洛谷 P1967 货车运输

原题

题面请查看洛谷 P1967 货车运输。

题解

思路

根据题面，假设我们有一个普通的图：

作图工具：Graph Editor

考虑从顶点\(1\)走到顶点\(3\)：

• 路径\(1 \to 3\)（最大运货量为\(1\)）；

• 路径\(1 \to 2 \to 3\)（最大运货量为\(3\)，更优）。

所以我们可以删掉\(1 \to 3\)这条边，形成了一棵树，通过多次观察发现，这是一颗原图的最大生成树。

问题就被转化成了求最大生成树和在树上解决原问题。

代码

• 求最大生成树：我们使用\(\text{Kruskal}\)算法；

• 在树上解决原问题比较简单，我们只需要通过最近公共祖先（倍增法求解）进行求解即可。

代码如下：

#include <algorithm>
#include <cstdio>
using std::sort;
#define INF 0X3F3F3F3F
#define min(a, b) ((a) < (b) ? (a) : (b))struct Tree
{bool vis[10001];int cnt, head[10001], to[20001], w[20001], Next[20001];int dep[10001], fa[10001][21], W[10001][21];void DFS(int);void Add_Edge(int, int, int);void LCA_Init(void);int LCA(int, int);
};struct Graph
{struct Kruskal{struct Edge{int f, t, val;bool operator<(const Edge &a) const{return val > a.val;}};struct Union_Find{int ID[10001];void Init(void);void connect(int, int);bool search(int, int);int find(int);};Union_Find B;Edge E[50001];void kruskal(void);};int n, m;Kruskal K;void Read(void);
};int q;
Tree T;
Graph G;int main(void)
{G.Read();G.K.B.Init();G.K.kruskal();T.LCA_Init();scanf("%d", &q);while (q--){static int x, y;scanf("%d%d", &x, &y);printf("%d\n", T.LCA(x, y));}return 0;
}void Tree::Add_Edge(int f, int t, int val)
{Next[++cnt] = head[f];to[cnt] = t;w[cnt] = val;head[f] = cnt;return;
}void Tree::DFS(int ID)
{register int i, To;vis[ID] = true;for (i = head[ID]; i; i = Next[i]){To = to[i];if (vis[To])continue;dep[To] = dep[ID] + 1;fa[To][0] = ID;W[To][0] = w[i];DFS(To);}return;
}void Tree::LCA_Init(void)
{register int i, j;for (i = 1; i <= G.n; ++i)if (!vis[i]){dep[i] = 1;DFS(i);fa[i][0] = i;W[i][0] = INF;}for (i = 1; i <= 20; ++i)for (j = 1; j <= G.n; ++j){fa[j][i] = fa[fa[j][i - 1]][i - 1];W[j][i] = min(W[j][i - 1], W[fa[j][i - 1]][i - 1]);}return;
}int Tree::LCA(int x, int y)
{if (!G.K.B.search(x, y))return -1;register int i, ans = INF;if (dep[x] > dep[y]){int temp = x;x = y;y = temp;}for (i = 20; i >= 0; --i)if (dep[fa[y][i]] >= dep[x]){ans = min(ans, W[y][i]);y = fa[y][i];}if (x == y)return ans;for (i = 20; i >= 0; --i)if (fa[x][i] != fa[y][i]){ans = min(ans, min(W[x][i], W[y][i]));x = fa[x][i];y = fa[y][i];}ans = min(ans, min(W[x][0], W[y][0]));return ans;
}void Graph::Kruskal::Union_Find::Init(void)
{register int i;for (i = 1; i <= G.n; ++i)ID[i] = i;return;
}void Graph::Kruskal::Union_Find::connect(int a, int b)
{register int ra = find(a), rb = find(b);if (ra != rb)ID[rb] = ra;return;
}bool Graph::Kruskal::Union_Find::search(int a, int b)
{return find(a) == find(b);
}int Graph::Kruskal::Union_Find::find(int x)
{if (x == ID[x])return x;elsereturn ID[x] = find(ID[x]);
}void Graph::Kruskal::kruskal(void)
{register int i, cnt = 0;sort(E + 1, E + G.m + 1);for (i = 1; i <= G.m && cnt < G.n - 1; ++i){if (!B.search(E[i].f, E[i].t)){B.connect(E[i].f, E[i].t);++cnt;T.Add_Edge(E[i].f, E[i].t, E[i].val);T.Add_Edge(E[i].t, E[i].f, E[i].val);}}return;
}void Graph::Read(void)
{register int i;scanf("%d%d", &n, &m);for (i = 1; i <= m; ++i)scanf("%d%d%d", &K.E[i].f, &K.E[i].t, &K.E[i].val);return;
}