{"id":76540,"date":"2026-02-22T18:17:38","date_gmt":"2026-02-22T10:17:38","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/76540.html"},"modified":"2026-02-22T18:17:38","modified_gmt":"2026-02-22T10:17:38","slug":"%e8%af%a6%e8%a7%a3%e6%9c%80%e7%9f%ad%e8%b7%af%e5%be%84","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/76540.html","title":{"rendered":"\u8be6\u89e3\u6700\u77ed\u8def\u5f84"},"content":{"rendered":"
\"\"<\/h5>\n

\u76ee\u5f55<\/p>\n

\u200b\u7f16\u8f91<\/p>\n

\u6700\u77ed\u8def\u5f84<\/p>\n

Dijkstra\u7b97\u6cd5（\u5355\u6e90）<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/p>\n

\u00a0\u4ee3\u7801\u6d4b\u8bd5<\/p>\n

\u7f3a\u70b9<\/p>\n

Bellman-Ford\u7b97\u6cd5（\u5355\u6e90）<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/p>\n

\u4ee3\u7801\u6d4b\u8bd5<\/p>\n

Floyd-Warshall\u7b97\u6cd5（\u591a\u6e90）<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/p>\n

\u4ee3\u7801\u6d4b\u8bd5<\/p>\n

\n
\u6700\u77ed\u8def\u5f84<\/h5>\n

\u6700\u77ed\u8def\u5f84\u95ee\u9898：\u4ece\u5728\u5e26\u6743\u56fe\u7684\u67d0\u4e00\u9876\u70b9\u51fa\u53d1，\u627e\u51fa\u4e00\u6761\u901a\u5f80\u53e6\u4e00\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84，\u6700\u77ed\u4e5f\u5c31\u662f\u6cbf\u8def\u5f84\u5404\u8fb9\u7684\u6743\u503c\u603b\u548c\u8fbe\u5230\u6700\u5c0f\u3002<\/p>\n

Dijkstra\u7b97\u6cd5（\u5355\u6e90）<\/h6>\n

Dijkstra\u7b97\u6cd5\u7531\u8377\u5170\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6Edsger W. Dijkstra\u57281956\u5e74\u63d0\u51fa，\u9002\u7528\u4e8e\u542b\u6709\u975e\u8d1f\u6743\u91cd<\/span>\u7684\u6709\u5411\u56fe\u6216\u65e0\u5411\u56fe\u3002<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/h6>\n

Dijkstra\u7b97\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u8d2a\u5fc3\u7b97\u6cd5\u3002\u5b83\u7ef4\u62a4\u4e00\u4e2a\u8ddd\u79bb\u6570\u7ec4，\u8be5\u6570\u7ec4\u8bb0\u5f55\u4e86\u4ece\u6e90\u70b9\u5230\u56fe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u4f30\u8ba1\u503c\u3002\u7b97\u6cd5\u91cd\u590d\u4ece\u672a\u5904\u7406\u7684\u9876\u70b9\u4e2d\u9009\u62e9\u5177\u6709\u6700\u5c0f\u8ddd\u79bb\u4f30\u8ba1\u503c\u7684\u9876\u70b9，\u5e76\u66f4\u65b0\u5176\u90bb\u5c45\u9876\u70b9\u7684\u8ddd\u79bb\u4f30\u8ba1\u503c，\u76f4\u5230\u6240\u6709\u9876\u70b9\u90fd\u88ab\u5904\u7406\u3002<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/h6>\n

