weight()));
                    else                                            // 边 v - w 与 s 或 t 无关，原样放进 H
                        H.addEdge(new Edge(v, w, e.weight()));
                }
            }
        }
        return H;
    }

    public static void main(String[] args)
    {
        In in = new In(args[0]);
        EdgeWeightedGraph G = new EdgeWeightedGraph(in);
        class01 mc = new class01(G);
        StdOut.print("Min cut: ");
        for (int v = 0; v < G.V(); v++)
        {
            if (mc.cut(v))
                StdOut.print(v + " ");
        }
        StdOut.println("
Min cut weight = " + mc.weight());
    }
}

● 最小权值二分图匹配

package package01;

import edu.princeton.cs.algs4.StdOut;
import edu.princeton.cs.algs4.StdRandom;
import edu.princeton.cs.algs4.EdgeWeightedDigraph;
import edu.princeton.cs.algs4.DirectedEdge;
import edu.princeton.cs.algs4.DijkstraSP;

public class class01
{
    private static final double FLOATING_POINT_EPSILON = 1E-14;
    private int n;              // 二分图一侧的顶点数，总顶点数2 * n
    private double[][] weight;  // 边权矩阵
    private double minWeight;   // 最小权值
    private double[] px;        // 每一行的对偶变量
    private double[] py;        // 每一列的对偶变量
    private int[] xy;           // 正向标记，xy[i] = j 表示 i-j 匹配
    private int[] yx;           // 反向标记，yx[j] = i 表示 i-j 匹配

    public class01(double[][] inputWeight)
    {
        n = inputWeight.length;
        weight = new double[n][n];
        minWeight = Double.MAX_VALUE;
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (Double.isNaN(weight[i][j]))
                    throw new IllegalArgumentException("weight " + i + "-" + j + " is NaN");
                weight[i][j] = inputWeight[i][j];
                minWeight = Math.min(minWeight, weight[i][j]);
            }
        }
        px = new double[n];
        py = new double[n];
        xy = new int[n];
        yx = new int[n];
        for (int i = 0; i < n; i++)
            xy[i] = yx[i] = -1;

        for (int k = 0; k < n; k++)             // 调整 n 次，每次添加一条边
        {
            assert isDualFeasibleAndComplementarySlack();
            augment();
        }
        assert isDualFeasibleAndComplementarySlack() && isPerfectMatching();    // 检查结果正确性，要求互补松弛，完美匹配
    }

    private void augment()                      // 寻找最小权值路径并更新
    {
        EdgeWeightedDigraph G = new EdgeWeightedDigraph(2 * n + 2);
        int s = 2 * n, t = 2 * n + 1;
        for (int i = 0; i < n; i++)             // 所有未匹配的顶点连到 s 和 t 上，s 侧权值为 0，t 侧有权值
        {
            if (xy[i] == -1)
                G.addEdge(new DirectedEdge(s, i, 0.0));
        }
        for (int j = 0; j < n; j++)
        {
            if (yx[j] == -1)
                G.addEdge(new DirectedEdge(n + j, t, py[j]));
        }
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)         // 已匹配的顶点对添加权值为 0 的反向边，未匹配的顶点对添加修正权值的正向边
            {
                if (xy[i] == j)
                    G.addEdge(new DirectedEdge(n + j, i, 0.0));
                else
                    G.addEdge(new DirectedEdge(i, n + j, reducedCost(i, j)));
            }
        }
        DijkstraSP spt = new DijkstraSP(G, s);  // 计算从 s 到各顶点最短距离
        for (DirectedEdge e : spt.pathTo(t))    // 研究从 s 到 t 的边
        {
            int i = e.from(), j = e.to() - n;
            if (i < n)                          // 去掉与顶点 s 和 t 有关的部分和反向边
            {
                xy[i] = j;
                yx[j] = i;
            }
        }
        for (int i = 0; i < n; i++)             // 垫起各顶点的距离
        {
            px[i] += spt.distTo(i);
            py[i] += spt.distTo(n + i);
        }
    }

    private double reducedCost(int i, int j)    // 顶点对之间的修正权值，用原始权值减去全局最小权值，再加上起点、终点的距离差
    {
        double reducedCost = (weight[i][j] - minWeight) + px[i] - py[j];
        assert reducedCost >= 0.0;
        if (Math.abs(reducedCost) <= FLOATING_POINT_EPSILON * (Math.abs(weight[i][j]) + Math.abs(px[i]) + Math.abs(py[j])))
            return 0.0;
        return reducedCost;
    }

    public double dualRow(int i)
    {
        return px[i];
    }

    public double dualCol(int j)
    {
        return py[j];
    }

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

    public double weight()                      // 输出解的权值总和
    {
        double total = 0.0;
        for (int i = 0; i < n; i++)
        {
            if (xy[i] != -1)
                total += weight[i][xy[i]];
        }
        return total;
    }

    private boolean isDualFeasibleAndComplementarySlack()   // 检查对偶可行性和互补松弛性
    {
        for (int i = 0; i < n; i++)             // 检查所有边的修正权值不小于 0
        {
            for (int j = 0; j < n; j++)
            {
                if (reducedCost(i, j) < 0)
                {
                    StdOut.println("Dual variables are not feasible");
                    return false;
                }
            }
        }
        for (int i = 0; i < n; i++)             // 检查原变量和对偶变量的互补松弛性，即已匹配的顶点对修正权值为 0，未匹配的非 0
        {
            if (xy[i] != -1 && reducedCost(i, xy[i]) != 0)
            {
                StdOut.println("Primal and dual variables are not complementary slack");
                return false;
            }
        }
        return true;
    }

    private boolean isPerfectMatching()// 检查是否为完美匹配
    {
        boolean[] perm = new boolean[n];
        for (int i = 0; i < n; i++)
        {
            if (perm[xy[i]])// ？perm[-1] 初始化为 true？
            {
                StdOut.println("Not a perfect matching");
                return false;
            }
            perm[xy[i]] = true;
        }
        for (int j = 0; j < n; j++)// 检查 xy[] 和 yx[] 对称性
        {
            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;
    }

    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(900) + 100;  // 权值 100 ~ 999
        }

        class01 assignment = new class01(weight);
        StdOut.printf("weight = %.0f

", assignment.weight());

        if (n >= 20)
            return;
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
                StdOut.printf("%c%.0f ", (j == assignment.sol(i)) ? '*' : ' ', weight[i][j]);
            StdOut.println();
        }
    }
}