Solution -「UOJ #87」mx 的仙人掌

(mathcal{Description})

  Link.

  给出含 (n) 个结点 (m) 条边的仙人掌图。(q) 次询问，每次询问给出一个点集 (S)，求 (S) 内两两结点最短距离的最大值。

  (n,sum|S|le3 imes10^5)。

(mathcal{Solution})

  圆方树 + 虚树 = 虚圆方树！

  首先，考虑对于整个仙人掌怎么求答案：建出圆方树，DP 记录子树最深结点深度，在方点处单调队列合并圆儿子的两条链贡献答案即可。

  接下来，只需要把“虚圆方树”给弄出来就好。关键即在于满足方点周围一定裹着它自己管辖的圆点的性质，那么在建虚树边时特殊考虑一下就完成啦。

(mathcal{Code})

/* Clearink */

#include <cstdio>
#include <vector>

#define rep( i, l, r ) for ( int i = l, repEnd##i = r; i <= repEnd##i; ++i )
#define per( i, r, l ) for ( int i = r, repEnd##i = l; i >= repEnd##i; --i )

inline char fgc() {
	static char buf[1 << 17], *p = buf, *q = buf;
	return p == q && ( q = buf + fread( p = buf, 1, 1 << 17, stdin ), p == q )
		? EOF : *p++;
}

inline int rint() {
	int x = 0; char s = fgc();
	for ( ; s < '0' || '9' < s; s = fgc() );
	for ( ; '0' <= s && s <= '9'; s = fgc() ) x = x * 10 + ( s ^ '0' );
	return x;
}

template<typename Tp>
inline void wint( Tp x ) {
	if ( x < 0 ) putchar( '-' ), x = -x;
	if ( 9 < x ) wint( x / 10 );
	putchar( x % 10 ^ '0' );
}

typedef long long LL;

template<typename Tp>
inline void chkmin( Tp& a, const Tp b ) { b < a && ( a = b ); }
template<typename Tp>
inline void chkmax( Tp& a, const Tp b ) { a < b && ( a = b ); }
inline LL lmin( const LL a, const LL b ) { return a < b ? a : b; }

const int MAXN = 3e5, MAXM = MAXN << 1, MAXLG = 20;
const LL LINF = 1ll << 60;
int n, m;

template<const int NODE, const int EDGE>
struct Graph {
	int ecnt, head[NODE], to[EDGE], nxt[EDGE];
	LL len[EDGE];
	Graph(): ecnt( 1 ) {}

	inline void operator () ( const int s, const int t, const LL w ) {
		#ifdef RYBY
			printf( "%d %d %lld
", s, t, w );
		#endif
		to[++ecnt] = t, len[ecnt] = w, nxt[ecnt] = head[s], head[s] = ecnt;
	}
};
#define adj( t, u, v ) 
	for ( int e = t.head[u], v; v = t.to[e], e; e = t.nxt[e] )

Graph<MAXN + 5, MAXM * 2 + 5> src;

int vnode, dfc, dfn[MAXN * 2 + 5], low[MAXN + 5];
LL pre[MAXN * 2 + 5];
Graph<MAXN * 2 + 5, MAXN * 2 + 5> cac;

inline void buildCactus( const int u, const int f ) {
	static int top = 0, stk[MAXN + 5];
	dfn[u] = low[u] = ++dfc, stk[++top] = u;
	adj( src, u, v ) if ( v != f ) {
		if ( !dfn[v] ) {
			 buildCactus( v, u );
			 chkmin( low[u], low[v] );

			 if ( low[v] >= dfn[u] ) {
				cac( u, ++vnode, 0 ), pre[vnode] = src.len[e];
				int las = u, ttop = top, cnt = 0;
				do {
					int w = stk[top];
					for ( int i = src.head[w]; i; i = src.nxt[i] ) {
						if ( i ^ e ^ 1 && src.to[i] == las ) {
							++cnt;
							pre[w] = pre[las] + src.len[i];
							pre[vnode] += src.len[i];
							break;
						}
					}
					las = w;
				} while ( stk[top--] != v );

				do {
					int w = stk[ttop];
					cac( vnode, w, cnt ?
						lmin( pre[w], pre[vnode] - pre[w] ) : pre[vnode] );
				} while ( stk[ttop--] != v );
			 }
		} else chkmin( low[u], dfn[v] );
	}
}