1.\u521d\u59cb\u5316： \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u521b\u5efa\u4e00\u4e2a\u4e00\u7ef4\u8ddd\u79bb\u6570\u7ec4dist[]，\u7528\u4e8e\u5b58\u50a8\u4ece\u6e90\u70b9\u5230\u56fe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u4f30\u8ba1\u8ddd\u79bb\u3002\u5c06\u6240\u6709\u9876\u70b9\u7684\u8ddd\u79bb\u521d\u59cb\u5316\u4e3a\u65e0\u7a77\u5927（\u9664\u4e86\u6e90\u70b9，\u5176\u8ddd\u79bb\u521d\u59cb\u5316\u4e3a0）\u3002 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u521b\u5efa\u4e00\u4e2a\u4e00\u7ef4\u5e03\u5c14\u6570\u7ec4visited[]，\u7528\u4e8e\u8bb0\u5f55\u9876\u70b9\u662f\u5426\u5df2\u88ab\u5904\u7406（\u5373\u662f\u5426\u5df2\u7ecf\u627e\u5230\u4e86\u4ece\u6e90\u70b9\u5230\u8be5\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84）\u3002\u521d\u59cb\u65f6，\u6240\u6709\u9876\u70b9\u90fd\u672a\u88ab\u5904\u7406\u3002 2.\u5faa\u73af\u5904\u7406：<\/p>\n

\u4ece\u672a\u88ab\u6807\u8bb0\u5904\u7406\u7684\u70b9\u8fde\u51fa\u53bb\u7684\u6240\u6709\u8fb9\u4e2d\u9009\u51fa\u8ddd\u79bb\u6700\u77ed\u7684\u8fb9，\u7136\u540e\u5c06\u8be5\u8fb9\u7684\u6e90\u8282\u70b9u\u6807\u8bb0\u4e3a\u5904\u7406\u8fc7\u3002 \u904d\u5386\u9876\u70b9u\u7684\u6240\u6709\u90bb\u5c45\u9876\u70b9v，\u5bf9\u4e8e\u6bcf\u4e2a\u90bb\u5c45\u9876\u70b9v，\u5982\u679c\u901a\u8fc7\u9876\u70b9u\u5230\u8fbe\u9876\u70b9v\u7684\u8ddd\u79bb（\u5373dist[u] + matrix(u, v)）\u5c0f\u4e8e\u5f53\u524d\u8bb0\u5f55\u7684dist[v]，\u5219\u66f4\u65b0dist[v]\u4e3a\u8fd9\u4e2a\u66f4\u5c0f\u7684\u503c\u3002 3.\u7ed3\u675f： \u5f53\u6240\u6709\u9876\u70b9\u90fd\u88ab\u5904\u7406\u540e，\u7b97\u6cd5\u7ed3\u675f\u3002\u6b64\u65f6，dist[]\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u5c31\u662f\u4ece\u6e90\u70b9\u5230\u56fe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u957f\u5ea6\u3002<\/p>\n

\u4e3a\u4e86\u8bb0\u5f55\u4e0b\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684，\u4f7f\u7528\u4e00\u4e2a\u4e00\u7ef4\u6570\u7ec4pPath[]\u6765\u6807\u8bb0\u6bcf\u4e2a\u70b9\u5728\u6700\u77ed\u8def\u5f84\u4e2d\u524d\u4e00\u4e2a\u8282\u70b9\u662f\u4ec0\u4e48，\u8fd9\u6837\u5c31\u80fd\u591f\u77e5\u9053\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684\u4e86（\u5e76\u67e5\u96c6\u601d\u60f3）\u3002<\/p>\n

\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(n^2)\u3002<\/p>\n

\u56fe\u89e3：<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/h6>\n

Constant.java<\/p>\n

public class Constant {
\n public static final int MAX = Integer.MAX_VALUE;
\n} <\/p>\n

Dijkstra.java<\/p>\n

public void dijkstra(char vSrc,int[] dist,int[] pPath) {
\n int srcIndex = getIndexOfV(vSrc);
\n \/\/\u8ddd\u79bb\u6570\u636e\u521d\u59cb\u5316
\n Arrays.fill(dist,Constant.MAX);
\n dist[srcIndex] = 0;
\n \/\/\u8def\u5f84\u6570\u7ec4\u521d\u59cb\u5316
\n Arrays.fill(pPath,-1);
\n pPath[srcIndex] = srcIndex;
\n \/\/\u5f53\u524d\u9876\u70b9\u662f\u5426\u88ab\u8bbf\u95ee\u8fc7
\n int n = arrayV.length;
\n boolean[] visted = new boolean[n];<\/p>\n

