Thank you for takeing part in our contest, we know leaves a lot to be desired. But we believe that you can find interesting among our problems.↵
↵
## [A. Nene's Game](https://codeforc.es/contest/1956/problem/A)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Tutorial">↵
Obviously, a person at place $p$ will be kicked out if and only if $p \ge a_1$. ↵
↵
Therefore, the answer is $\min(n,a_1-1)$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
int a[105];↵
void solve(){↵
int q,k,n;cin>>k>>q;↵
for(int i=1;i<=k;i++) cin>>a[i];↵
for(int i=1;i<=q;i++){↵
cin>>n;↵
cout<<min(a[1]-1,n)<<' ';↵
}↵
cout<<endl;↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve(); ↵
}↵
↵
```↵
</spoiler>↵
↵
## [B. Nene and the Card Game](https://codeforc.es/contest/1956/problem/B)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
The number of pairs on each side should be the same.↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
For each color (in the following text, "this point" refers to the point someone got by playing a card with this color):↵
↵
- If you have both cards of this color in your hand, you will be able to get this point.↵
↵
- If Nene has both cards, you will not be able to get this point.↵
↵
- If you have only one card, you cannot get this point when Nene is using the following strategy:↵
↵
>When you play one of your paired cards, Nene also plays one of her paired cards; Otherwise, Nene will have the card with the same color. She can play it and get this point. ↵
↵
Therefore, the answer will be the amount of pairs in your hand.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
const int MAXN=4e5+5;↵
int cnt[MAXN];↵
void solve() {↵
int n,ans=0;↵
scanf("%d",&n);↵
fill(cnt+1,cnt+n+1,0);↵
for(int i=1,a;i<=n;++i) scanf("%d",&a),ans+=(++cnt[a]==2);↵
printf("%d\n",ans);↵
}↵
signed main() {↵
int T;↵
scanf("%d",&T);↵
while(T--) solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
## [C. Nene's Magical Matrix](https://codeforc.es/contest/1956/problem/C)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
When $n=3$, the optimal matrix is the following:↵
↵
```↵
1 2 3↵
2 2 3↵
3 3 3↵
```↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
The optimal matrix would be:↵
↵
```↵
1 2 3 ... n↵
2 2 3 ... n↵
3 3 3 ... n↵
..... ... n↵
n n n n n n↵
```↵
↵
Construction method:↵
↵
```text↵
for i in [n, n-1, ..., 1]:↵
set the i'th column to [1, 2, ..., n];↵
set the i'th row to [1, 2, ..., n];↵
```↵
↵
This takes exactly $2n$ operations.↵
</spoiler>↵
↵
<spoiler summary="Proof">↵
For the final matrix, we define $f(x)$ as the number of elements greater or equal to $x$. The sum of all elements in the matrix is $\sum_{i=1}^n f(i)$ because an element with value $x$ will be counted $x$ times in the formula before.↵
↵
Now, we proof that $f(x) \le n^2-(x-1)^2$:↵
↵
Let's rewrite the problem to make it a little simpler:↵
↵
>You have an $n\times n$ matrix. In each operation, you can paint exactly $x-1$ cells white and $n-(x-1)$ cells black in a row or a column. Proof that there will be at most $n^2-(x-1)^2$ black cells.↵
↵
Try to strengthen the conclusion by stating that:↵
↵
