#include <bits/stdc++.h>
#define int long long
#define fi first
#define se second

using namespace std;

int32_t main()
{
        ios_base::sync_with_stdio(false);
        cin.tie(NULL);
        
        int t;
        cin >> t;

        while (t --> 0) {
                int n;
                cin >> n;

                vector<int> a(n);
                vector<int> b(n);

                for (int i = 0; i < n; i++) cin >> a[i];
                for (int i = 0; i < n; i++) cin >> b[i];

                vector<int> ord(n);
                iota(ord.begin(), ord.end(), 0);
                sort(ord.begin(), ord.end(), [&](int x, int y) {
                        if (b[x] == b[y]) return a[x] > a[y];
                        return b[x] == 1;
                });

                int ans = a[0] * b[0];
                int suma = 0, sumb = 0;

                for (int i = 0; i < n; i++) {
                        suma += a[ord[i]];
                        sumb += b[ord[i]];

                        ans = max(ans, suma * sumb);
                        ans = max(ans, a[i] * b[i]);
                }

                cout << ans << '\n';
        }

        return 0;
}

The probability of Alice's winning is strictly higher than Bob's iff $$$l_1+r_1>l_2+r_2$$$.

Submitting this code directly will get ILE, please delete cerr and submit.

#include<bits/stdc++.h>
using namespace std;
#pragma GCC optimize("Ofast","inline","-ffast-math")
#pragma GCC target("avx,sse2,sse3,sse4,mmx")
namespace Fact {
   vector<char> isPrime;
   vector<int> Prime,OneFac;
   void GetPrime(int n){for(int i=0;i<n;++i)isPrime[i]=1;isPrime[1]=1;for(int i=2;i<=n;i++){if(isPrime[i])Prime.push_back(i);for(int j:Prime){if(i*j>n) break;isPrime[i*j]=0;OneFac[i*j]=j;if(i%j==0)break;}}}
   void Fac_dfs(int x,int y,vector<int>& ans,vector<pair<int,int>>& FacList){if(x==(long long)(FacList.size())){ans.push_back(y);return;}int k=1;for(int i=0;i<=FacList[x].second;++i){Fac_dfs(x+1,y*k,ans,FacList);k*=FacList[x].first;}}
   vector<pair<int,int>>GetPrimeFac(int a){vector<pair<int,int>>answer;if(isPrime[a])return(vector<pair<int,int>>{pair<int,int>{a,1}});answer.push_back({OneFac[a],1});a/=OneFac[a];while(!isPrime[a]){if(OneFac[a]==answer.back().first)answer.back().second++;else answer.push_back({OneFac[a],1});a/=OneFac[a];}if(a==answer.back().first)answer.back().second++;else answer.push_back({a,1});return(answer);}
   vector<int>GetFacList(int a){vector<int> ans;vector<pair<int,int>>FacList=GetPrimeFac(a);Fac_dfs(0,1,ans,FacList);return(ans);}
   void init(int n){isPrime.resize(n+2);OneFac.resize(n+2);GetPrime(n);}
   void init(){init(10000005);}
};
#define int long long

const int MOD=998244353;
void Delta() {
   int a,b;
   cin >> a >> b;
   if(Fact::isPrime[a]||(a/Fact::OneFac[a]<b)) cout << "NO" << endl;
   // in code "OneFac" is The smallest non-1 factor.
   else {
      cout << "YES" << endl;
      for(int i=1;i<b;++i) cerr << Fact::OneFac[a] << ' ';
      cerr << a-(b-1)*Fact::OneFac[a] << endl;
   }
}
signed main() {
   Fact::init(100001);
   ios_base::sync_with_stdio(0);
   cin.tie(0);cout.tie(0);
   int T;cin >> T;
   while(T--) Delta();
}

Let's say we finally get the gcd $$$x$$$, so all $$$a_i$$$ are multiples of $$$x$$$.

We want all $$$a_i$$$to be positive, so every $$$a_i$$$ is at least $$$x$$$, and all sums are at least $$$kx$$$ and sum is divisible by $$$x$$$ (You have some $$$a_i$$$ that is divisible by $$$x$$$, add it up, and it's still divisible by $$$x$$$).

Since we already know sum (is $$$n$$$), $$$x$$$ can only be a factor of $$$n$$$, Since $$$x>1$$$, just to take the smallest non-1 factor of $$$n$$$ and check. If $$$kx$$$ is less than $$$n$$$, then just put all the rest on $$$a_k$$$, which still guarantees that gcd=x.