\/\/n\u4e2a\u9876\u70b9,\u8981\u66f4\u65b0n\u6b21，\u6bcf\u6b21\u90fd\u8981\u4ece0\u4e0b\u6807\u5f00\u59cb，\u627e\u5230\u4e00\u4e2a\u6700\u5c0f\u503c
\n for (int k = 0; k < n; k++) {
\n int min = Constant.MAX;
\n int u = srcIndex;
\n for (int i = 0; i < n; i++) {
\n if(visted[i] == false && dist[i] < min) {
\n min = dist[i];
\n u = i;\/\/\u66f4\u65b0u\u4e0b\u6807
\n }
\n }
\n visted[u] = true;\/\/u:s
\n \/\/\u677e\u5f1bu\u8fde\u63a5\u51fa\u53bb\u7684\u6240\u6709\u7684\u9876\u70b9 v
\n for (int v = 0; v < n; v++) {
\n if(visted[v] == false && matrix[u][v] != Constant.MAX
\n && dist[u] + matrix[u][v] < dist[v]) {
\n dist[v] = dist[u] + matrix[u][v];
\n pPath[v] = u;\/\/\u66f4\u65b0\u5f53\u524d\u7684\u8def\u5f84
\n }
\n }
\n }
\n} <\/p>\n

\u00a0\u4ee3\u7801\u6d4b\u8bd5<\/h6>\n

public void printShortPath(char vSrc,int[] dist,int[] pPath) {
\n int srcIndex = getIndexOfV(vSrc);<\/p>\n

int n = arrayV.length;<\/p>\n

for (int i = 0; i < n; i++) {
\n \/\/i\u4e0b\u6807\u6b63\u597d\u662f\u8d77\u70b9 \u5219\u4e0d\u8fdb\u884c\u8def\u5f84\u7684\u6253\u5370
\n if(i != srcIndex) {
\n ArrayList<Integer> path = new ArrayList<>();
\n int pathI = i;
\n while (pathI != srcIndex) {
\n path.add(pathI);
\n pathI = pPath[pathI];
\n }
\n path.add(srcIndex);<\/p>\n

Collections.reverse(path);<\/p>\n

for (int pos : path) {
\n System.out.print(arrayV[pos]+" -> ");
\n }
\n System.out.println(dist[i]);
\n }
\n }
\n}<\/p>\n

public static void testGraphDijkstra() {
\n String str = "syztx";
\n char[] array = str.toCharArray();
\n GraphByMatrix g = new GraphByMatrix(str.length(),true);
\n g.initArrayV(array);
\n g.addEdge('s', 't', 10);
\n g.addEdge('s', 'y', 5);
\n g.addEdge('y', 't', 3);
\n g.addEdge('y', 'x', 9);
\n g.addEdge('y', 'z', 2);
\n g.addEdge('z', 's', 7);
\n g.addEdge('z', 'x', 6);
\n g.addEdge('t', 'y', 2);
\n g.addEdge('t', 'x', 1);
\n g.addEdge('x', 'z', 4);<\/p>\n

int[] dist = new int[array.length];
\n int[] parentPath = new int[array.length];
\n g.dijkstra('s', dist, parentPath);<\/p>\n

g.printShortPath('s', dist, parentPath);
\n} <\/p>\n

\u00a0\u8fd0\u884c\u7ed3\u679c：<\/p>\n

\"\"<\/p>\n

\u7f3a\u70b9<\/h6>\n

\u65e0\u6cd5\u89e3\u51b3\u8fb9\u6743\u4e3a\u8d1f\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n