// `dfc` and `dfn` was used by `buildCactus`, pay attention.
int dep[MAXN * 2 + 5], fa[MAXN * 2 + 5][MAXLG + 5];
LL dis[MAXN * 2 + 5];

inline void initCactus( const int u ) {
	dfn[u] = ++dfc;
	for ( int i = 1; fa[u][i - 1]; fa[u][i] = fa[fa[u][i - 1]][i - 1], ++i );
	adj( cac, u, v ) {
		dep[v] = dep[u] + 1, dis[v] = dis[u] + cac.len[e], fa[v][0] = u;
		initCactus( v );
	}
}

inline int lca( int u, int v ) {
	if ( dep[u] < dep[v] ) u ^= v ^= u ^= v;
	per ( i, MAXLG, 0 ) if ( dep[fa[u][i]] >= dep[v] ) u = fa[u][i];
	if ( u == v ) return u;
	per ( i, MAXLG, 0 ) if ( fa[u][i] != fa[v][i] ) u = fa[u][i], v = fa[v][i];
	return fa[u][0];
}

inline int climb( int u, const int par ) {
	per ( i, MAXLG, 0 ) if ( dep[fa[u][i]] > dep[par] ) u = fa[u][i];
	return u;
}

Graph<MAXN * 2 + 5, MAXN * 2 + 5> virc;

inline void vlink( int s, int t ) {
	// s is t's ancestor in cactus tree.
	if ( s > n ) {
		int is = climb( t, s );
		virc( s, is, dis[is] - dis[s] ), s = is;
	}
	if ( t > n ) {
		virc( fa[t][0], t, dis[t] - dis[fa[t][0]] ), t = fa[t][0];
	}
	if ( s != t ) virc( s, t, dis[t] - dis[s] );
}

inline void buildVirCac( std::vector<int>& vec ) {
	static int top, stk[MAXN * 2 + 5];
	virc.ecnt = 0, stk[top = 1] = 1;
	
	std::sort( vec.begin(), vec.end(), []( const int a, const int b ) {
		return dfn[a] < dfn[b];
	} );

	for ( int u: vec ) if ( u != 1 ) {
		int anc = lca( stk[top], u );
		while ( dep[stk[top]] > dep[anc] ) {
			int a = stk[top--], b = dep[stk[top]] < dep[anc] ? anc : stk[top];
			vlink( b, a );
		}
		if ( stk[top] != anc ) stk[++top] = anc;
		stk[++top] = u;
	}

	while ( top > 1 ) {
		int a = stk[top--], b = stk[top];
		vlink( b, a );
	}
}

LL ans, f[MAXN * 2 + 5];
bool book[MAXN + 5];

inline void contri( const int u, const std::vector<int>& cir ) {
	static int que[MAXN + 5], hd, tl;
	int sz = int( cir.size() ); LL half = pre[u] >> 1;
	
#define val( i ) ( pre[cir[i]] + ( i >= sz >> 1 ? pre[u] : 0 ) )

	que[hd = tl = 1] = 0;
	rep ( i, 1, sz - 1 ) {
		while ( hd <= tl && val( i ) - val( que[hd] ) > half ) ++hd;

		if ( hd <= tl ) {
			chkmax( ans, f[cir[que[hd]]] - val( que[hd] )
				+ f[cir[i]] + val( i ) );
		}

		while ( hd <= tl && f[cir[que[tl]]] - val( que[tl] )
			<= f[cir[i]] - val( i ) ) --tl;
		que[++tl] = i;
	}

#undef val
}

inline void solve( const int u, const int par ) {
	f[u] = -LINF;
	adj( virc, u, v ) solve( v, u );
	if ( u <= n ) {
		LL mx = book[u] ? 0 : -LINF, sx = -LINF;
		adj( virc, u, v ) {
			if ( LL d = f[v] + virc.len[e]; d > mx ) sx = mx, mx = d;
			else if ( d > sx ) sx = d;
		}
		chkmax( ans, mx + sx ), f[u] = mx;
	} else {
		static std::vector<int> cir; cir.clear();

		LL tmpp = pre[par]; pre[par] = 0;
		cir.push_back( par );
		adj( virc, u, v ) cir.push_back( v );
		int sz = int( cir.size() );
		cir.resize( sz << 1 );
		rep ( i, 0, sz - 1 ) cir[sz + i] = cir[i];

		contri( u, cir );
		pre[par] = tmpp;

		adj( virc, u, v ) chkmax( f[u], f[v] + virc.len[e] );
	}
	virc.head[u] = 0;
}

int main() {
	n = rint(), m = rint();
	rep ( i, 1, m ) {
		int u = rint(), v = rint(), w = rint();
		src( u, v, w ), src( v, u, w );
	}

	#ifdef RYBY
		puts( "+++ +++ +++" );
	#endif

	vnode = n, buildCactus( 1, 0 );
	dfc = 0, dep[1] = 1, initCactus( 1 );

	#ifdef RYBY
		puts( "--- --- ---" );
	#endif

	std::vector<int> vec;
	for ( int q = rint(), k; q--; vec.clear() ) {
		k = rint(), vec.resize( k );
		rep ( i, 0, k - 1 ) book[vec[i] = rint()] = true;
		buildVirCac( vec );
		
		ans = 0, solve( 1, 0 );
		wint( ans ), putchar( '
' );

		for ( int u: vec ) book[u] = false;
	}

	return 0;
}