(The constructional output is given in cerr of code for help understand)

#include<bits/stdc++.h>
using namespace std;
#pragma GCC optimize("Ofast","inline","-ffast-math")
#pragma GCC target("avx,sse2,sse3,sse4,mmx")
#define int long long
const int MOD=998244353;
int n,m,sum;
map<int,int> a,b,c;
vector<pair<int,int>> Q;
void dfs(int idx,int A,int B,int C) {
   if(idx==(int)Q.size()) {
      sum+=a[A]*b[B]*c[C];
      return;
   }
   for(int i=0,t1=1;i<=Q[idx].second;++i,t1*=Q[idx].first)
      for(int j=i,t2=1;j<=Q[idx].second;++j,t2*=Q[idx].first)
         for(int k=j,t3=1;k<=Q[idx].second;++k,t3*=Q[idx].first)
            if(k==Q[idx].second)
               dfs(idx+1,A*t1,B*t2,C*t3);
}
void Delta() {
   cin >> n >> m;
   for(auto i:{&a,&b,&c})
      for(int j=1,x;j<=n;++j) {
         cin >> x;
         ++(*i)[x];
      }
   for(int i=2;i*i<=m;++i)
      while(m%i==0) {
         if(Q.empty()||Q.end()[-1].first!=i) Q.push_back({i,1});
         else Q.end()[-1].second++;
         m/=i;
      }
   if(m!=1)
      if(Q.empty()||Q.end()[-1].first!=m) Q.push_back({m,1});
      else Q.end()[-1].second++;
   else;
   dfs(0,1,1,1);
   cout << sum << endl;
}
signed main() {
   ios_base::sync_with_stdio(0);
   cin.tie(0);cout.tie(0);
   Delta();
}

If a number is less than $$$10^9$$$, it has a maximum possible factor of $$$1344$$$, enumerate two factors of $$$m$$$, then divide the third factor, and then use map to do count to pass.

In this table you can find all the values of $$$\text{d}(n)$$$ and $$$ω(n)$$$ for your future use :)

upd:

$$$\text{d}(n)$$$ = the number of different factors, and $$$ω(n)$$$ = the number of different prime factors

#include<bits/stdc++.h>
using namespace std;
#pragma GCC optimize("Ofast","inline","-ffast-math")
#pragma GCC target("avx,sse2,sse3,sse4,mmx")
 
#define int long long
 
const int MOD=998244353;
int a[500001];
void Delta() {
   int n,x; cin >> n >> x;
   if(x<n-1||x>(n/2)*(n-n/2)) {
      cout << -1 << endl;
      return;
   }
   for(int i=2;i<=n/2+1;++i) cout << i-1 << ' ' << i << endl;
   x-=n-1;
   for(int i=n/2+2;i<=n;++i) {
      cout << min(x,n/2-1)+1 << ' ' << i << endl;
      x-=min(x,n/2-1);
   }
}
signed main() {
   ios_base::sync_with_stdio(0);
   cin.tie(0);cout.tie(0);
   int T;cin >> T;
   while(T--) Delta();
}

The key to solving the problem is — consider constructing a tree with the highest score first.

It's obvious all leaf have the same LCA (othwise,link leaves with smaller depths next to leaves with larger depths, and the score will not decrease) .Note $$$x$$$ as the number of leaf nodes,then the score is $$$x(n-x)$$$,when $$$x=\lfloor\frac{n}{2}\rfloor$$$,it reaches the maximum.

From the tree with the highest score, we can easily construct a tree with any score (by moving each leaf node up).

For convenience,note $$$A={a_1,a_2,...,a_n}$$$ and $$$B={a_{n+1},a_{n+2},...,a_{2n}}$$$($$$A,B$$$ are circular).

Calculate $$$preA$$$ and $$$sufB$$$:

• $$$\text{preA}_i:\max(a_j+...+a{i-1}+a_{i})$$$;

• $$$\text{sufB}_i:\max(b_i+b_{i+1}+...+b_{j})$$$;

This can be solved using monotonic stacks in $$$\Theta(n)$$$.

Then reverse $$$A,B$$$ and do it again.