> For any matrix of size $n\times m$, each operation can paint a row into $x$ white cells and $m-x$ black cells, or a column into $y$ white cells and $n-y$ black cells. No matter how we paint, the final matrix will have at most $nm-xy$ black cells.↵
↵
We will prove this by induction.↵
↵
If $x=n$ or $y=m$, the conclusion holds.↵
↵
Otherwise, if the last operation is to paint a row, then this row has exactly $m-x$ black cells. And, by induction, other rows will contain at most $(n-1)m-x(y-1)$ black cells. Painting a column in the last step is similar.↵
↵
Then, we have proven the conclusion above.↵
↵
Since the construction above maximizes each $f(x)$, it is the optimal answer.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
void solve(){↵
int n;cin>>n;↵
int ans=0;↵
for(int i=1;i<=n;i++) ans+=(2*i-1)*i;↵
cout<<ans<<' '<<2*n<<endl;↵
for(int i=n;i>=1;i--){↵
cout<<"1 "<<i<<' ';↵
for(int j=1;j<=n;j++) cout<<j<<' ';cout<<endl;↵
cout<<"2 "<<i<<' ';↵
for(int j=1;j<=n;j++) cout<<j<<' ';cout<<endl;↵
}↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve(); ↵
}↵
```↵
</spoiler>↵
↵
## [D. Nene and the Mex Operator](https://codeforc.es/contest/1956/problem/D)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
What is the answer when $a_i = 0$?↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
When $a_i =0$, the sum can hit $n^2$ with making $a_i = n$ at last. Construction:↵
↵
```↵
//! a function which sets a_1 ... a_k into k.↵
function solve(k):↵
if k is 1:↵
operate [1,1]↵
return↵
end if↵
solve(k-1);↵
for i in [k-2, ..., 1]:↵
operate [1,i] //! this sets a_1 ... a_i into 0↵
solve(i)↵
//! here, a should be [i, i, ..., i, i+1, i+2, ..., k-1, 0]↵
end for↵
//! here, a should be [1, 2, 3, ..., k-1, 0]↵
operate [1,k]↵
return↵
```↵
↵
Here, `solve(k)` will take about $2^k$ operations.↵
↵
Since doing operation $[l,r]$ will make $a_l,\cdots,a_r \le r-l+1$, if for all $l\le i\le r$, $a_i$ is included in at least one of the operations and $a_{l-1},a_{r+1}$ are not, the optimal strategy will be setting $a_i = r-l+1$ for $i\in[l,r]$ using the construction above.↵
↵
Finally, we can use DFS or DP to determine whether each element is included in operations.↵
↵
The number of operations used will not exceed $2^n$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
int n,a[20],cnt[20];↵
vector <array<int,2>> I;↵
void oper(int l,int r) {↵
fill(cnt,cnt+n+1,0);↵
for(int i=l;i<=r;++i) if(a[i]<=n) ++cnt[a[i]];↵
int mex=0;↵
while(cnt[mex]) ++mex;↵
for(int i=l;i<=r;++i) a[i]=mex;↵
I.push_back({l,r});↵
}↵
void build(int l,int r) {↵
if(l==r) {↵
if(a[l]) oper(l,r);↵
return ;↵
}↵
build(l,r-1);↵
if(a[r]!=r-l) oper(l,r),build(l,r-1);↵
}↵
void solve() {↵
scanf("%d",&n),I.clear(),memset(a,0,sizeof(a));↵
for(int i=0;i<n;++i) scanf("%d",&a[i]);↵
int cur=0,ans=0;↵
for(int s=0;s<(1<<n);++s) {↵
int tmp=0;↵
for(int l=0;l<n;++l) {↵
if(s&(1<<l)) {↵
int r=l;↵
while(r+1<n&&(s&(1<<(r+1)))) ++r;↵
tmp+=(r-l+1)*(r-l+1);↵
l=r;↵
} else tmp+=a[l];↵
}↵
if(tmp>ans) ans=tmp,cur=s;↵
}↵
for(int l=0;l<n;++l) if(cur&(1<<l)) {↵
int r=l;↵
while(r+1<n&&(cur&(1<<(r+1)))) ++r;↵
build(l,r),oper(l,r),l=r;↵
}↵
printf("%d %d\n",ans,(int)I.size());↵
