AtCoder Beginner Contest 133

目录

• Contest Info
• Solutions
  • A. T or T
  • B.Good Distance
  • C. Remainder Minimization 2019
  • D. Rain Flows into Dams
  • E. Virus Tree 2
  • F. Colorful Tree

Contest Info

---

[Practice Link](https://atcoder.jp/contests/abc133/tasks)

Solved	A	B	C	D	E	F
6/6	O	O	O	O	O	O

• O 在比赛中通过
• Ø 赛后通过
• ! 尝试了但是失败了
• - 没有尝试

Solutions

---

A. T or T

#include <bits/stdc++.h>
using namespace std;

int main() {
	int n, a, b;
	while (cin >> n >> a >> b) {
		cout << min(n * a, b) << "\n";
	}
	return 0;
}

B.Good Distance

#include <bits/stdc++.h>
using namespace std;

#define N 110
int n, d, x[N][N];

int dis(int i, int j) {
	int res = 0;
	for (int o = 1; o <= d; ++o) {
		res += (x[i][o] - x[j][o]) * (x[i][o] - x[j][o]);
	}
	return res;
}

int main() {
	set <int> se; se.insert(0);
	for (int i = 1; i <= 30000; ++i) {
		se.insert(i * i);
	}
	while (scanf("%d%d", &n, &d) != EOF) {
		for (int i = 1; i <= n; ++i) {
			for (int j = 1; j <= d; ++j) {
				scanf("%d", x[i] + j);
			}
		}
		int res = 0;
		for (int i = 1; i <= n; ++i) {
			for (int j = 1; j < i; ++j) {
				int Dis = dis(i, j);
				if (se.count(Dis)) {
					++res;
				}
			}
		}
		printf("%d\n", res);
	}
	return 0;
}

C. Remainder Minimization 2019

#include <bits/stdc++.h>
using namespace std;

#define p 2019

int main() {
	int l, r;
	while (scanf("%d%d", &l, &r) != EOF) {
		if (r - l + 1 >= 3000) {
			puts("0");
			continue;
		} else {
			int res = 1e9;
			for (int i = l; i <= r; ++i) {
				for (int j = l; j < i; ++j) {
					res = min(res, (i % p) * (j % p) % p);
				}
			}
			printf("%d\n", res);
		}
	}
	return 0;
}

D. Rain Flows into Dams

题意：

给出两个序列\(a_i, b_i\)，知道\(a_i\)和\(b_i\)的关系如下：

\[a_i =
\begin{cases}
\frac{b_n}{2} + \frac{b_1}{2} &&i = n \\
\frac{b_i}{2} + \frac{b_{i + 1}}{2} && i \neq n
\end{cases}
\]

现在给出\(a_i\)，要求还原\(b_i\)。

思路：

\(n = 3\)时：

\[\begin{eqnarray}
a_1 = \frac{b_1}{2} + \frac{b_2}{2} \\
a_2 = \frac{b_2}{2} + \frac{b_3}{2} \\
a_3 = \frac{b_3}{2} + \frac{b_1}{2}
\end{eqnarray}
\]

我们发现\(b_1 = (1) - (2) + (3)\)

然后就可以还原\(b_i\)了。

#include <bits/stdc++.h>
using namespace std;

#define ll long long
#define N 100010
int n, a[N];
ll b[N];

int main() {
	while (scanf("%d", &n) != EOF) {
		for (int i = 1; i <= n; ++i) {
			scanf("%d", a + i);
		}
		b[1] = 0;
		for (int i = 1; i <= n; ++i) {
			b[1] += a[i] * ((i & 1) ? 1 : -1);
		}
		b[n] = 2ll * a[n] - b[1];
		for (int i = n - 1; i > 1; --i) {
			b[i] = 2ll * a[i] - b[i + 1];
		}
		for (int i = 1; i <= n; ++i) printf("%lld%c", b[i], " \n"[i == n]);
	}
	return 0;
}

E. Virus Tree 2

题意：

用\(K\)种颜色个一棵树染色，要求距离小于等于\(2\)的两个点的颜色不同。

求方案数。

思路：

如果距离小于等于\(1\)的两个点的颜色不同怎么求?

考虑每个点跟父亲不同就好了，答案为\(k \cdot (k - 1)^{n - 1}\)

那距离小于等于\(2\)呢？

考虑跟父亲往上走的距离小于等于\(2\)的点的颜色，因为这个点任意两个点的距离也是小于等于\(2\)的，也就是说这个点集中的每个点的颜色都不同，假设大小为\(sze\)，那么当前点的选择总数为\(k - sze\)。

乘一乘就好了。

#include <bits/stdc++.h>
using namespace std;

#define ll long long
#define N 100010
const ll p = 1e9 + 7;
int n, k;
ll res;
vector <vector<int>> G;

void DFS(int u, int fa) {
	int sze = 1 + (u != 1);
	for (auto v : G[u]) if (v != fa) {
		res = res * max(0, k - sze) % p;
	    ++sze;
		DFS(v, u);
	}
}

int main() {
	while (scanf("%d%d", &n, &k) != EOF) {
		G.clear(); G.resize(n + 1);
		for (int i = 1, u, v; i < n; ++i) {
			scanf("%d%d", &u, &v);
			G[u].push_back(v);
			G[v].push_back(u);
		}
		res = k;
		DFS(1, 1);
		printf("%lld\n", res);
	}
	return 0;
}

