九校联考-DL24凉心模拟Day2总结

T1 锻造 forging#

题目描述#

“欢迎啊,老朋友。”

一阵寒暄过后,厂长带他们参观了厂子四周,并给他们讲锻造的流程。

“我们这里的武器分成若干的等级,等级越高武器就越厉害,并且对每一等级的武器都有两种属性值 b 和 c,但是我们初始只能花$a$个金币来生产$1$把$0$级剑......”

“所以你们厂子怎么这么垃圾啊,不能一下子就造出来$999$级的武器吗?”勇者不耐烦的打断了厂长的话。
“别着急,还没开始讲锻造呢......那我们举例你手中有一把$x$级武器和一把$y$级武器$(y = max(x − 1, 0))$,我们令锻造附加值$k = min(c_x , b_y )$,则你有 $\frac{k}{c_x}$ 的概率将两把武器融合成一把$x + 1$级的武器。”
“......但是,锻造不是一帆风顺的,你同样有 $1−\frac{k}{c_x}$ 的概率将两把武器融合成一把$max(x − 1, 0)$ 级的武器......”

勇者听完后暗暗思忖,他知道厂长一定又想借此机会坑骗他的零花钱,于是求助这个村最聪明的智者——你,来告诉他,想要强化出一把$n$级的武器,其期望花费为多少?
由于勇者不精通高精度小数,所以你只需要将答案对$998244353$取模即可。

分析#

期望DP+线性逆元。
设$f[i]$表示打造成等级为$i$的武器的期望花费，那么考虑转移：

$$f[i]=f[i-1]+f[i-2]+P\times (f[i] - f[i - 2])$$

其中,$P=(1-\frac{k}{c_{i-1}})$，为打造失败的概率。

这个方程表示，若打造成功，那么花费即为$f[i-1]+f[i-2]$，若未成功，那么我们就有了一件等级为$i-2$的武器，所以格外的花费就应该是$f[i]-f[i-2]$，然后整理可得：

$$f[i]=\frac{c_{i-1}}{k} \times f[i-1]+f[i-2]$$

然后再加个线性求逆元就行了。

时间复杂度$O(n)$。

Copy
#include<cstdio>
#include<cstdlib>
#define ll long long
#define Re register
const int N = 1e7 + 5;
const int P = 998244353;
inline int read() {
	int f = 1, x = 0; char ch;
	do { ch = getchar(); if (ch == '-') f = -1; } while (ch < '0' || ch > '9');
	do {x = (x << 3) + (x << 1) + ch - '0'; ch = getchar(); } while (ch >= '0' && ch <= '9'); 
	return f * x;
}
inline int min(int a, int b) { return a < b ? a : b; }
inline void hand_in() {
	freopen("forging.in", "r", stdin);
	freopen("forging.out", "w", stdout); 
}
int n, a, bx, by, cx, cy, p, b[N], c[N], inv[N], f[N];
int main() {
	hand_in();
	inv[0] = inv[1] = 1;
	for (Re int i = 2;i <= 10000000; ++i) {
		inv[i] = (ll)(P - (P / i)) * (ll)inv[P % i] % P;
	}
	n = read(), a = read(), bx = read(), by = read(), cx = read(), cy = read(), p = read();
	b[0] = by + 1, c[0] = cy + 1;
	for (Re int i = 1;i < n; ++i) {
		b[i] = ((ll)b[i - 1] * bx + by) % p + 1;
		c[i] = ((ll)c[i - 1] * cx + cy) % p + 1;
	}
	f[0] = a, f[1] = ((ll)(((ll)c[0] * inv[min(c[0], b[0])]) % P + 1) * f[0]) % P;
	for (int i = 2;i <= n; ++i) {
		f[i] =((((ll)c[i - 1] * inv[min(c[i - 1], b[i - 2])]) % P * f[i - 1]) % P + f[i - 2]) % P;
	}
	printf("%d\n", f[n]);
	return 0;
}

T2 整除 division#

题目大意#

求解$x^m\equiv x (mod p_1\times p_2\times \dots \times p_c)$在$[1, p_1\times p_2\times \dots \times p_c]$区间取值间的个数。

分析#

懒得写了，这里很清楚

Copy
#include<queue>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<algorithm>
#define ll long long
#define Re register
const int N = 1e5 + 5;
const int P = 998244353;
inline int read() {
	int f = 1, x = 0; char ch;
	do { ch = getchar(); if (ch == '-') f = -1; } while (ch < '0' || ch > '9');
	do {x = (x << 3) + (x << 1) + ch - '0'; ch = getchar(); } while (ch >= '0' && ch <= '9'); 
	return f * x;
}

inline int min(int a, int b) { return a < b ? a : b; }

inline int max(int a, int b) { return a < b ? b : a; }

inline void swap(int &a, int &b) { a ^= b ^= a ^= b; }

inline void hand_in() {
	freopen("division.in", "r", stdin);
	freopen("division.out", "w", stdout); 
}

inline ll mi(ll a, ll b, ll p) {
	ll ret = 1;
	while (b) {
		if (b & 1) ret *= a, ret %= p;
		a *= a, a %= p;
		b >>= 1;
	}
	return ret;
}

int prim[N], vis[N], tot;
inline void Prim() {
	vis[1] = 1;
	for (int i = 2;i <= 10000; ++i) {
		if (!vis[i]) prim[++tot] = i;
		for (int j = 1;j <= tot; ++j) {
			if (prim[j] * i > 10000) break;
			vis[prim[j] * i] = 1;
			if (!(i % prim[j])) break;
		}
	}
}

int pw[N];