for(auto i:I) printf("%d %d\n",i[0]+1,i[1]+1);↵
}↵
signed main() {↵
solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
## [E. Nene vs. Monsters](https://codeforc.es/contest/1956/problem/E2)↵
↵
idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint1">↵
If three consecutive monsters have energy level $0,x,y\ (x,y>0)$, the monster with energy lever $y$ will "die"(have energy level $0$) at last.↵
</spoiler>↵
↵
<spoiler summary="Hint2">↵
If four consecutive monsters have energy levels $0,x,y,z\ (x,y,z>0)$, what will happen to the monster $z$?↵
</spoiler>↵
↵
<spoiler summary="Hint3">↵
If four consecutive monsters have energy level $x,y,z,w\ (x,y,z,w>0)$, how many rounds of spells must be used to make at least one of these monsters die?↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
If four consecutive monsters have energy level $x,y,z,w\ (x,y,z,w>0)$ and they did not die after $t$ rounds of spells, then $y$ will receive at least $t$ points of damage, $z$ will receive at least $(t-1) +(t-2)+\cdots=O(t^2)$ of damage, and $w$ will receive at least $O(t^3)$ of damage.↵
↵
That is to say, let $V=\max_{i=1}^n a_i$, after $O(\sqrt[3]{V})$ rounds, at least one of $x,y,z,w$ will die.↵
↵
So, we can simulate the process by brute force until there are no four consecutive alive monsters, and then the problem is reduced to the one described in Hint 2.↵
↵
If four consecutive monster have energy level $0,x,y,z\ (x,y,z>0)$, $x$ will remain alive, $y$ will die at last and sending $D=(y-x)+(y-2x)+\cdots+(y\bmod x)$ damage to $z$ before that. Therefore, $z$ will remain alive if and only if $z>D$.↵
↵
The time complexity is $O(n\sqrt[3]{V})$.↵
↵
Bonus: Actually, it can be shown that after $O(\sqrt[k]{V})$ rounds, there will be no $k$ consecutive alive monsters. Making $k$ bigger than $3$ can further reduce the time complexity, but it will be harder to implement and optimize little on actual performance.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
const int MAXN=2e5+5;↵
int n,a[MAXN];bool b[MAXN];↵
bool check(){↵
a[n+1]=a[1],a[n+2]=a[2],a[n+3]=a[3];↵
for(int i=1;i<=n;i++)↵
if(a[i]&&a[i+1]&&a[i+2]&&a[i+3]) return true;↵
return false;↵
} ↵
void solve(){↵
cin>>n;↵
for(int i=1;i<=n;i++) cin>>a[i];↵
if(n==2){↵
while(a[1]&&a[2]){↵
a[2]=max(0,a[2]-a[1]);↵
a[1]=max(0,a[1]-a[2]);↵
}↵
b[1]=(a[1]>0);b[2]=(a[2]>0);↵
}else if(n==3){↵
while(a[1]&&a[2]&&a[3]){↵
a[2]=max(0,a[2]-a[1]);↵
a[3]=max(0,a[3]-a[2]);↵
a[1]=max(0,a[1]-a[3]);↵
}↵
b[1]=(!a[3]&&a[1]);↵
b[2]=(!a[1]&&a[2]);↵
b[3]=(!a[2]&&a[3]);↵
}else{↵
while(check()){↵
for(int i=1;i<=n;i++) a[i%n+1]=max(0,a[i%n+1]-a[i]);↵
}↵
for(int i=1;i<=n;i++) b[i]=false;↵
auto attack=[&](ll x,ll y){↵
ll k=x/y;↵
return (2*x-(k+1)*y)*k/2;↵
};↵
for(int p=1;p<=n;p++) ↵
if(a[p]&&a[p%n+1]) a[p%n+1]=max(0,a[p%n+1]-a[p]);↵
else break;↵
for(int i=1;i<=n;i++) if(!a[i]&&a[i%n+1]){↵
b[i%n+1]=true;↵
b[(i+2)%n+1]=(a[(i+2)%n+1]>attack(a[(i+1)%n+1],a[i%n+1]));↵
}↵
}↵
int cnt=0;↵
for(int i=0;i<=n;i++) if(b[i]) cnt++;↵
cout<<cnt<<endl;↵
for(int i=1;i<=n;i++) if(b[i]) cout<<i<<' ';↵
cout<<endl;↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve();↵
}↵
```↵
</spoiler>↵
↵