\u4f8b\u5982：<\/p>\n

\"\"<\/p>\n

\u5bf9\u4e8e\u4e0a\u9762\u7684\u60c5\u51b5，\u6211\u4eec\u77e5\u9053，Dijkstra\u7b97\u6cd5\u4e00\u5b9a\u4f1a\u9009\u62e9\u201c\u7eff\u8272\u201d\u7684\u90a3\u6761\u8def，\u4f46\u662f\u201c\u7ea2\u8272\u201d\u7684\u90a3\u6761\u8def\u662f\u6700\u77ed\u7684，\u8fd9\u5c31\u662fDijkstra\u7b97\u6cd5\u65e0\u6cd5\u89e3\u51b3\u7684\u8fb9\u6743\u4e3a\u8d1f\u7684\u60c5\u51b5，\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898，\u53ef\u4ee5\u4f7f\u7528\u4e0b\u9762\u7684Bellman-Ford\u7b97\u6cd5\u3002<\/p>\n

Bellman-Ford\u7b97\u6cd5（\u5355\u6e90）<\/h6>\n

Bellman-Ford\u7b97\u6cd5\u7531\u7406\u67e5\u5fb7\u00b7\u8d1d\u5c14\u66fc\u548c\u83b1\u65af\u7279\u00b7\u798f\u7279\u5171\u540c\u521b\u7acb\u3002\u8be5\u7b97\u6cd5\u7279\u522b\u9002\u7528\u4e8e\u56fe\u4e2d\u5b58\u5728\u8d1f\u6743\u8fb9\u7684\u60c5\u51b5，\u8fd9\u662f\u5176\u76f8\u5bf9\u4e8e\u5176\u4ed6\u7b97\u6cd5（\u5982Dijkstra\u7b97\u6cd5）\u7684\u4e3b\u8981\u4f18\u52bf\u3002<\/p>\n

\u7b97\u6cd5\u601d\u60f3<\/h6>\n

Bellman-Ford\u7b97\u6cd5\u901a\u8fc7\u591a\u6b21\u8fed\u4ee3\u6765\u9010\u6b65\u903c\u8fd1\u6700\u77ed\u8def\u5f84\u7684\u771f\u5b9e\u503c\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d，\u7b97\u6cd5\u4f1a\u5c1d\u8bd5\u901a\u8fc7\u6240\u6709\u8fb9\u6765\u66f4\u65b0\u8d77\u70b9\u5230\u5176\u4ed6\u6240\u6709\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u3002\u5982\u679c\u56fe\u4e2d\u4e0d\u5b58\u5728\u8d1f\u6743\u56de\u8def，\u90a3\u4e48\u7ecf\u8fc7\u6709\u9650\u6b21\u8fed\u4ee3\u540e，\u7b97\u6cd5\u5c06\u6536\u655b\u5230\u6b63\u786e\u7684\u6700\u77ed\u8def\u5f84\u89e3\u3002<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/h6>\n

1.\u521d\u59cb\u5316\u8ddd\u79bb\u6570\u7ec4dist[]，\u5c06\u8d77\u70b9\u5230\u81ea\u8eab\u7684\u8ddd\u79bb\u8bbe\u4e3a0，\u8d77\u70b9\u5230\u5176\u4ed6\u6240\u6709\u70b9\u7684\u8ddd\u79bb\u8bbe\u4e3a\u65e0\u7a77\u5927\u3002 2.\u6267\u884cn\u6b21\u8fed\u4ee3（n\u4e3a\u56fe\u4e2d\u9876\u70b9\u7684\u6570\u91cf），\u6bcf\u6b21\u8fed\u4ee3\u90fd\u5bf9\u6240\u6709\u8fb9\u6267\u884c\u677e\u5f1b\u64cd\u4f5c\u3002 3.\u5728\u7b2cn+1\u6b21\u8fed\u4ee3\u4e2d，\u518d\u6b21\u5bf9\u6240\u6709\u8fb9\u6267\u884c\u677e\u5f1b\u64cd\u4f5c，\u4f46\u8fd9\u6b21\u662f\u4e3a\u4e86\u68c0\u6d4b\u56fe\u4e2d\u662f\u5426\u5b58\u5728\u8d1f\u6743\u56de\u8def\u3002\u5982\u679c\u8fd8\u80fd\u66f4\u65b0\u67d0\u4e9b\u70b9\u7684\u8ddd\u79bb，\u5219\u8bf4\u660e\u56fe\u4e2d\u5b58\u5728\u8d1f\u6743\u56de\u8def\u3002<\/p>\n

