Link.
给定一个 \(n\) 个点 \(m\) 条边的无向连通图,\(q\) 次询问,每次给出一个点集 \(s\),求至少在原图中删去多少个点,使得 \(s\) 中存在两点不连通。多组数据。
每组数据 \(n,q\le10^5\),\(m,\sum|s|\le2\times10^5\)。
看到 \(\sum|s|\) 的限制,不难联想到虚树或者其它与 DFN 相关的算法。
所以,先建出圆方树,并处理好 DFN,LCA 的一系列信息。考虑到答案就是 \(s\) 中的点在树上构成的连通块中不在 \(s\) 集合里的圆点个数,可以求一个树上前缀和:\(sum_u\) 表示从 \(u\) 到根路径上的圆点个数。处理询问时,先将 \(s\) 按 DFN 从小到大排序,两两 LCA 计算贡献(\(s_1\) 和 \(s_{|s|}\) 也要算一次),最后除以 \(2\),得到 \(cnt\)。不过整个连通块的根(\(\operatorname{LCA}(s_1,s_{|s|})\))这个点的贡献被遗漏了,所以要再加上这个点单独的贡献。最终答案就是 \(cnt-|s|\)。
复杂度 \(\mathcal O(T(n\log n+\sum|s|\log\sum|s|))\)。
就是练练码力的一道题嘛 qwq。
#include <queue>
#include <cstdio>
#include <algorithm>
#define adj( g, u, v ) \
for ( int eid = g.head[u], v; v = g.to[eid], eid; eid = g.nxt[eid] )
const int MAXN = 2e5, MAXM = 4e5;
int n, m, q, snode;
int dfc, top, dfn[MAXN + 5], low[MAXN + 5], stk[MAXN + 5];
int lg[MAXN * 2 + 5], dep[MAXN + 5], st[MAXN * 2 + 5][20], sum[MAXN + 5];
struct Graph {
int ecnt, head[MAXN + 5], to[MAXM + 5], nxt[MAXM + 5];
inline void link ( const int s, const int t ) {
to[++ ecnt] = t, nxt[ecnt] = head[s];
head[s] = ecnt;
}
inline void add ( const int u, const int v ) {
link ( u, v ), link ( v, u );
}
inline void clear () {
ecnt = 0;
for ( int i = 1; i <= n << 1; ++ i ) head[i] = 0;
}
} src, tre;
inline int rint () {
int x = 0; char s = getchar ();
for ( ; s < '0' || '9' < s; s = getchar () );
for ( ; '0' <= s && s <= '9'; s = getchar () ) x = x * 10 + ( s ^ '0' );
return x;
}
inline bool chkmin ( int& a, const int b ) { return b < a ? a = b, true : false; }
inline void clear () {
src.clear (), tre.clear ();
dfc = top = snode = 0;
for ( int i = 1; i <= n << 1; ++ i ) dfn[i] = low[i] = 0;
}
inline void Tarjan ( const int u, const int f ) {
dfn[u] = low[u] = ++ dfc, stk[++ top] = u;
adj ( src, u, v ) if ( v ^ f ) {
if ( ! dfn[v] ) {
Tarjan ( v, u ), chkmin ( low[u], low[v] );
if ( low[v] >= dfn[u] ) {
tre.add ( u, ++ snode );
do tre.add ( snode, stk[top] ); while ( stk[top --] ^ v );
}
} else chkmin ( low[u], dfn[v] );
}
}
inline void initDFN ( const int u, const int f ) {
dep[st[dfn[u] = ++ dfc][0] = u] = dep[f] + 1, sum[u] = sum[f] + ( u <= n );
adj ( tre, u, v ) if ( v ^ f ) initDFN ( v, u ), st[++ dfc][0] = u;
}
inline void initST () {
for ( int i = 2; i <= n << 2; ++ i ) lg[i] = lg[i >> 1] + 1;
for ( int j = 1; 1 << j <= dfc; ++ j ) {
for ( int i = 1; i + ( 1 << j ) - 1 <= dfc; ++ i ) {
if ( dep[st[i][j - 1]] < dep[st[i + ( 1 << j >> 1 )][j - 1]] ) st[i][j] = st[i][j - 1];
else st[i][j] = st[i + ( 1 << j >> 1 )][j - 1];
}
}
}
inline int calcLCA ( int u, int v ) {
if ( dfn[u] > dfn[v] ) u ^= v ^= u ^= v;
int k = lg[dfn[v] - dfn[u] + 1];
return dep[st[dfn[u]][k]] < dep[st[dfn[v] - ( 1 << k ) + 1][k]] ?
st[dfn[u]][k] : st[dfn[v] - ( 1 << k ) + 1][k];
}
inline void solve () {
static int cnts, s[MAXN + 5];
cnts = rint ();
for ( int i = 1; i <= cnts; ++ i ) s[i] = rint ();
std::sort ( s + 1, s + cnts + 1, []( const int a, const int b ) { return dfn[a] < dfn[b]; } );
int cnt = 0;
for ( int i = 1; i < cnts; ++ i ) {
int u = s[i], v = s[i + 1], t = calcLCA ( u, v );
cnt += sum[u] + sum[v] - 2 * sum[t];
}
int u = s[1], v = s[cnts], t = calcLCA ( u, v );
cnt += sum[u] + sum[v] - 2 * sum[t];
cnt /= 2, cnt += ( t <= n ) - cnts;
printf ( "%d\n", cnt );
}
int main () {
for ( int T = rint (); T --; clear () ) {
n = snode = rint (), m = rint ();
for ( int i = 1, u, v; i <= m; ++ i ) {
u = rint (), v = rint ();
src.add ( u, v );
}
Tarjan ( 1, 0 ), dfc = 0;
initDFN ( 1, 0 ), initST ();
for ( q = rint (); q --; ) solve ();
}
return 0;
}
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