## [F. Nene and the Passing Game](https://codeforc.es/contest/1956/problem/F)↵
↵
idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
Two personople $i$ and $j\ (i<j)$ can pass the ball to each other directly if and only if $[i+l_i,i+r_i]\cap[j-r_j,j-l_j]\neq \varnothing$↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
According to the hint above, we can build the following graph:↵
↵
- There are $2n$ vertices in the graph. Vertice $i$ links to vertices $([n+i-r_i,n+i-l_1]\cup[n+i+l_i,n+i+r_i])\cap[n+1,2n]\cap Z$.↵
↵
That is, assuming vertices $1$ to $n$ are players, vertices $n+1$ to $2n$ are temporary spots, and player $i$ links to all the spots where his/her arm can reach.↵
↵
Then, the answer will be the number of connected components in this graph which contains at least one vertice with an index less than or equal to $n$.↵
↵
But there's still a little problemonwith the solution. For two players $i, j\ (i<j)$ satisfying $[i+l_i,i+r_i]\cap[j+l_i,j+r_i]\neq \varnothing$ (that is, both players reaching out their "right arm"s), they are incorrectly counted as connected.↵
↵
To solve that, we can delete all the vertices $n+x$ such that $\forall i,\ x\not\in[i-r_i,i-l_i]$ or $\forall i,\ x\not\in[i+l_i,i+r_i]$ (that is, nobody's left/right arm can reach $x$). Finding such $x$ can be done easily in $O(n)$.↵
↵
The last issue is, the graph contains $O(n^2)$ edges. but since we only care about connectivity, operation "link $x$ to $[y,z]$" can be changed to "link $x$ to $y$, and link $i$ to $i+1$ for all $i$ in $[y,z-1]$". After that and removing multiple edges, the number of edges is reducesd to $O(n)$.↵
↵
Finally, counting connected components in a graph can be easily done in $O(n)$, so the time complexity is $O(n)$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
const int MAXN=2e6+5;↵
int n,tot,le[MAXN],ri[MAXN];↵
int fa[MAXN<<1],s[MAXN],t[MAXN],pre[MAXN],suf[MAXN];↵
inline int find(int x){↵
while(x^fa[x]) x=fa[x]=fa[fa[x]];↵
return x;↵
}↵
void solve(){↵
cin>>n;↵
for(int i=1;i<=2*n+1;i++) fa[i]=i;↵
for(int i=0;i<=n+1;i++) s[i]=t[i]=pre[i]=suf[i]=0;↵
for(int i=1;i<=n;i++){↵
cin>>le[i]>>ri[i];↵
s[max(1,i-ri[i])]++;s[max(0,i-le[i])+1]--;↵
t[min(n+1,i+le[i])]++;t[min(n,i+ri[i])+1]--;↵
}↵
tot=n;↵
for(int i=1;i<=n;i++){↵
s[i]+=s[i-1],t[i]+=t[i-1];↵
if(s[i]&&t[i]) suf[i]=pre[i]=++tot;↵
}↵
suf[n+1]=0;↵
for(int i=1;i<=n;i++) pre[i]=(pre[i]?pre[i]:pre[i-1]);↵
for(int i=n;i>=1;i--) suf[i]=(suf[i]?suf[i]:suf[i+1]);↵
for(int i=1;i<=n;i++){↵
int l=max(1,i-ri[i]),r=max(0,i-le[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) for(int i=find(l);i<r;i=find(i)) fa[i]=i+1; ↵
}↵
l=min(n+1,i+le[i]),r=min(n,i+ri[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) for(int i=find(l);i<r;i=find(i)) fa[i]=i+1; ↵
}↵
}↵
for(int i=1;i<=n;i++){↵
int l=max(1,i-ri[i]),r=max(0,i-le[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) fa[find(i)]=find(l);↵
}↵
l=min(n+1,i+le[i]),r=min(n,i+ri[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) fa[find(i)]=find(l);↵
}↵
}↵
int ans=0;↵
for(int i=1;i<=tot;i++) if(fa[i]==i) ans++; ↵
cout<<ans<<'\n';↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