\u4e3a\u4e86\u8bb0\u5f55\u4e0b\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684，\u4f7f\u7528\u4e00\u4e2a\u4e00\u7ef4\u6570\u7ec4pPath[]\u6765\u6807\u8bb0\u6bcf\u4e2a\u70b9\u5728\u6700\u77ed\u8def\u5f84\u4e2d\u524d\u4e00\u4e2a\u8282\u70b9\u662f\u4ec0\u4e48，\u8fd9\u6837\u5c31\u80fd\u591f\u77e5\u9053\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684\u4e86（\u5e76\u67e5\u96c6\u601d\u60f3）\u3002<\/p>\n

\u65f6\u95f4\u590d\u6742\u5ea6\u8f83\u9ad8，\u4e3aO(n^3)，\u4e0d\u9002\u5408\u5904\u7406\u5927\u89c4\u6a21\u56fe\u3002<\/p>\n

\u56fe\u89e3：<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/h6>\n

public boolean bellmanFord(char vSrc,int[] dist,int[] pPath) {
\n int srcIndex = getIndexOfV(vSrc);
\n \/\/\u8ddd\u79bb\u6570\u636e\u521d\u59cb\u5316
\n Arrays.fill(dist,Constant.MAX);
\n dist[srcIndex] = 0;
\n \/\/\u8def\u5f84\u6570\u7ec4\u521d\u59cb\u5316
\n Arrays.fill(pPath,-1);
\n pPath[srcIndex] = 0;<\/p>\n

int n = arrayV.length;<\/p>\n

for (int k = 0; k < n; k++) {
\n for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n if(matrix[i][j] != Constant.MAX &&
\n dist[i] + matrix[i][j] < dist[j]) {
\n dist[j] = dist[i] + matrix[i][j];
\n pPath[j] = i;
\n }
\n }
\n }
\n }<\/p>\n

for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n if(matrix[i][j] != Constant.MAX &&
\n dist[i] + matrix[i][j] < dist[j]) {
\n return false;
\n }
\n }
\n }
\n return true;
\n} <\/p>\n

\u4ee3\u7801\u6d4b\u8bd5<\/h6>\n

\u4e0d\u5e26\u8d1f\u6743\u56de\u8def\u7684\u60c5\u51b5<\/p>\n

public static void testGraphBellmanFord() {
\n String str = "syztx";
\n char[] array = str.toCharArray();
\n GraphByMatrix g = new GraphByMatrix(str.length(),true);
\n g.initArrayV(array);
\n g.addEdge('s', 't', 6);
\n g.addEdge('s', 'y', 7);
\n g.addEdge('y', 'z', 9);
\n g.addEdge('y', 'x', -3);
\n g.addEdge('z', 's', 2);
\n g.addEdge('z', 'x', 7);
\n g.addEdge('t', 'x', 5);
\n g.addEdge('t', 'y', 8);
\n g.addEdge('t', 'z', -4);
\n g.addEdge('x', 't', -2);<\/p>\n

int[] dist = new int[array.length];
\n int[] parentPath = new int[array.length];
\n boolean flg = g.bellmanFord('s', dist, parentPath);
\n if(flg) {
\n g.printShortPath('s', dist, parentPath);
\n }else {
\n System.out.println("\u5b58\u5728\u8d1f\u6743\u56de\u8def");
\n }
\n} <\/p>\n

\u8fd0\u884c\u7ed3\u679c：<\/p>\n

\"\"<\/p>\n

\u5e26\u8d1f\u6743\u56de\u8def\u7684\u60c5\u51b5<\/p>\n

public static void testGraphBellmanFord() {
\n String str = "syztx";
\n char[] array = str.toCharArray();
\n GraphByMatrix g = new GraphByMatrix(str.length(),true);
\n g.initArrayV(array);<\/p>\n

