AtCoder Beginner Contest 334

C. Socks 2

• 若 $2\times n-k$ 为偶数，那么直接从小到大一对一对选即可。

• 若 $2\times n-k$ 为奇数，则必定剩下一只。考虑不好知道到底剩下哪一只，那么直接暴力枚举剩第 $i$ 只，则 $1\sim i-1$ 和 $i+1\sim n$ 的袜子搭配起来，可以用前缀和后缀和预处理。

E. Christmas Color Grid 1

对于每一个格子，考虑 . 变成 # 会增加什么贡献，其实就会使周围本来 互相不连通的连通块 变得连通。

然后就看周围四个格子的变化量，不太好说，看代码吧。

注：期望只是虚晃一枪，骗人的。

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

#define int long long
const int N = 4e6 + 5, M = 1005, mod = 998244353;
int dir[4][2] = {-1, 0, 0, -1, 1, 0, 0, 1};

int n, m;
char a[M][M];
int p[N], p2[N];

int find(int x)  {
	if (p[x] != x) p[x] = find(p[x]);
	return p[x];
}

void merge(int x, int y) {
	int fx = find(x), fy = find(y);
	if (fx != fy) p[fx] = fy;
}

int find2(int x)  {
	if (p2[x] != x) p2[x] = find2(p2[x]);
	return p2[x];
}

void merge2(int x, int y) {
	int fx = find2(x), fy = find2(y);
	if (fx != fy) p2[fx] = fy;
}


int get(int x, int y) {
	return (x - 1) * m + y;
}

int qpow(int a, int k, int p ){
	int res = 1;
	while(k) {
		if (k & 1) res = res * a % p;
		a = a * a % p;
		k >>= 1;
	}
	return res;
}
map<pair<int, int>, int> vis;
map<int, int> vs, vs2;
int res, ss;

signed main() {
	cin >> n >> m;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) cin >> a[i][j];
	}
	for (int i = 0; i <= N - 5; i++) p[i] = i;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			for (int k = 0; k < 4; k++) {
				int dx = i + dir[k][0], dy = j + dir[k][1];
				if (dx < 1 || dx > n || dy < 1 || dy > m) continue;
				if (a[dx][dy] == a[i][j] && a[i][j] == '#') merge(get(dx, dy), get(i, j)); 
			}
		}
	}
	int qq = 0;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			int t = get(i, j);
			if (a[i][j]=='#'&&find(t) == t) qq++; .// 算出原来连通块数量
		}
	}
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			if (a[i][j] == '#') continue;
			vis.clear();
			vs.clear();
			vs2.clear();
			ss++;
			int idx = 0, vvv = 0;
			vs[get(i,j)] = ++idx;
			for (int k = 0; k < 4; k++) {
				int dx = i + dir[k][0], dy = j + dir[k][1];
				if (dx < 1 || dx > n || dy < 1 || dy > m) continue;
				if (a[dx][dy] == '#') vs[get(dx, dy)] = ++idx;
				if (a[dx][dy] == '#' && !vs2[find(get(dx, dy))]) vvv++, vs2[find(get(dx, dy))] = 1;
			}  // vs是给"自己和周围4个格子"编号，vvv统计原来周围4个格子的连通块数量
			
			for (int k= 1; k <= idx; k++) p2[k] = k;
			for (int k = 0; k < 4; k++) {
				int dx = i + dir[k][0], dy = j + dir[k][1];
				if (dx < 1 || dx > n || dy < 1 || dy > m) continue;
				if (x < 1 || x > n || y < 1 || y > m) continue;
				if (a[dx][dy] == '#') {
					merge2(vs[get(i, j)], vs[get(dx, dy)]); 
				}
			}
			int now = 0;
			for (int k = 1; k <= idx; k++) {
				if (find2(k) == k) now++; // 统计 .变#之后的周围4个格子的连通块数量
			}
			res += qq - (vvv - now) ; // vvv-now就是局部变化量
			res%=mod; // 这里可能要取模
		}
	}
	cout << res * qpow(ss, mod - 2, mod) % mod<<endl;
}