inline int calc(int m, int p) {
	pw[1] = 1, pw[p] = 0;
	for (int i = 2;i < p; ++i) {
		if (!vis[i]) pw[i] = mi(i, m, p);
		for (int j = 1;prim[j] <= i; ++j) {
			if (prim[j] * i > p) break;
			pw[prim[j] * i] = pw[prim[j]] * pw[i];
			if (!(i % prim[j])) break;
		}
	}
	int ret = 1;
	for (int i = 1;i < p; ++i) ret += (pw[i] == i);
	return ret;
}

inline ll gcd(ll a, ll b) {
	return b == 0 ? a : gcd(b, a % b);
}

int id, T, c, m, ans;

int main() {
	hand_in();
	id = read();
	T = read();
	Prim();
	while (T --) {
		c = read(), m = read();
		ans = 1;
		for (int i = 1, p;i <= c; ++i) {
			p = read();
//			ans = ((ll)ans * calc(m, p)) % P;
			ans = ((ll)ans * (gcd(m - 1, p - 1) + 1)) % P; 
		}
		printf("%d\n", ans);
	}
	return 0;
}

T3 欠钱 money#

分析#

将有向有根树改成无向无根树存下来,树上倍增+启发式合并,每次合并时暴力重构倍增数组,倍增数组多存一个到$2^i$的父节点的方向,全向上为$1$,全向下为$2$,两个都有为$3$,询问时判断两个点的方向关系就可以了。

时间复杂度$O(nlog^2n + mlogn)$。

Copy
#include<queue>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<algorithm>
#define ll long long
#define Re register
const int N = 1e5 + 5;
const int INF = 0x7fffffff;
const int BASE = 16;
inline int read() {
	int f = 1, x = 0; char ch;
	do { ch = getchar(); if (ch == '-') f = -1; } while (ch < '0' || ch > '9');
	do {x = (x << 3) + (x << 1) + ch - '0'; ch = getchar(); } while (ch >= '0' && ch <= '9'); 
	return f * x;
}

inline void write(int x) {
	if (x < 0) putchar('-'), x = -x;
	if (x > 9) write(x / 10);
	putchar(x % 10 + '0');
}

inline int min(int a, int b) { return a < b ? a : b; }

inline void swap(int &a, int &b) { a ^= b ^= a ^= b; }

inline void hand_in() {
	freopen("money.in", "r", stdin);
	freopen("money.out", "w", stdout); 
}

int n, m, last;

struct Graph {
	int to[N << 1], nxt[N << 1], w[N << 1], dir[N << 1], head[N], cnt;
	inline void add(int x, int y, int z, int d) {
		++cnt;
		to[cnt] = y, w[cnt] = z, dir[cnt] = d, nxt[cnt] = head[x], head[x] = cnt;
	}
}G;

int f[N], sz[N];
int anc[N][BASE + 1], mn[N][BASE + 1], dir[N][BASE + 1], dep[N], up[N];
inline void init() {for (Re int i = 1;i <= n; ++i) f[i] = i, sz[i] = 1, up[i] = 1; }

inline void dfs(int u, int fa, int rt) {
	f[u] = rt, dep[u] = dep[fa] + 1;
	for (Re int i = 1;i <= BASE; ++i) {
		anc[u][i] = anc[anc[u][i - 1]][i - 1];
		mn[u][i] = min(mn[u][i - 1], mn[anc[u][i - 1]][i - 1]);
		dir[u][i] = (dir[u][i - 1] | dir[anc[u][i - 1]][i - 1]);
	}
	for (Re int i = G.head[u], v;i;i = G.nxt[i]) {
		v = G.to[i];
		if (v == fa) continue;
		anc[v][0] = u;
		mn[v][0] = G.w[i];
		dir[v][0] = 3 ^ G.dir[i];
		dfs(v, u, rt);
	}
}

inline void merge(int u, int v, int w) {
	G.add(u, v, w, 1), G.add(v, u, w, 2);
	int dirs = sz[f[u]] < sz[f[v]] ? 1 : 2;
	if (dirs == 2) swap(u, v);
	anc[u][0] = v, mn[u][0] = w, dir[u][0] = dirs;
	sz[f[v]] += sz[f[u]];
	dfs(u, v, f[v]);
}

inline int ask(int u, int v) {
	if (f[u] != f[v]) return 0;
	int dirs = 0, res = INF;
	if (dep[u] < dep[v]) swap(u, v), dirs = 3;
	for (int i = BASE; i >= 0; --i) {
		if (dep[u] - (1 << i) >= dep[v]) {
			if (dir[u][i] != (1 ^ dirs)) return 0;
			res = min(res, mn[u][i]);
			u = anc[u][i];
		}
	}
	if (u == v) return res;
	for (int i = BASE;i >= 0; --i) {
		if (anc[u][i] != anc[v][i]) {
			if (dir[u][i] != (1 ^ dirs) || dir[v][i] != (2 ^ dirs)) return 0;
			res = min(res, min(mn[u][i], mn[v][i]));
			u = anc[u][i], v = anc[v][i];
		}
	}
	if (dir[u][0] != (1 ^ dirs) || dir[v][0] != (2 ^ dirs)) return 0;
	res = min(res, min(mn[u][0], mn[v][0]));
	return res;
}

int main() {
	hand_in();
	n = read(), m = read(), init();
	for (Re int i = 1, op, a, b, c;i <= m; ++i) {
		op = read(), a = read(), b = read();
		a = (a + last) % n + 1;
		b = (b + last) % n + 1;
		if (!op) {
			c = read();
			c = (c + last) % n + 1;
			merge(a, b, c);
		}
		else {
			write(last = ask(a, b));
			puts("");
		}
	}
	return 0;
}