\/\/\u8d1f\u6743\u56de\u8def\u5b9e\u4f8b
\n g.addEdge('s', 't', 6);
\n g.addEdge('s', 'y', 7);
\n g.addEdge('y', 'z', 9);
\n g.addEdge('y', 'x', -3);
\n g.addEdge('y', 's', 1);
\n g.addEdge('z', 's', 2);
\n g.addEdge('z', 'x', 7);
\n g.addEdge('t', 'x', 5);
\n g.addEdge('t', 'y', -8);
\n g.addEdge('t', 'z', -4);
\n g.addEdge('x', 't', -2);<\/p>\n

int[] dist = new int[array.length];
\n int[] parentPath = new int[array.length];
\n boolean flg = g.bellmanFord('s', dist, parentPath);
\n if(flg) {
\n g.printShortPath('s', dist, parentPath);
\n }else {
\n System.out.println("\u5b58\u5728\u8d1f\u6743\u56de\u8def");
\n }
\n} <\/p>\n

\u8fd0\u884c\u7ed3\u679c：<\/p>\n

\"\"<\/p>\n

Floyd-Warshall\u7b97\u6cd5（\u591a\u6e90）<\/h6>\n
\u7b97\u6cd5\u601d\u60f3<\/h6>\n

Floyd-Warshall\u7b97\u6cd5\u91c7\u7528\u52a8\u6001\u89c4\u5212\u7684\u601d\u60f3，\u901a\u8fc7\u4e09\u91cd\u5faa\u73af\u904d\u5386\u56fe\u4e2d\u7684\u6bcf\u4e2a\u9876\u70b9，\u5e76\u5c1d\u8bd5\u5c06\u5176\u4f5c\u4e3a\u4e2d\u95f4\u70b9\u6765\u66f4\u65b0\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002\u7b97\u6cd5\u7684\u6838\u5fc3\u5728\u4e8e\u4e0d\u65ad\u6bd4\u8f83\u548c\u66f4\u65b0\u8def\u5f84\u957f\u5ea6，\u76f4\u5230\u627e\u5230\u6240\u6709\u9876\u70b9\u5bf9\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\u7b97\u6cd5\u6b65\u9aa4<\/h6>\n

1.\u521d\u59cb\u5316： \u4f7f\u7528\u56fe\u7684\u5e26\u6743\u90bb\u63a5\u77e9\u9635A\u4f5c\u4e3a\u8d77\u70b9，\u5176\u4e2dA[i][j]\u8868\u793a\u9876\u70b9i\u5230\u9876\u70b9j\u7684\u76f4\u63a5\u8ddd\u79bb\u3002\u5982\u679ci\u548cj\u4e4b\u95f4\u6ca1\u6709\u76f4\u63a5\u8fde\u63a5，\u5219A[i][j]\u88ab\u8bbe\u7f6e\u4e3a\u65e0\u7a77\u5927（\u6216\u4e00\u4e2a\u8db3\u591f\u5927\u7684\u6570）\u3002 2.\u4e09\u91cd\u5faa\u73af： \u5916\u5c42\u5faa\u73af\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u4e2d\u95f4\u70b9k（1\u5230n）\u3002 \u4e2d\u95f4\u4e24\u5c42\u5faa\u73af\u5206\u522b\u904d\u5386\u6240\u6709\u8d77\u70b9i\u548c\u7ec8\u70b9j（1\u5230n）\u3002 \u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d，\u68c0\u67e5\u662f\u5426\u53ef\u4ee5\u901a\u8fc7\u9876\u70b9k\u6765\u7f29\u77ed\u4ecei\u5230j\u7684\u8def\u5f84\u957f\u5ea6\u3002\u5982\u679c\u53ef\u4ee5，\u5219\u66f4\u65b0A[i][j]\u4e3aA[i][k] + A[k][j]\u3002 3.\u7ed3\u675f： \u5f53\u6240\u6709\u9876\u70b9\u90fd\u88ab\u7528\u4f5c\u4e2d\u95f4\u70b9\u65f6，\u7b97\u6cd5\u7ed3\u675f\u3002\u6b64\u65f6，A\u77e9\u9635\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20A[i][j]\u90fd\u8868\u793a\u4ece\u9876\u70b9i\u5230\u9876\u70b9j\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\u3002<\/p>\n

