算法: 树的直径

1. 二叉树的直径

int height(Node* root) {    // 二叉树的高度
    int ret = 0;
    if (root != null) {
        ret = 1 + max(height(root->left), height(root->right));
    }
    return ret;
}

int diameter(Node* root) { 
    int ret = 0;
    if (root != null) {
        int lefth = height(root->left);
        int righth = height(root->right);
        ret = max(lefth + 1 + righth, 
                diameter(root->left),
                diameter(root->right));
    }
    return ret;
}

2. 无向树的直径

A standard algorithm for finding longest path in undirected trees using two depth-first searches:
  (1) Start DFS from a random vertex v and find the farthest vertex from it; say it is v′.
  (2) Now start a DFS from v′ to find the vertex farthest from it. This path is the longest path in the graph.

算法要求树中边的权值非负。

这里有一个观察： 在树中, 与一个节点距离最远的节点, 一定是树的直径的端点。(尚不知如何证明。)

#include <stdio.h>  

#define N   (10 + 1)    
#define M   2*N        

typedef struct s_edge {
    int v, w;
    int next;
} Edge;

Edge edges[M]; int ne = 0;
int adj[N]; int nv;
int visited[N];

int find_furthest(int u, int* pv) { // 求一个与 u 距离最远的节点, 并返回最远距离
    int dist = 0;
    int i, flag = 0;
    visited[u] = 1;
    for (i = adj[u]; i != -1; i = edges[i].next) {
        int v = edges[i].v, x;
        if (visited[v] == 0) {
            int tmp = edges[i].w + find_furthest(v, &x);
            if (tmp > dist) {
                dist = tmp, *pv = x;
            }
            flag = 1;
        }
    }
    if (flag == 0) *pv = u;
    return dist;
}

int furthest_vertex(int u) { // 求一个与 u 距离最远的节点
    int v;
    memset(visited, 0, sizeof(visited));
    find_furthest(u, &v);
    return v;
}

void print_graph() {    // 打印图的邻接表
    int u, i, v;
    for (u = 1; u <= nv; u++) {
        printf("%d : ", u);
        for (i = adj[u]; i != -1; i = edges[i].next) {
            printf("%d ", edges[i].v);
        }
        printf("
");
    }
}

int main() {
    int i, u, v, w;
    scanf("%d", &nv); // 输入节点数
    for (i = 1; i <= nv; i++) {
        adj[i] = -1;
    }
    for (i = 1; i < nv; i++) {
        scanf("%d%d%d", &u, &v, &w); // 输入边， 端点u、v, 权值 w (w >= 0)
        edges[ne].v = v, edges[ne].w = w;
        edges[ne].next = adj[u], adj[u] = ne++;
        edges[ne].v = u, edges[ne].w = w;
        edges[ne].next = adj[v], adj[v] = ne++;
    }
    //print_graph();

    u = furthest_vertex(1);
    v = furthest_vertex(u);

    printf("(%d, %d)
", u, v);

    return 0;
}