虚树
文章目录
题目描述
给定一棵树,割断每一条边都有代价,每次询问会给定一些点,求用最少的代价使所有给定点都和1号节点不连通
虚树这东西就是每次只把有用的点留下,不用的点删了
首先你树形 dp 要好,不然这东西学了也没用
关于如何构建虚树
单调栈建虚树
每次用单调栈维护一条链
OI wiki 上讲的很详细,还带图片
这做法虽然跑得快,但是写着好不爽,我选择欧拉序
#include <stdio.h>
#include <algorithm>
#include <memory.h>
typedef long long ll;
const int N = 25e4 + 5;
const int M = N << 1;
const ll INF = 1LL << 60;
int n, m;
int dfn[N], idx[N], top[N], son[N], fa[N], depth[N], size[N];
ll min[N], f[N];
int a[N], stack[N], ceil;
struct Graph
{
int head[N], num_edge;
struct Node
{
int next, to, dis;
} edge[M];
void add_edge(int u, int v, int dis = 0) { edge[++num_edge] = Node{head[u], v, dis}, head[u] = num_edge; }
void dfs1(int u, int fa)
{
::fa[u] = fa, size[u] = 1, depth[u] = depth[fa] + 1;
for (int i = head[u], v; i; i = edge[i].next)
if ((v = edge[i].to) != fa)
min[v] = std::min(min[u], (ll)edge[i].dis), dfs1(v, u), size[u] += size[v], son[u] = size[v] > size[son[u]] ? v : son[u];
}
void dfs2(int u)
{
dfn[u] = ++dfn[0], idx[dfn[u]] = u, top[u] = u == son[fa[u]] ? top[fa[u]] : u;
if (son[u])
dfs2(son[u]);
for (int i = head[u], v; i; i = edge[i].next)
if ((v = edge[i].to) != son[u] and v != fa[u])
dfs2(v);
}
int lca(int x, int y)
{
for (; top[x] != top[y]; x = fa[top[x]])
if (dfn[top[x]] < dfn[top[y]])
std::swap(x, y);
return dfn[x] < dfn[y] ? x : y;
}
void insert(int u)
{
if (ceil == 1)
{
if (u != 1)
stack[++ceil] = u;
return;
}
int x = lca(u, stack[ceil]);
if (x == stack[ceil])
return;
for (; ceil > 1 and dfn[stack[ceil - 1]] >= dfn[x]; ceil--)
add_edge(stack[ceil - 1], stack[ceil]);
if (stack[ceil] != x)
add_edge(x, stack[ceil]), stack[ceil] = x;
stack[++ceil] = u;
}
void dp(int u)
{
if (!head[u])
{
f[u] = min[u];
return;
}
f[u] = 0;
for (int i = head[u], v; i; i = edge[i].next)
dp(v = edge[i].to), f[u] += f[v];
f[u] = std::min(f[u], min[u]);
head[u] = 0;
}
} G, VT;
bool cmp(int a, int b) { return dfn[a] < dfn[b]; }
int read()
{
int x = 0, f = 1;
char ch = getchar();
while ('0' > ch or ch > '9')
f = ch == '-' ? -1 : 1, ch = getchar();
while ('0' <= ch and ch <= '9')
x = x * 10 + ch - 48, ch = getchar();
return x * f;
}
int main()
{
n = read();
for (int i = 1, u, v, dis; i < n; i++)
u = read(), v = read(), dis = read(), G.add_edge(u, v, dis), G.add_edge(v, u, dis);
min[1] = INF, G.dfs1(1, 0), G.dfs2(1);
for (m = read(); m--;)
{
int k = read();
for (int i = 1; i <= k; i++)
a[i] = read();
std::sort(a + 1, a + 1 + k, cmp);
VT.num_edge = 0;
stack[ceil = 1] = 1;
for (int i = 1; i <= k; i++)
VT.insert(a[i]);
for (; ceil > 1; ceil--)
