D. Denouncing Mafia
给定一颗树，然后给定\(k\)个起点，对于每个起点来说，从该点到根节点的一条链都会被染色，求最多有几个点会被染色
\(3 \leq n \leq 1e5, 1 \leq k \leq n\)
题解
- 我们贪心的来看，起点一定会选择在叶子节点，假设叶子节点的数量为\(cnt\)，所以如果\(k \geq cnt\)，那么答案一定为\(n\)
- 否则我们可以长链剖分，将每条链剖出来后排序取前\(k\)条链即可
const int N = 1e5 + 10, M = 4e5 + 10;
int n, k;
vector<int> g[N];
int hson[N], dep[N], mx_dep[N], fa[N], sz[N];
int l[N], r[N], id[N], idx, top[N];
int tot, w[N];
void dfs1(int u, int par)
{
sz[u] = 1;
hson[u] = -1;
fa[u] = par;
mx_dep[u] = dep[u] = dep[par] + 1;
for (auto v : g[u])
{
if (v == par)
continue;
dfs1(v, u);
sz[u] += sz[v];
mx_dep[u] = max(mx_dep[u], mx_dep[v]);
if (hson[u] == -1 || mx_dep[v] > mx_dep[hson[u]])
hson[u] = v;
}
}
void dfs2(int u, int head)
{
top[u] = head;
l[u] = ++idx;
id[idx] = u;
if (hson[u] != -1)
dfs2(hson[u], head);
for (auto v : g[u])
{
if (v != fa[u] && v != hson[u])
dfs2(v, v);
}
r[u] = idx;
}
void solve()
{
cin >> n >> k;
for (int i = 2; i <= n; ++i)
{
int u;
cin >> u;
g[u].push_back(i);
g[i].push_back(u);
}
dfs1(1, 0);
dfs2(1, 1);
int cnt_son = 0;
for (int i = 1; i <= n; ++i)
{
if (sz[i] == 1)
{
cnt_son++;
w[++tot] = dep[i] - dep[top[i]] + 1;
}
}
if (k >= cnt_son)
{
cout << n << endl;
return;
}
sort(w + 1, w + tot + 1, greater<int>());
int ans = 0;
for (int i = 1; i <= k; ++i)
ans += w[i];
cout << ans << endl;
}
L. Less Coin Tosses
给定\(n, 2 \leq n \leq 1e18\)，将\(2^n\)个二进制字符串分别分配给两个人，使得两个人获胜的概率相同的情况下未被分配的字符串数量最少
题解
显然只要某一次分配的字符串中\(1\)的个数相同，那么两个人获胜的胜率能够始终相同
所以答案为\(\sum (C_n^i \& 1==1)\)
显然对于组合数的奇偶性，我们知道结论：若\((n \& k==k)\)，则\(C_n^k\)为奇数，否则为偶数
也就是说如果\(k\)的二进制是\(n\)的二进制的子集，那么答案贡献\(+1\)
那么显然答案为\(n\)的二进制中\(1\)个数为\(x\)，答案为\(2^x\)
const int N = 2e5 + 10, M = 4e5 + 10;
int n;
void solve()
{
cin >> n;
int x = __builtin_popcountll(n);
cout << (1ll << x) << endl;
}
M. Maratona Brasileira de Popcorn
题解
- 考虑二分答案
- \(check\)时检查需要几个人才能将所有的玉米吃完，然后与原有人数比较即可
const int N = 2e6 + 10;
int n,m,cnt;
int a[N];
bool check(int mid)
{
int w=1;
int x=mid*cnt;
for(int i=1;i<=n;)
{
if(a[i]-x<=0)
{
x-=a[i];
i++;
}
else
{
x=mid*cnt;
w++;
if(w>m) return false;
}
}
if(w>m ) return false;
else return true;
}
void solve()
{
cin>>n>>m>>cnt;
for(int i=1;i<=n;i++) cin>>a[i];
int l=0,r=2e15;
while(l<=r)
{
int mid=l+r>>1;
if(check(mid)) r=mid-1;
else l=mid+1;
}
cout<<l<<endl;
}
J. Jar of Water Game
题解
- 模拟即可
- 注意细节：
- 拿到鬼牌后不能立刻传递
- 一开始的时候有可能已经存在获胜的玩家
- 每次判断胜利条件在传递牌之后，判断所有队伍是否胜利
const int N = 1e2 + 10, M = 4e5 + 10;
int n, k;
map<char, int> mp[N];
string cmp = "A23456789DQJK";
void solve()
{
cin >> n >> k;
for (int i = 1; i <= n; ++i)
{
string s;
cin >> s;
for (auto ch : s)
mp[i][ch]++;
if (i == k)