F. Christmas Present 2

典中典。$dp_i$ 表示前 $i$ 个走完的最小花费。

枚举 $j$，$dp_i=\min\{dp_j+d(j+1)+d(i)+sum(j+1\to i)\}(i-j\leq k)$

$d(i)$ 表示从家走到第 $i$ 个点的距离，$sum(i\to j)$ 表示路程上的距离。

单调队列模板题。

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

#define int long long
const int N = 3e5 + 5;

int n, c, q[N];
double dp[N], sx, sy, s[N], sum[N], d[N];

struct edge {
	double x, y;
}ed[N];

double dist(double a, double b, double c, double d) {
	return sqrt((a - c) * (a - c) + (b - d) * (b - d));
}

signed main() {
	cin >> n >> c >> sx >> sy;
	for (int i = 1; i <= n; i++) cin >> ed[i].x >> ed[i].y;
	for (int i = 1; i <= n; i++) dp[i] = 2e18;
	dp[0] = 0;
	ed[0].x = sx, ed[0].y = sy;
	for (int i = 1; i <= n; i++) {
		sum[i] = sum[i - 1] + dist(ed[i - 1].x, ed[i - 1].y, ed[i].x, ed[i].y);
		d[i] = dist(sx, sy, ed[i].x, ed[i].y);
	} 
	int hd = 1, tl = 1;
	q[1] = 0;
	for (int i = 1; i <= n; i++) {
		while (hd <= tl && i - q[hd] > c) hd++;
		if (hd <= tl) dp[i] = dp[q[hd]] + d[i] + d[q[hd] + 1] + sum[i] - sum[q[hd] + 1];
		while (hd <= tl && dp[q[tl]] + d[q[tl] + 1] - sum[q[tl] + 1] > dp[i] + d[i + 1] - sum[i + 1]) tl--;
		q[++tl] = i;
	}
	printf("%.10lf", dp[n]);
}

G. Christmas Color Grid 2

把一个 # 变成 . 有啥变化？

不太好考虑，考虑对原数组建图（相邻的 # 连边）。

一个 # 变成 . 等价于删掉这个点和与它相连的所有边。等价于割点。

若一个割点被 $x$ 个点双分量所包含，那么删掉他就会增加 $x-1$ 个连通分量。

注意特判孤立点的情况。

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

#define int long long
const int N = 4e6 + 5, M = 1005, mod = 998244353;
int dir[4][2] = {-1, 0, 0, -1, 1, 0, 0, 1};
vector<int> G[N];

int n, m, low[N], dfn[N], cut[N], idx, sv[N];
stack<int> stk;
char a[M][M];
int p[N], cc[N];

int find(int x)  {
	if (p[x] != x) p[x] = find(p[x]);
	return p[x];
}

void merge(int x, int y) {
	int fx = find(x), fy = find(y);
	if (fx != fy) p[fx] = fy;
}

int get(int x, int y) {
	return (x - 1) * m + y;
}

int qpow(int a, int k, int p ){
	int res = 1;
	while(k) {
		if (k & 1) res = res * a % p;
		a = a * a % p;
		k >>= 1;
	}
	return res;
}

void tarjan(int root, int u) {
	sv[u] = root;
	dfn[u] = low[u] = ++idx;
	stk.push(u);
	if (u == root && G[u].size() == 0) {
		cc[u]++;
		return;
	}
	
	int cnt = 0;
	for (auto v : G[u]) {
		if (!dfn[v]) {
			tarjan(root, v);
			low[u] = min(low[u], low[v]);
			if (low[v] >= dfn[u]) {
				cnt++;
				if (u != root || cnt > 1) cut[u] = 1;
				int y;	
				do {
					y = stk.top(); stk.pop();
					cc[y]++;
				} while (v != y);
				cc[u]++;
			}
		}
		else low[u] = min(low[u], dfn[v]);
	}
} 

signed main() {
	cin >> n >> m;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) cin >> a[i][j];
	}
	for (int i = 0; i <= N - 5; i++) p[i] = i;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			for (int k = 0; k < 4; k++) {
				int dx = i + dir[k][0], dy = j + dir[k][1];
				if (dx < 1 || dx > n || dy < 1 || dy > m) continue;
				if (a[dx][dy] == a[i][j] && a[i][j] == '#') G[get(dx, dy)].push_back(get(i, j)), G[get(i, j)].push_back(get(dx, dy));
			}
		}
	}
	int qq = 0;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			if (!dfn[get(i, j)] && a[i][j] == '#') tarjan(get(i, j), get(i, j)), qq++;
		}
		
	}
	int cnt = 0, ss = 0;
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
		 
			if (a[i][j] == '#') {
				ss++;
				if (cut[get(i, j)]) cnt += qq + cc[get(i, j)] - 1;
				else {
					if (G[get(i, j)].size() == 0 && sv[get(i,j)] == get(i, j)) cnt += qq - 1;
					else cnt += qq;
				}
				cnt %= mod;
			}
		} 
	}
	cout << cnt * qpow(ss, mod - 2, mod) % mod << endl;
}