// freopen("Otomachi_Una.in","r",stdin);↵
// freopen("Otomachi_Una.out","w",stdout);↵
int _;cin>>_;↵
while(_--) solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
↵
## [A. Nene's Game](https://codeforc.es/contest/1956/problem/A)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Tutorial">↵
Obviously, a person at place $p$ will be kicked out if and only if $p \ge a_1$. ↵
↵
Therefore, the answer is $\min(n,a_1-1)$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
int a[105];↵
void solve(){↵
int q,k,n;cin>>k>>q;↵
for(int i=1;i<=k;i++) cin>>a[i];↵
for(int i=1;i<=q;i++){↵
cin>>n;↵
cout<<min(a[1]-1,n)<<' ';↵
}↵
cout<<endl;↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve(); ↵
}↵
↵
```↵
</spoiler>↵
↵
## [B. Nene and the Card Game](https://codeforc.es/contest/1956/problem/B)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
The number of pairs on each side should be the same.↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
For each color (in the following text, "this point" refers to the point someone got by playing a card with this color):↵
↵
- If you have both cards of this color in your hand, you will be able to get this point.↵
↵
- If Nene has both cards, you will not be able to get this point.↵
↵
- If you have only one card, you cannot get this point when Nene is using the following strategy:↵
↵
>When you play one of your paired cards, Nene also plays one of her paired cards; Otherwise, Nene will have the card with the same color. She can play it and get this point. ↵
↵
Therefore, the answer will be the amount of pairs in your hand.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
const int MAXN=4e5+5;↵
int cnt[MAXN];↵
void solve() {↵
int n,ans=0;↵
scanf("%d",&n);↵
fill(cnt+1,cnt+n+1,0);↵
for(int i=1,a;i<=n;++i) scanf("%d",&a),ans+=(++cnt[a]==2);↵
printf("%d\n",ans);↵
}↵
signed main() {↵
int T;↵
scanf("%d",&T);↵
while(T--) solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
## [C. Nene's Magical Matrix](https://codeforc.es/contest/1956/problem/C)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
When $n=3$, the optimal matrix is the following:↵
↵
```↵
1 2 3↵
2 2 3↵
3 3 3↵
```↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
The optimal matrix would be:↵
↵
```↵
1 2 3 ... n↵
2 2 3 ... n↵
3 3 3 ... n↵
..... ... n↵
n n n n n n↵
```↵
↵
Construction method:↵
↵
```text↵
for i in [n, n-1, ..., 1]:↵
set the i'th column to [1, 2, ..., n];↵
set the i'th row to [1, 2, ..., n];↵
```↵
↵
This takes exactly $2n$ operations.↵
</spoiler>↵
↵
<spoiler summary="Proof">↵
For the final matrix, we define $f(x)$ as the number of elements greater or equal to $x$. The sum of all elements in the matrix is $\sum_{i=1}^n f(i)$ because an element with value $x$ will be counted $x$ times in the formula before.↵
↵
Now, we proof that $f(x) \le n^2-(x-1)^2$:↵
↵
Let's rewrite the problem to make it a little simpler:↵
↵
>You have an $n\times n$ matrix. In each operation, you can paint exactly $x-1$ cells white and $n-(x-1)$ cells black in a row or a column. Proof that there will be at most $n^2-(x-1)^2$ black cells.↵
↵
Try to strengthen the conclusion by stating that:↵