mp[i]['W']++;
}
int now = k;
bool flag = true;
int cur = 4;
while (true)
{
int nxt = now + 1;
if (nxt > n)
nxt = 1;
if (mp[now]['W'] > 0 && flag == false)
mp[nxt]['W']++, mp[now]['W']--, flag = true;
else
{
if (mp[now]['W'] > 0 && flag == true)
flag = false;
int mi = INF;
char ch;
for (auto [x, y] : mp[now])
{
if (y == 0)
continue;
if (x == 'W')
continue;
if (y < mi)
{
mi = y;
ch = x;
}
else if (y == mi)
{
if (cmp.find(x) < cmp.find(ch))
ch = x;
}
}
mp[now][ch]--;
mp[nxt][ch]++;
}
int ans = -1;
char ans_ch;
for (int i = 1; i <= n; ++i)
{
int cnt = 0;
bool ok = false;
char c;
for (auto [x, y] : mp[i])
{
cnt += (y > 0);
if (y == 4)
{
ok = true;
c = x;
}
}
if (ok && cnt == 1)
{
if (ans == -1)
{
ans = i;
ans_ch = c;
}
else if (cmp.find(c) < cmp.find(ans_ch))
{
ans = i;
ans_ch = c;
}
}
}
if (ans != -1)
{
cout << ans << endl;
return;
}
now = nxt;
}
}
A. Artwork
题解
- \(O(n^2)\)将所有相交的监控用并查集合并，然后并查集中维护四种信息：是否与左墙接触\(left\)，\(right\)，\(up\)，\(down\)
- 然后判断是否有监控集合同时接触左墙和右墙，上墙和下墙，左墙和上墙，右墙和下墙之一即可
const int N = 1e4 + 10;
const int K = 1010;
int n, m, k, x[K], y[K], R[K];
bool f[K][5]; //up 1 down 2 left 3 right 4
const int UP = 1;
const int DOW = 2;
const int LE = 3;
const int RI = 4;
bool check(int i){
return ((f[i][UP] && f[i][LE]) || (f[i][UP] && f[i][DOW]) || (f[i][LE] && f[i][RI]) || (f[i][RI] && f[i][DOW]));
}
bool xiangjiao(int e1, int e2){
int dis=(x[e1]-x[e2])*(x[e1]-x[e2])+(y[e1]-y[e2])*(y[e1]-y[e2]);
int rr=(R[e1]+R[e2])*(R[e1]+R[e2]);
return dis<=rr;
}
int fa[K];
int find(int x){
return x == fa[x] ? x : fa[x] = find(fa[x]);
}
void merg(int a, int b){
a = find(a);
b = find(b);
if(a == b) return;
f[b][UP] |= f[a][UP];
f[b][DOW] |= f[a][DOW];
f[b][LE] |= f[a][LE];
f[b][RI] |= f[a][RI];
fa[a] = b;
}
void solve()
{
for(int i = 1;i < K;i++) fa[i] = i;
bool ans = false;
cin >> m >> n >> k;
for(int i = 1;i <= k;i++){
cin >> x[i] >> y[i] >> R[i];
if(x[i] - R[i] <= 0) f[i][UP] = true;
if(x[i] + R[i] >= m) f[i][DOW] = true;
if(y[i] - R[i] <= 0) f[i][LE] = true;
if(y[i] + R[i] >= n) f[i][RI] = true;
if(check(i)) ans = true;
}
if(ans){
cout << "N";
return;
}
for(int i = 1;i <= k;i++){
for(int j = i + 1;j <= k;j++){
if(xiangjiao(i, j)){
merg(i, j);
if(check(find(j))){
cout << "N";
return;
}
}
}
}
cout << "S";
}
F. Forests in Danger
给定\(n\)条平行于\(x\)轴或\(y\)轴的线段，每条线段能够覆盖的范围是一个矩阵，线段上每个点到矩阵边的最小距离都为\(r\)，给定一块矩阵区域\(p\)，请你确定一个最小的\(r\)，能够使得\(n\)条线段产生的矩阵与矩阵\(p\)的重合部分的面积至少是矩阵\(p\)面积的\(\%P\)
![image-20230910150632558]()
题解：二分答案 + 矩形面积并
- 显然二分答案
- \(check\)时我们可以求出每个矩阵与矩阵\(p\)的交，交集同样也是一个矩阵
- 然后对每一个交的矩阵求矩形面积并即可，最后和矩阵\(p\)的面积比较一下即可