\u4e3a\u4e86\u8bb0\u5f55\u4e0b\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684，\u4f7f\u7528\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4pPath[]\u6765\u6807\u8bb0\u6bcf\u4e2a\u70b9\u5728\u6700\u77ed\u8def\u5f84\u4e2d\u524d\u4e00\u4e2a\u8282\u70b9\u662f\u4ec0\u4e48，\u8fd9\u6837\u5c31\u80fd\u591f\u77e5\u9053\u6700\u77ed\u8def\u5f84\u7684\u6574\u6761\u8def\u5f84\u662f\u600e\u6837\u7684\u4e86（\u5e76\u67e5\u96c6\u601d\u60f3）\u3002\u00a0<\/p>\n

\u65f6\u95f4\u590d\u6742\u5ea6\u8f83\u9ad8，\u4e3aO(n^3)，\u4e0d\u9002\u5408\u5904\u7406\u5927\u89c4\u6a21\u56fe\u3002<\/p>\n

\u4ee3\u7801\u5b9e\u73b0<\/h6>\n

public void floydWarShall(int[][] dist,int[][] pPath) {
\n int n = arrayV.length;<\/p>\n

for (int i = 0; i < n; i++) {
\n Arrays.fill(dist[i],Constant.MAX);
\n Arrays.fill(pPath[i],-1);
\n }<\/p>\n

for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n if(matrix[i][j] != Constant.MAX) {
\n dist[i][j] = matrix[i][j];
\n pPath[i][j] = i;
\n }else {
\n pPath[i][j] = -1;
\n }
\n if(i == j) {
\n dist[i][j] = 0;
\n pPath[i][j] = -1;
\n }
\n }
\n }<\/p>\n

for (int k = 0; k < n; k++) {
\n for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n if(dist[i][k] != Constant.MAX &&
\n dist[k][j] != Constant.MAX &&
\n dist[i][k] + dist[k][j] < dist[i][j]) {
\n dist[i][j] = dist[i][k] + dist[k][j];
\n \/\/\u66f4\u65b0\u7236\u8282\u70b9\u4e0b\u6807
\n \/\/pPath[i][j] = k;\/\/\u4e0d\u5bf9
\n \/\/\u5982\u679c\u7ecf\u8fc7\u4e86 i->k k->j \u6b64\u65f6\u662fk
\n \/\/i->x->s->k k->..t->…x->j
\n pPath[i][j] = pPath[k][j];
\n }
\n }
\n }
\n \/\/ \u6d4b\u8bd5 \u6253\u5370\u6743\u503c\u548c\u8def\u5f84\u77e9\u9635\u89c2\u5bdf\u6570\u636e
\n for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n if(dist[i][j] == Constant.MAX) {
\n System.out.print(" * ");
\n }else{
\n System.out.print(dist[i][j]+" ");
\n }
\n }
\n System.out.println();
\n }
\n System.out.println("=========\u6253\u5370\u8def\u5f84==========");
\n for (int i = 0; i < n; i++) {
\n for (int j = 0; j < n; j++) {
\n System.out.print(pPath[i][j]+" ");
\n }
\n System.out.println();
\n }
\n System.out.println("=================");
\n }
\n} <\/p>\n

\u4ee3\u7801\u6d4b\u8bd5<\/h6>\n

public static void testGraphFloydWarShall() {
\n String str = "12345";
\n char[] array = str.toCharArray();
\n GraphByMatrix g = new GraphByMatrix(str.length(),true);
\n g.initArrayV(array);
\n g.addEdge('1', '2', 3);
\n g.addEdge('1', '3', 8);
\n g.addEdge('1', '5', -4);
\n g.addEdge('2', '4', 1);
\n g.addEdge('2', '5', 7);
\n g.addEdge('3', '2', 4);
\n g.addEdge('4', '1', 2);
\n g.addEdge('4', '3', -5);
\n g.addEdge('5', '4', 6);<\/p>\n

int[][] dist = new int[array.length][array.length];
\n int[][] parentPath = new int[array.length][array.length];
\n g.floydWarShall(dist,parentPath);<\/p>\n

for (int i = 0; i < array.length; i++) {
\n g.printShortPath(array[i],dist[i],parentPath[i]);
\n System.out.println("************************");
\n }
\n} <\/p>\n