↵
> For any matrix of size $n\times m$, each operation can paint a row into $x$ white cells and $m-x$ black cells, or a column into $y$ white cells and $n-y$ black cells. No matter how we paint, the final matrix will have at most $nm-xy$ black cells.↵
↵
We will prove this by induction.↵
↵
If $x=n$ or $y=m$, the conclusion holds.↵
↵
Otherwise, if the last operation is to paint a row, then this row has exactly $m-x$ black cells. And, by induction, other rows will contain at most $(n-1)m-x(y-1)$ black cells. Painting a column in the last step is similar.↵
↵
Then, we have proven the conclusion above.↵
↵
Since the construction above maximizes each $f(x)$, it is the optimal answer.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
void solve(){↵
int n;cin>>n;↵
int ans=0;↵
for(int i=1;i<=n;i++) ans+=(2*i-1)*i;↵
cout<<ans<<' '<<2*n<<endl;↵
for(int i=n;i>=1;i--){↵
cout<<"1 "<<i<<' ';↵
for(int j=1;j<=n;j++) cout<<j<<' ';cout<<endl;↵
cout<<"2 "<<i<<' ';↵
for(int j=1;j<=n;j++) cout<<j<<' ';cout<<endl;↵
}↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve(); ↵
}↵
```↵
</spoiler>↵
↵
## [D. Nene and the Mex Operator](https://codeforc.es/contest/1956/problem/D)↵
↵
Idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
What is the answer when $a_i = 0$?↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
When $a_i =0$, the sum can hit $n^2$ with making $a_i = n$ at last. Construction:↵
↵
```↵
//! a function which sets a_1 ... a_k into k.↵
function solve(k):↵
if k is 1:↵
operate [1,1]↵
return↵
end if↵
solve(k-1);↵
for i in [k-2, ..., 1]:↵
operate [1,i] //! this sets a_1 ... a_i into 0↵
solve(i)↵
//! here, a should be [i, i, ..., i, i+1, i+2, ..., k-1, 0]↵
end for↵
//! here, a should be [1, 2, 3, ..., k-1, 0]↵
operate [1,k]↵
return↵
```↵
↵
Here, `solve(k)` will take about $2^k$ operations.↵
↵
Since doing operation $[l,r]$ will make $a_l,\cdots,a_r \le r-l+1$, if for all $l\le i\le r$, $a_i$ is included in at least one of the operations and $a_{l-1},a_{r+1}$ are not, the optimal strategy will be setting $a_i = r-l+1$ for $i\in[l,r]$ using the construction above.↵
↵
Finally, we can use DFS or DP to determine whether each element is included in operations.↵
↵
The number of operations used will not exceed $2^n$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
int n,a[20],cnt[20];↵
vector <array<int,2>> I;↵
void oper(int l,int r) {↵
fill(cnt,cnt+n+1,0);↵
for(int i=l;i<=r;++i) if(a[i]<=n) ++cnt[a[i]];↵
int mex=0;↵
while(cnt[mex]) ++mex;↵
for(int i=l;i<=r;++i) a[i]=mex;↵
I.push_back({l,r});↵
}↵
void build(int l,int r) {↵
if(l==r) {↵
if(a[l]) oper(l,r);↵
return ;↵
}↵
build(l,r-1);↵
if(a[r]!=r-l) oper(l,r),build(l,r-1);↵
}↵
void solve() {↵
scanf("%d",&n),I.clear(),memset(a,0,sizeof(a));↵
for(int i=0;i<n;++i) scanf("%d",&a[i]);↵
int cur=0,ans=0;↵
for(int s=0;s<(1<<n);++s) {↵
int tmp=0;↵
for(int l=0;l<n;++l) {↵
if(s&(1<<l)) {↵
int r=l;↵
while(r+1<n&&(s&(1<<(r+1)))) ++r;↵
tmp+=(r-l+1)*(r-l+1);↵
l=r;↵
} else tmp+=a[l];↵
}↵
if(tmp>ans) ans=tmp,cur=s;↵
}↵
for(int l=0;l<n;++l) if(cur&(1<<l)) {↵
int r=l;↵
while(r+1<n&&(cur&(1<<(r+1)))) ++r;↵
build(l,r),oper(l,r),l=r;↵
}↵