F. Colorful Tree

题意：

有一棵树，每条边有颜色和边权。

每次询问要求将所有颜色为\(x_i\)的边的边权都改成\(y_i\)，然后询问\(u_i \rightarrow v_i\)的距离。

思路：

将询问拆成原来的距离 - 路径上颜色为\(x_i\)的边的边权和 + 路径上颜色为\(x_i\)的边的条数 * \(y_i\)

然后离线即可。

其实不用写树剖和树状数组，只要将询问拆成三个询问丢在点那里，\(DFS\)下去的时候分别维护一下三个信息就好了。

#include <bits/stdc++.h>
using namespace std;

#define N 100010
#define pii pair <int, int>
#define fi first
#define se second
int n, q;
struct node {
	int u, v, w, id;
	node() {}
	node(int v, int w) : v(v), w(w) {}
	node (int u, int v, int w, int id = 0) : u(u), v(v), w(w), id(id) {}
};
vector <vector<node>> E, Q, G;
int res[N];  

int fa[N], deep[N], dis[N], sze[N], son[N], top[N], in[N], cnt;
void DFS(int u) {
	sze[u] = 1;
	for (auto it : G[u]) if (it.v != fa[u]) {
		int v = it.v;
		fa[v] = u;
		deep[v] = deep[u] + 1;
		dis[v] = dis[u] + it.w;
		DFS(v);
		sze[u] += sze[v];
		if (son[u] == -1 || sze[v] > sze[son[u]]) son[u] = v;
	}
}
void gettop(int u, int sp) {
	top[u] = sp;
	in[u] = ++cnt;
	if (son[u] == -1) return;
	gettop(son[u], sp);
	for (auto it : G[u]) {
		int v = it.v;
		if (v == fa[u] || v == son[u]) continue;
		gettop(v, v);
	}
}
int querylca(int u, int v) {
	while (top[u] != top[v]) {
		if (deep[top[u]] < deep[top[v]]) {
			swap(u, v);
		}
		u = fa[top[u]];
	}
	if (deep[u] > deep[v]) swap(u, v);
	return u;
}

pii add(pii x, pii y) {
	return pii(x.fi + y.fi, x.se + y.se);
}
pii sub(pii x, pii y) {
	return pii(x.fi - y.fi, x.se - y.se);
}
struct BIT {
	pii a[N];
	void init() {
		memset(a, 0, sizeof a);
	}
	void update(int x, int s, int t) {
		for (; x < N; x += x & -x) {
			a[x] = add(a[x], pii(s, t));
		}
	}
	pii query(int x) {
		pii res = pii(0, 0);
		for (; x > 0; x -= x & -x) {
			res = add(res, a[x]);
		}
		return res;
	}
	pii query(int l, int r) {
		return sub(query(r), query(l - 1));
	}
}bit;

pii query(int u, int v) {
	pii res = pii(0, 0);
	while (top[u] != top[v]) {
		if (deep[top[u]] < deep[top[v]]) swap(u, v);
		res = add(res, bit.query(in[top[u]], in[u]));
		u = fa[top[u]];
	}
	if (u == v) return res;
	if (deep[u] > deep[v]) swap(u, v);
	return add(res, bit.query(in[son[u]], in[v]));
}

void init() {
	bit.init();
	cnt = 0; dis[1] = 0; fa[1] = 0;
	E.clear(); E.resize(n + 1);
	Q.clear(); Q.resize(n + 1);
	G.clear(); G.resize(n + 1);
	memset(son, -1, sizeof son);
	memset(res, 0, sizeof res);
}
int main() {
	while (scanf("%d%d", &n, &q) != EOF) {
		init();
		for (int i = 1, u, v, c, d; i < n; ++i) {
			scanf("%d%d%d%d", &u, &v, &c, &d);
			G[u].push_back(node(v, d));
			G[v].push_back(node(u, d));
			E[c].push_back(node(u, v, d));
		}
		for (int i = 1, c, u, v, w; i <= q; ++i) {
			scanf("%d%d%d%d", &c, &w, &u, &v);
			Q[c].push_back(node(u, v, w, i));
		}
		DFS(1); gettop(1, 1);
	//	for (int i = 1; i <= n; ++i) printf("%d %d %d %d\n", i, fa[i], son[i], in[i]);
		for (int i = 1; i < n; ++i) {
			if (Q[i].empty()) continue;
			for (auto &it : E[i]) {
				if (fa[it.u] == it.v) swap(it.u, it.v);
				bit.update(in[it.v], 1, it.w);
			}
			for (auto it : Q[i]) {
				int u = it.u, v = it.v, w = it.w, id = it.id;
				int lca = querylca(u, v);
			//	cout << u << " " << v << " " << lca << endl;
				pii tmp = query(u, v);
				res[id] = dis[u] + dis[v] - 2 * dis[lca] - tmp.se + w * tmp.fi;
			}
			for (auto it : E[i]) {
				bit.update(in[it.v], -1, -it.w);
			}
		}
		for (int i = 1; i <= q; ++i) printf("%d\n", res[i]);
	}
	return 0;
}