T1：[GXOI/GZOI2019]旧词

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solution

考虑$k=1$的情况
由于dep[lca(x,y)]=∣{z,zdep[lca(x,y)]=|\{z,zdep[lca(x,y)]=∣{z,z是$x$和$y$的祖先$\}|$
按$x$从小到大离线处理询问，每当$x$向右一个时，将$x$的祖先权值都$+1$
然后询问就相当于求$y$的所有祖先的权值之和
非常简单的板树链剖分$+$线段树

如果$k>1$，那么权值不是加$1$，而是对于深度为$i$的点，权值加上$i^k-(i-1{)}^k$
本质是树上差分，发现这样后从根到$x$的距离刚好是$dep[x{]}^k$
同样用线段树维护

code

#include <cstdio>
#include <vector>
#include <algorithm>
using namespace std;
#define int long long
#define maxn 50005
#define mod 998244353
struct node {int x, y, id;
}q[maxn];
vector < int > G[maxn];
int n, Q, k, cnt;
int ans[maxn], mi[maxn];
int son[maxn], siz[maxn], top[maxn], dep[maxn], id[maxn], f[maxn], rnk[maxn];
int val[maxn << 2], t[maxn << 2], flag[maxn << 2];bool cmp( node s, node t ) {return s.x < t.x;
}int qkpow( int x, int y ) {int res = 1;while( y ) {if( y & 1 ) res = res * x % mod;x = x * x % mod;y >>= 1;}return res;
}void dfs1( int u ) {siz[u] = 1, dep[u] = dep[f[u]] + 1;for( int i = 0;i < G[u].size();i ++ ) {int v = G[u][i];dfs1( v );siz[u] += siz[v];if( ! son[u] || siz[v] > siz[son[u]] )son[u] = v;}
}void dfs2( int u, int t ) {top[u] = t, id[u] = ++ cnt, rnk[cnt] = u;if( ! son[u] ) return;dfs2( son[u], t );for( int i = 0;i < G[u].size();i ++ ) {int v = G[u][i];if( v == son[u] ) continue;dfs2( v, v );}
}void pushdown( int num ) {if( ! flag[num] ) return;t[num << 1] = ( t[num << 1] + flag[num] * val[num << 1] % mod ) % mod;t[num << 1 | 1] = ( t[num << 1 | 1] + flag[num] * val[num << 1 | 1] % mod ) % mod; flag[num << 1] += flag[num];flag[num << 1 | 1] += flag[num];flag[num] = 0;
}void modify( int num, int l, int r, int L, int R ) {if( L <= l && r <= R ) {t[num] = ( t[num] + val[num] ) % mod;flag[num] ++;return;}pushdown( num );int mid = ( l + r ) >> 1;if( L <= mid ) modify( num << 1, l, mid, L, R );if( mid < R ) modify( num << 1 | 1, mid + 1, r, L, R );t[num] = ( t[num << 1] + t[num << 1 | 1] ) % mod;
}void change( int x ) {while( top[x] ) {modify( 1, 1, cnt, id[top[x]], id[x] );x = f[top[x]];}
}int query( int num, int l, int r, int L, int R ) {if( R < l || r < L ) return 0;if( L <= l && r <= R ) return t[num];pushdown( num );int mid = ( l + r ) >> 1;return ( query( num << 1, l, mid, L, R ) + query( num << 1 | 1, mid + 1, r, L, R ) ) % mod;
}int solve( int x ) {int sum = 0;while( top[x] ) {sum = ( sum + query( 1, 1, cnt, id[top[x]], id[x] ) ) % mod;x = f[top[x]];}return sum;
}void build( int num, int l, int r ) {if( l == r ) {val[num] = ( mi[dep[rnk[l]]] - mi[dep[rnk[l]] - 1] + mod ) % mod;return;}int mid = ( l + r ) >> 1;build( num << 1, l, mid ), build( num << 1 | 1, mid + 1, r );val[num] = ( val[num << 1] + val[num << 1 | 1] ) % mod;
}signed main() {scanf( "%lld %lld %lld", &n, &Q, &k );for( int i = 1;i <= n;i ++ ) mi[i] = qkpow( i, k );for( int i = 2;i <= n;i ++ ) {scanf( "%lld", &f[i] );G[f[i]].push_back( i );}dfs1( 1 );dfs2( 1, 1 );build( 1, 1, cnt );	for( int i = 1;i <= Q;i ++ ) {scanf( "%lld %lld", &q[i].x, &q[i].y );q[i].id = i;}sort( q + 1, q + Q + 1, cmp );int last = 1;for( int i = 1;i <= Q;i ++ ) {while( last <= q[i].x )change( last ++ );ans[q[i].id] = solve( q[i].y );}for( int i = 1;i <= Q;i ++ )printf( "%lld\n", ans[i] );return 0;
}

T2：GRE Words Once More!