printf("%d %d\n",ans,(int)I.size());↵
for(auto i:I) printf("%d %d\n",i[0]+1,i[1]+1);↵
}↵
signed main() {↵
solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
## [E. Nene vs. Monsters](https://codeforc.es/contest/1956/problem/E2)↵
↵
idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint1">↵
If three consecutive monsters have energy level $0,x,y\ (x,y>0)$, the monster with energy lever $y$ will "die"(have energy level $0$) at last.↵
</spoiler>↵
↵
<spoiler summary="Hint2">↵
If four consecutive monsters have energy levels $0,x,y,z\ (x,y,z>0)$, what will happen to the monster $z$?↵
</spoiler>↵
↵
<spoiler summary="Hint3">↵
If four consecutive monsters have energy level $x,y,z,w\ (x,y,z,w>0)$, how many rounds of spells must be used to make at least one of these monsters die?↵
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
If four consecutive monsters have energy level $x,y,z,w\ (x,y,z,w>0)$ and they did not die after $t$ rounds of spells, then $y$ will receive at least $t$ points of damage, $z$ will receive at least $(t-1) +(t-2)+\cdots=O(t^2)$ of damage, and $w$ will receive at least $O(t^3)$ of damage.↵
↵
That is to say, let $V=\max_{i=1}^n a_i$, after $O(\sqrt[3]{V})$ rounds, at least one of $x,y,z,w$ will die.↵
↵
So, we can simulate the process by brute force until there are no four consecutive alive monsters, and then the problem is reduced to the one described in Hint 2.↵
↵
If four consecutive monster have energy level $0,x,y,z\ (x,y,z>0)$, $x$ will remain alive, $y$ will die at last and sending $D=(y-x)+(y-2x)+\cdots+(y\bmod x)$ damage to $z$ before that. Therefore, $z$ will remain alive if and only if $z>D$.↵
↵
The time complexity is $O(n\sqrt[3]{V})$.↵
↵
Bonus: Actually, it can be shown that after $O(\sqrt[k]{V})$ rounds, there will be no $k$ consecutive alive monsters. Making $k$ bigger than $3$ can further reduce the time complexity, but it will be harder to implement and optimize little on actual performance.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
mt19937 rnd(time(0));↵
const int MAXN=2e5+5;↵
int n,a[MAXN];bool b[MAXN];↵
bool check(){↵
a[n+1]=a[1],a[n+2]=a[2],a[n+3]=a[3];↵
for(int i=1;i<=n;i++)↵
if(a[i]&&a[i+1]&&a[i+2]&&a[i+3]) return true;↵
return false;↵
} ↵
void solve(){↵
cin>>n;↵
for(int i=1;i<=n;i++) cin>>a[i];↵
if(n==2){↵
while(a[1]&&a[2]){↵
a[2]=max(0,a[2]-a[1]);↵
a[1]=max(0,a[1]-a[2]);↵
}↵
b[1]=(a[1]>0);b[2]=(a[2]>0);↵
}else if(n==3){↵
while(a[1]&&a[2]&&a[3]){↵
a[2]=max(0,a[2]-a[1]);↵
a[3]=max(0,a[3]-a[2]);↵
a[1]=max(0,a[1]-a[3]);↵
}↵
b[1]=(!a[3]&&a[1]);↵
b[2]=(!a[1]&&a[2]);↵
b[3]=(!a[2]&&a[3]);↵
}else{↵
while(check()){↵
for(int i=1;i<=n;i++) a[i%n+1]=max(0,a[i%n+1]-a[i]);↵
}↵
for(int i=1;i<=n;i++) b[i]=false;↵
auto attack=[&](ll x,ll y){↵
ll k=x/y;↵
return (2*x-(k+1)*y)*k/2;↵
};↵
for(int p=1;p<=n;p++) ↵
if(a[p]&&a[p%n+1]) a[p%n+1]=max(0,a[p%n+1]-a[p]);↵
else break;↵
for(int i=1;i<=n;i++) if(!a[i]&&a[i%n+1]){↵
b[i%n+1]=true;↵
b[(i+2)%n+1]=(a[(i+2)%n+1]>attack(a[(i+1)%n+1],a[i%n+1]));↵
}↵
}↵
int cnt=0;↵
for(int i=0;i<=n;i++) if(b[i]) cnt++;↵
cout<<cnt<<endl;↵
for(int i=1;i<=n;i++) if(b[i]) cout<<i<<' ';↵
cout<<endl;↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
