diff --git a/problems/kamacoder/0096.城市间货物运输III.md b/problems/kamacoder/0096.城市间货物运输III.md index a3e9e840..08fd010c 100644 --- a/problems/kamacoder/0096.城市间货物运输III.md +++ b/problems/kamacoder/0096.城市间货物运输III.md @@ -702,6 +702,125 @@ public class Main { ``` +```java +class Edge { + public int u; // 边的端点1 + public int v; // 边的端点2 + public int val; // 边的权值 + + public Edge() { + } + + public Edge(int u, int v) { + this.u = u; + this.v = v; + this.val = 0; + } + + public Edge(int u, int v, int val) { + this.u = u; + this.v = v; + this.val = val; + } +} + +/** + * SPFA算法(版本3):处理含【负权回路】的有向图的最短路径问题 + * bellman_ford(版本3) 的队列优化算法版本 + * 限定起点、终点、至多途径k个节点 + */ +public class SPFAForSSSP { + + /** + * SPFA算法 + * + * @param n 节点个数[1,n] + * @param graph 邻接表 + * @param startIdx 开始节点(源点) + */ + public static int[] spfa(int n, List> graph, int startIdx, int k) { + // 定义最大范围 + int maxVal = Integer.MAX_VALUE; + // minDist[i] 源点到节点i的最短距离 + int[] minDist = new int[n + 1]; // 有效节点编号范围:[1,n] + Arrays.fill(minDist, maxVal); // 初始化为maxVal + minDist[startIdx] = 0; // 设置源点到源点的最短路径为0 + + // 定义queue记录每一次松弛更新的节点 + Queue queue = new LinkedList<>(); + queue.offer(startIdx); // 初始化:源点开始(queue和minDist的更新是同步的) + + + // SPFA算法核心:只对上一次松弛的时候更新过的节点关联的边进行松弛操作 + while (k + 1 > 0 && !queue.isEmpty()) { // 限定松弛 k+1 次 + int curSize = queue.size(); // 记录当前队列节点个数(上一次松弛更新的节点个数,用作分层统计) + while (curSize-- > 0) { //分层控制,限定本次松弛只针对上一次松弛更新的节点,不对新增的节点做处理 + // 记录当前minDist状态,作为本次松弛的基础 + int[] minDist_copy = Arrays.copyOfRange(minDist, 0, minDist.length); + + // 取出节点 + int cur = queue.poll(); + // 获取cur节点关联的边,进行松弛操作 + List relateEdges = graph.get(cur); + for (Edge edge : relateEdges) { + int u = edge.u; // 与`cur`对照 + int v = edge.v; + int weight = edge.val; + if (minDist_copy[u] + weight < minDist[v]) { + minDist[v] = minDist_copy[u] + weight; // 更新 + // 队列同步更新(此处有一个针对队列的优化:就是如果已经存在于队列的元素不需要重复添加) + if (!queue.contains(v)) { + queue.offer(v); // 与minDist[i]同步更新,将本次更新的节点加入队列,用做下一个松弛的参考基础 + } + } + } + } + // 当次松弛结束,次数-1 + k--; + } + + // 返回minDist + return minDist; + } + + public static void main(String[] args) { + // 输入控制 + Scanner sc = new Scanner(System.in); + System.out.println("1.输入N个节点、M条边(u v weight)"); + int n = sc.nextInt(); + int m = sc.nextInt(); + + System.out.println("2.输入M条边"); + List> graph = new ArrayList<>(); // 构建邻接表 + for (int i = 0; i <= n; i++) { + graph.add(new ArrayList<>()); + } + while (m-- > 0) { + int u = sc.nextInt(); + int v = sc.nextInt(); + int weight = sc.nextInt(); + graph.get(u).add(new Edge(u, v, weight)); + } + + System.out.println("3.输入src dst k(起点、终点、至多途径k个点)"); + int src = sc.nextInt(); + int dst = sc.nextInt(); + int k = sc.nextInt(); + + // 调用算法 + int[] minDist = SPFAForSSSP.spfa(n, graph, src, k); + // 校验起点->终点 + if (minDist[dst] == Integer.MAX_VALUE) { + System.out.println("unreachable"); + } else { + System.out.println("最短路径:" + minDist[n]); + } + } +} +``` + + + ### Python ```python def main(): diff --git a/problems/kamacoder/0097.小明逛公园.md b/problems/kamacoder/0097.小明逛公园.md index 8b3078fc..27ad0eb9 100644 --- a/problems/kamacoder/0097.小明逛公园.md +++ b/problems/kamacoder/0097.小明逛公园.md @@ -424,6 +424,71 @@ floyd算法的时间复杂度相对较高,适合 稠密图且源点较多的 ### Java +- 基于三维数组的Floyd算法 + +```java +public class FloydBase { + + // public static int MAX_VAL = Integer.MAX_VALUE; + public static int MAX_VAL = 10005; // 边的最大距离是10^4(不选用Integer.MAX_VALUE是为了避免相加导致数值溢出) + + public static void main(String[] args) { + // 输入控制 + Scanner sc = new Scanner(System.in); + System.out.println("1.输入N M"); + int n = sc.nextInt(); + int m = sc.nextInt(); + + System.out.println("2.输入M条边"); + + // ① dp定义(grid[i][j][k] 节点i到节点j 可能经过节点K(k∈[1,n]))的最短路径 + int[][][] grid = new int[n + 1][n + 1][n + 1]; + for (int i = 1; i <= n; i++) { + for (int j = 1; j <= n; j++) { + for (int k = 0; k <= n; k++) { + grid[i][j][k] = grid[j][i][k] = MAX_VAL; // 其余设置为最大值 + } + } + } + + // ② dp 推导:grid[i][j][k] = min{grid[i][k][k-1] + grid[k][j][k-1], grid[i][j][k-1]} + while (m-- > 0) { + int u = sc.nextInt(); + int v = sc.nextInt(); + int weight = sc.nextInt(); + grid[u][v][0] = grid[v][u][0] = weight; // 初始化(处理k=0的情况) ③ dp初始化 + } + + // ④ dp推导:floyd 推导 + for (int k = 1; k <= n; k++) { + for (int i = 1; i <= n; i++) { + for (int j = 1; j <= n; j++) { + grid[i][j][k] = Math.min(grid[i][k][k - 1] + grid[k][j][k - 1], grid[i][j][k - 1]); + } + } + } + + System.out.println("3.输入[起点-终点]计划个数"); + int x = sc.nextInt(); + + System.out.println("4.输入每个起点src 终点dst"); + + while (x-- > 0) { + int src = sc.nextInt(); + int dst = sc.nextInt(); + // 根据floyd推导结果输出计划路径的最小距离 + if (grid[src][dst][n] == MAX_VAL) { + System.out.println("-1"); + } else { + System.out.println(grid[src][dst][n]); + } + } + } +} +``` + + + ### Python 基于三维数组的Floyd