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solution

但这都不是我的做法，因为上面的口胡题解着实不是很会
所以我学习了网上的奇技淫巧
讲解很详细，不想重复了

code

#include <cstdio>
#include <vector>
#include <cstring>
#include <algorithm>
using namespace std;
#define maxn 100005
#define MAX 100000000 
struct node {int v, c;node() {}node( int V, int C ) {v = V, c = C;}bool operator < ( node &t ) const {return c < t.c;}
};
vector < node > G[maxn];
int T, n, m, Q, tot;
int s[maxn], ans[MAX + 5], dep[maxn], pre[maxn], word[maxn];void dfs( int u, int depth ) {if( tot >= MAX ) return;if( ~ word[u] ) {for( int i = 1;i <= word[u];i ++ ) {if( tot >= MAX ) return;ans[++ tot] = ans[pre[u] + i] - dep[u] + depth;}return;}else {pre[u] = tot;dep[u] = depth;if( s[u] ) ans[++ tot] = depth;for( int i = 0;i < G[u].size();i ++ ) {int v = G[u][i].v;dfs( v, depth + 1 );}word[u] = tot - pre[u];}
}int main() {scanf( "%d", &T );for( int Case = 1;Case <= T;Case ++ ) {scanf( "%d %d %d", &n, &m, &Q );for( int i = 1;i <= n;i ++ ) G[i].clear();memset( word, -1, sizeof( word ) );tot = 0;for( int i = 2;i <= n;i ++ )scanf( "%d", &s[i] );for( int i = 1, u, v, c;i <= m;i ++ ) {scanf( "%d %d %d", &u, &v, &c );G[u].push_back( node( v, c ) );}for( int i = 1;i <= n;i ++ )sort( G[i].begin(), G[i].end() );dfs( 1, 0 ); printf( "Case #%d:\n", Case );int k;while( Q -- ) {scanf( "%d", &k );if( k > tot ) printf( "-1\n" );else printf( "%d\n", ans[k] );}}return 0;
}

T3：Problem B. Harvest of Apples

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solution

一看就知道跟组合数挂钩，考虑莫队
$$ans=\sum _{i=0}^m{C}_n^i$$
只需要处理四种操作即可
$m++:ans+C_n^{m+1}$
$m--:ans-C_n^m$
$n++:ans=(ans<<1)-C_n^m$
$$\sum _{i=0}^m{C}_{n+1}^i=\sum _{i=0}^m(C_n^i+C_n^{i-1})=2\ast \sum _{i=0}^m{C}_n^i-C_n^m$$
$n--:ans=(ans+C_{n-1}^m)/2$

code

#include <cmath>
#include <cstdio>
#include <iostream>
#include <algorithm>
using namespace std;
#define maxn 100000
#define int long long 
#define mod 1000000007
struct node {int n, m, id;
}query[maxn + 5];
int T, sqt, result;
int ans[maxn + 5], fac[maxn + 5], inv[maxn + 5];int qkpow( int x, int y ) {int res = 1;while( y ) {if( y & 1 ) res = res * x % mod;x = x * x % mod;y >>= 1;}return res;
}bool cmp( node x, node y ) {return ( x.n / sqt == y.n / sqt ) ? x.m < y.m : x.n < y.n;
}int C( int n, int m ) {return fac[n] * inv[m] % mod * inv[n - m] % mod;
}signed main() {fac[0] = inv[0] = 1;for( int i = 1;i <= maxn;i ++ )fac[i] = fac[i - 1] * i % mod;inv[maxn] = qkpow( fac[maxn], mod - 2 ) % mod;for( int i = maxn - 1;i;i -- )inv[i] = inv[i + 1] * ( i + 1 ) % mod;scanf( "%lld", &T );for( int i = 1;i <= T;i ++ ) {scanf( "%lld %lld", &query[i].n, &query[i].m );query[i].id = i;sqt = max( sqt, ( int ) sqrt( query[i].n ) );}sort( query + 1, query + T + 1, cmp );int curn = 0, curm = -1;for( int i = 1;i <= T;i ++ ) {int n =	query[i].n, m = query[i].m;while( n < curn ) result = ( result + C( -- curn, curm ) ) % mod * inv[2] % mod;while( curn < n )result = ( ( result << 1 ) - C( curn ++, curm ) + mod ) % mod;while( curm < m )result = ( result + C( curn, ++ curm ) ) % mod;while( m < curm )result = ( result - C( curn, curm -- ) + mod ) % mod;ans[query[i].id] = result;}for( int i = 1;i <= T;i ++ )printf( "%lld\n", ans[i] );return 0;
}