VT.add_edge(stack[ceil - 1], stack[ceil]);
VT.dp(1);
printf("%lld\n", f[1]);
}
return 0;
}
欧拉序建虚树
好理解,写起来容易
先把给定的点按照欧拉序排个序
那么虚树里肯定有且只有这些点和数组里相邻两个点的 lca 和 根节点 (记得去重)
然后再把所有点复制一遍,用负数表示它是出栈的点
然后按照欧拉序再排个序,这数组就变成我们在虚树上 dfs 时的栈啦
按照这个栈做就行了,连虚树的邻接表都不用搞,方便的 eb
稍微慢了一点,应该不会卡吧(大概
#include <stdio.h>
#include <algorithm>
#include <memory.h>
typedef long long ll;
const int N = 3e6 + 5;
const int M = N << 1;
const ll INF = 1LL << 60;
int n, m;
int dfin[N], dfou[N], top[N], son[N], fa[N], depth[N], size[N];
long long f[N], min[N];
int head[N], num_edge;
int a[N], stack[N];
bool b[N];
struct Node
{
int next, to, dis;
} edge[M];
void add_edge(int u, int v, int dis = 0) { edge[++num_edge] = Node{head[u], v, dis}, head[u] = num_edge; }
int read()
{
int x = 0, f = 1;
char ch = getchar();
while ('0' > ch or ch > '9')
f = ch == '-' ? -1 : 1, ch = getchar();
while ('0' <= ch and ch <= '9')
x = x * 10 + ch - 48, ch = getchar();
return x * f;
}
void dfs1(int u, int fa)
{
::fa[u] = fa, size[u] = 1, depth[u] = depth[fa] + 1;
for (int i = head[u], v; i; i = edge[i].next)
if ((v = edge[i].to) != fa)
min[v] = std::min(min[u], (ll)edge[i].dis), dfs1(v, u), size[u] += size[v], son[u] = size[v] > size[son[u]] ? v : son[u];
}
void dfs2(int u)
{
dfin[u] = ++dfin[0], top[u] = u == son[fa[u]] ? top[fa[u]] : u;
if (son[u])
dfs2(son[u]);
for (int i = head[u], v; i; i = edge[i].next)
if ((v = edge[i].to) != fa[u] and v != son[u])
dfs2(v);
dfou[u] = ++dfin[0];
}
int lca(int x, int y)
{
for (; top[x] != top[y]; x = fa[top[x]])
if (dfin[top[x]] < dfin[top[y]])
std::swap(x, y);
return dfin[x] < dfin[y] ? x : y;
}
bool cmp(int x, int y)
{
int k1 = x > 0 ? dfin[x] : dfou[-x];
int k2 = y > 0 ? dfin[y] : dfou[-y];
return k1 < k2;
}
int main()
{
n = read();
for (int i = 1, u, v, dis; i < n; i++)
u = read(), v = read(), dis = read(), add_edge(u, v, dis), add_edge(v, u, dis);
min[1] = INF, dfs1(1, 0), dfs2(1);
for (m = read(); m--;)
{
int k = read();
for (int i = 1; i <= k; i++)
a[i] = read(), b[a[i]] = true, f[a[i]] = min[a[i]];
std::sort(a + 1, a + 1 + k, cmp);
for (int i = 1; i < k; i++)
{
int LCA = lca(a[i], a[i + 1]);
if (!b[LCA])
a[++k] = LCA, b[LCA] = true;
}
for (int i = 1, tmp = k; i <= tmp; i++)
a[++k] = -a[i];
if (!b[1])
a[++k] = 1, a[++k] = -1;
std::sort(a + 1, a + 1 + k, cmp);
for (int i = 1, top = 0; i <= k; i++)
if (a[i] > 0)
stack[++top] = a[i];
else
{
int u = stack[top--];
if (u != 1)
{
int fa = stack[top];
f[fa] += std::min(f[u], min[u]);
}
else
printf("%lld\n", f[1]);
f[u] = b[u] = 0;
}
}
return 0;
}