int _;cin>>_;↵
while(_--) solve();↵
}↵
```↵
</spoiler>↵
↵
## [F. Nene and the Passing Game](https://codeforc.es/contest/1956/problem/F)↵
↵
idea: [user:Otomachi_Una,2024-04-13]↵
↵
<spoiler summary="Hint">↵
Two pe
</spoiler>↵
↵
<spoiler summary="Tutorial">↵
According to the hint above, we can build the following graph:↵
↵
- There are $2n$ vertices in the graph. Vertice $i$ links to vertices $([n+i-r_i,n+i-l_1]\cup[n+i+l_i,n+i+r_i])\cap[n+1,2n]\cap Z$.↵
↵
That is, assuming vertices $1$ to $n$ are players, vertices $n+1$ to $2n$ are temporary spots, and player $i$ links to all the spots where his/her arm can reach.↵
↵
Then, the answer will be the number of connected components in this graph which contains at least one vertice with an index less than or equal to $n$.↵
↵
But there's still a little problem
↵
To solve that, we can delete all the vertices $n+x$ such that $\forall i,\ x\not\in[i-r_i,i-l_i]$ or $\forall i,\ x\not\in[i+l_i,i+r_i]$ (that is, nobody's left/right arm can reach $x$). Finding such $x$ can be done easily in $O(n)$.↵
↵
The last issue is, the graph contains $O(n^2)$ edges. but since we only care about connectivity, operation "link $x$ to $[y,z]$" can be changed to "link $x$ to $y$, and link $i$ to $i+1$ for all $i$ in $[y,z-1]$". After that and removing multiple edges, the number of edges is reduce
↵
Finally, counting connected components in a graph can be easily done in $O(n)$, so the time complexity is $O(n)$.↵
</spoiler>↵
↵
<spoiler summary="Solution">↵
```cpp↵
#include<bits/stdc++.h>↵
using namespace std;↵
#define ll long long↵
#define MP make_pair↵
const int MAXN=2e6+5;↵
int n,tot,le[MAXN],ri[MAXN];↵
int fa[MAXN<<1],s[MAXN],t[MAXN],pre[MAXN],suf[MAXN];↵
inline int find(int x){↵
while(x^fa[x]) x=fa[x]=fa[fa[x]];↵
return x;↵
}↵
void solve(){↵
cin>>n;↵
for(int i=1;i<=2*n+1;i++) fa[i]=i;↵
for(int i=0;i<=n+1;i++) s[i]=t[i]=pre[i]=suf[i]=0;↵
for(int i=1;i<=n;i++){↵
cin>>le[i]>>ri[i];↵
s[max(1,i-ri[i])]++;s[max(0,i-le[i])+1]--;↵
t[min(n+1,i+le[i])]++;t[min(n,i+ri[i])+1]--;↵
}↵
tot=n;↵
for(int i=1;i<=n;i++){↵
s[i]+=s[i-1],t[i]+=t[i-1];↵
if(s[i]&&t[i]) suf[i]=pre[i]=++tot;↵
}↵
suf[n+1]=0;↵
for(int i=1;i<=n;i++) pre[i]=(pre[i]?pre[i]:pre[i-1]);↵
for(int i=n;i>=1;i--) suf[i]=(suf[i]?suf[i]:suf[i+1]);↵
for(int i=1;i<=n;i++){↵
int l=max(1,i-ri[i]),r=max(0,i-le[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) for(int i=find(l);i<r;i=find(i)) fa[i]=i+1; ↵
}↵
l=min(n+1,i+le[i]),r=min(n,i+ri[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) for(int i=find(l);i<r;i=find(i)) fa[i]=i+1; ↵
}↵
}↵
for(int i=1;i<=n;i++){↵
int l=max(1,i-ri[i]),r=max(0,i-le[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) fa[find(i)]=find(l);↵
}↵
l=min(n+1,i+le[i]),r=min(n,i+ri[i]);↵
if(l<=r){↵
l=suf[l],r=pre[r];↵
if(l&&r&&l<=r) fa[find(i)]=find(l);↵
}↵
}↵
int ans=0;↵
for(int i=1;i<=tot;i++) if(fa[i]==i) ans++; ↵
cout<<ans<<'\n';↵
}↵
int main(){↵
ios::sync_with_stdio(false);↵
// freopen("Otomachi_Una.in","r",stdin);↵
// freopen("Otomachi_Una.out","w",stdout);↵
int _;cin>>_;↵
while(_--) solve();↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