\u8fd0\u884c\u7ed3\u679c：<\/p>\n

\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

\u76ee\u5f55
\n\u200b\u7f16\u8f91
\n\u6700\u77ed\u8def\u5f84
\nDijkstra\u7b97\u6cd5（\u5355\u6e90）
\n\u7b97\u6cd5\u601d\u60f3
\n\u7b97\u6cd5\u6b65\u9aa4
\n\u4ee3\u7801\u5b9e\u73b0 \u4ee3\u7801\u6d4b\u8bd5
\n\u7f3a\u70b9
\nBellman-Ford\u7b97\u6cd5（\u5355\u6e90）
\n\u7b97\u6cd5\u601d\u60f3
\n\u7b97\u6cd5\u6b65\u9aa4
\n\u4ee3\u7801\u5b9e\u73b0
\n\u4ee3\u7801\u6d4b\u8bd5
\nFloyd-Warshall\u7b97\u6cd5（\u591a\u6e90）
\n\u7b97\u6cd5\u601d\u60f3
\n\u7b97\u6cd5\u6b65\u9aa4
\n\u4ee3\u7801\u5b9e\u73b0
\n\u4ee3\u7801\u6d4b\u8bd5 \u6700\u77ed\u8def\u5f84
\n\u6700\u77ed\u8def\u5f84\u95ee\u9898：\u4ece\u5728\u5e26\u6743\u56fe\u7684\u67d0\u4e00\u9876\u70b9\u51fa\u53d1，\u627e\u51fa\u4e00\u6761\u901a\u5f80\u53e6\u4e00\u9876\u70b9\u7684<\/p>\n","protected":false},"author":2,"featured_media":76527,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[99],"topic":[],"class_list":["post-76540","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-java"],"yoast_head":"\n\u8be6\u89e3\u6700\u77ed\u8def\u5f84 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/76540.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u8be6\u89e3\u6700\u77ed\u8def\u5f84 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u76ee\u5f55 \u200b\u7f16\u8f91 \u6700\u77ed\u8def\u5f84 Dijkstra\u7b97\u6cd5（\u5355\u6e90） \u7b97\u6cd5\u601d\u60f3 \u7b97\u6cd5\u6b65\u9aa4 \u4ee3\u7801\u5b9e\u73b0 \u4ee3\u7801\u6d4b\u8bd5 \u7f3a\u70b9 Bellman-Ford\u7b97\u6cd5（\u5355\u6e90） \u7b97\u6cd5\u601d\u60f3 \u7b97\u6cd5\u6b65\u9aa4 \u4ee3\u7801\u5b9e\u73b0 \u4ee3\u7801\u6d4b\u8bd5 Floyd-Warshall\u7b97\u6cd5（\u591a\u6e90） \u7b97\u6cd5\u601d\u60f3 \u7b97\u6cd5\u6b65\u9aa4 \u4ee3\u7801\u5b9e\u73b0 \u4ee3\u7801\u6d4b\u8bd5 \u6700\u77ed\u8def\u5f84 \u6700\u77ed\u8def\u5f84\u95ee\u9898：\u4ece\u5728\u5e26\u6743\u56fe\u7684\u67d0\u4e00\u9876\u70b9\u51fa\u53d1，\u627e\u51fa\u4e00\u6761\u901a\u5f80\u53e6\u4e00\u9876\u70b9\u7684\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.wsisp.com\/helps\/76540.html\" \/>\n<meta property=\"og:site_name\" content=\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"article:published_time\" content=\"2026-02-22T10:17:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222101735-699ad7bf3de03.gif\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/76540.html\",\"url\":\"https:\/\/www.wsisp.com\/helps\/76540.html\",\"name\":\"\u8be6\u89e3\u6700\u77ed\u8def\u5f84 - 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