- \(tips\)：利用线段树计算矩形的长时，需要对长度\(-1\)，因为\([2,5]\)的长度在计算矩形面积时应该为\(3\)，但是线段树中会返回\(4\)
const int N = 2e5 + 10, M = 4e5 + 10;
int n, P, x3, x4, y3, y4, m = 2e5 + 5;
struct Line
{
int x1, y1, x2, y2;
} line[N];
struct Mat : Line
{
} mat[N];
struct info
{
int mi, cnt;
friend info operator+(const info &a, const info &b)
{
info c;
if (a.mi == b.mi)
{
c.mi = a.mi;
c.cnt = a.cnt + b.cnt;
}
else if (a.mi < b.mi)
{
c.mi = a.mi;
c.cnt = a.cnt;
}
else
{
c.mi = b.mi;
c.cnt = b.cnt;
}
return c;
}
info(int mi = 0, int cnt = 0) : mi(mi), cnt(cnt) {}
};
struct SEG
{
int lazy;
info val;
} seg[N << 2];
void up(int id)
{
seg[id].val = seg[lson].val + seg[rson].val;
}
void settag(int id, int tag)
{
seg[id].val.mi += tag;
seg[id].lazy += tag;
}
void down(int id)
{
if (seg[id].lazy == 0)
return;
settag(lson, seg[id].lazy);
settag(rson, seg[id].lazy);
seg[id].lazy = 0;
}
void build(int id, int l, int r)
{
seg[id].lazy = 0;
if (l == r)
{
seg[id].val = info(0, 1);
return;
}
int mid = l + r >> 1;
build(lson, l, mid);
build(rson, mid + 1, r);
up(id);
}
void modify(int id, int l, int r, int ql, int qr, int val)
{
if (ql <= l && r <= qr)
{
settag(id, val);
return;
}
down(id);
int mid = l + r >> 1;
if (qr <= mid)
modify(lson, l, mid, ql, qr, val);
else if (ql > mid)
modify(rson, mid + 1, r, ql, qr, val);
else
{
modify(lson, l, mid, ql, qr, val);
modify(rson, mid + 1, r, ql, qr, val);
}
up(id);
}
bool check(int r)
{
build(1, 1, m);
vector<array<int, 4>> evt;
vector<int> vec;
for (int i = 1; i <= n; ++i)
{
auto [x1, y1, x2, y2] = line[i];
mat[i] = {x1 - r, y1 - r, x2 + r, y2 + r};
}
for (int i = 1; i <= n; ++i)
{
auto [x1, y1, x2, y2] = mat[i];
int lx = max(x1, x3), ly = max(y1, y3);
int rx = min(x2, x4), ry = min(y2, y4);
if (lx < rx && ly < ry)
{
// 别忘记右端点 -1
evt.push_back({ly, -1, lx, rx - 1}); // 入边
evt.push_back({ry, 1, lx, rx - 1}); // 出边
}
}
sort(all(evt));
int ans = 0;
int sum = seg[1].val.cnt;
int preY = 0;
for (auto [y, op, l, r] : evt)
{
int len = sum;
if (seg[1].val.mi == 0)
len -= seg[1].val.cnt;
ans += (y - preY) * len;
if (op == -1)
modify(1, 1, m, l, r, 1);
else
modify(1, 1, m, l, r, -1);
preY = y;
}
int area = (x4 - x3) * (y4 - y3);
return ans * 100 >= area * P;
}
void solve()
{
cin >> n;
for (int i = 1; i <= n; ++i)
{
int x1, y1, x2, y2;
cin >> x1 >> y1 >> x2 >> y2;
line[i] = {x1 + 1, y1 + 1, x2 + 1, y2 + 1};
}
cin >> P;
cin >> x3 >> y3 >> x4 >> y4;
x3++, y3++, x4++, y4++;
int l = 0, r = 1e9;
while (l <= r)
{
int mid = l + r >> 1;
if (check(mid))
r = mid - 1;
else
l = mid + 1;
}
cout << l << endl;
}
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