The answer is the maximum of the following values:

• $$$\max(\text{preA}_i);$$$

• $$$\max(\text{sufB}_i);$$$

• $$$\max(\text{preA}_i+\text{sufB}_{i+1}).$$$

#include <map>
#include <set>
#include <cmath>
#include <ctime>
#include <queue>
#include <stack>
#include <cstdio>
#include <cstdlib>
#include <vector>
#include <cstring>
#include <algorithm>
#include <iostream>
using namespace std;
typedef double db; 
typedef long long ll;
typedef unsigned long long ull;
const int N=3000010;
const int LOGN=28;
const ll  TMD=0;
const ll  INF=2147483647;
int T,n;
ll  ans;
int a[N],b[N],Q[N];
ll  Sa[N],Sb[N],prea[N],sufb[N];

void solve()
{
	for(int i=1;i<=n;i++)   a[i+n]=a[i],b[i+n]=b[i];
	for(int i=1;i<=n*2;i++) Sa[i]=Sa[i-1]+a[i],Sb[i]=Sb[i-1]+b[i];
	int head=1,tail=0;
	for(int i=1;i<=n*2;i++)
	{
		if(i>n)
		{
			while(i-Q[head]>n) head++;
			prea[i-n]=Sa[i]-Sa[Q[head]];
		}	
		while(head<=tail&&Sa[Q[tail]]>Sa[i]) tail--;
		Q[++tail]=i;
	}
	head=1;tail=0;
	for(int i=n*2;i;i--)
	{	
		while(head<=tail&&Sb[Q[tail]]<Sb[i]) tail--;
		Q[++tail]=i;
		if(i<=n)
		{
			while(Q[head]-i+1>n) head++;
			sufb[i]=Sb[Q[head]]-Sb[i-1];
		}
	}
	for(int i=1;i<=n;i++) ans=max(ans,prea[i]);
	for(int i=1;i<=n;i++) ans=max(ans,sufb[i]);
	for(int i=1;i<n;i++)  ans=max(ans,prea[i]+sufb[i+1]);
	ans=max(ans,prea[n]+sufb[1]);
}

int main()
{
	scanf("%d",&T);
	while(T--)
	{
		scanf("%d",&n);
		for(int i=1;i<=n;i++) scanf("%d",&a[i]);
		for(int i=1;i<=n;i++) scanf("%d",&b[i]);
		ans=-INF;
		solve();
		for(int i=1,j=n;i<j;i++,j--) swap(a[i],a[j]),swap(b[i],b[j]);
		solve();
		cout<<ans<<'\n';
	}
	
	return 0;
}

We can see how to merge the largest subsegment of two intervals and by GSS1 in $$$\Theta(1)$$$ time, so using a seg tree and just enumerates all possible k and computs it in $$$O(\log n)$$$ time. This is $$$O(n\log n)$$$.

As you can see, each query is a prefix or suffix of a single array, so we can use the prefix sum and the suffix sum, this is $$$\Theta(n)$$$.

I mentioned a minimum-memory solution in the announcement comment, so now explain how to do $$$\text{12,000 KB}$$$.

For this $$$\Theta(n)$$$ level's memory we can only save int32 a[].

For the prefix sum, just save the value of the previous one, and then add it up, you can do $$$\Theta(1)$$$ memory.

For suffix sum, do the normal suffix sum first, and save the suffix sum values on all $$$k·\sqrt{n}$$$ locations. Then compute each sqrt block separately, $$$\Theta(\sqrt{n})$$$ saves its suffix, computes its perfix recursively, and releases the memory after the calculation is complete.

This allows us to achieve memory complexity $$$\Theta(\text{sqrt})$$$ and time complexity $$$\Theta(n)$$$ in addition to "int32 a[]".

It's $$$\text{12,000 KB}$$$.

#include<bits/stdc++.h>
using namespace std;
#pragma GCC optimize("Ofast","inline","-ffast-math")
#pragma GCC target("avx,sse2,sse3,sse4,mmx")
int a[3000010];
#define int long long
#define B 1000
class node{public:int sum,lm,rm,ans;}temp;
inline void merge1(node&a,node&l,node&r){a={l.sum+r.sum,max(l.lm,l.sum+r.lm),max(r.rm,r.sum+l.rm),max({l.ans,r.ans,l.rm+r.lm})};}
inline void merge2(node&a,node&l,node&r){a={l.sum+r.sum,min(l.lm,l.sum+r.lm),min(r.rm,r.sum+l.rm),min({l.ans,r.ans,l.rm+r.lm})};}
inline node i2node(int x){return {x,x,x,x};}
node sufb[1500010/B+10],sufnb[1500010/B+10],tsuff[B+10],tsufnf[B+10];
void Delta() {
   int n,s=-2147483648,sum=0,flag=true,mx=-2147483648;cin >> n;
   for(int i=1;i<=n*2;++i) {
      cin >> a[i];sum+=(int)a[i];
      flag&=a[i]<0;mx=max(mx,(int)a[i]);
   } 
   node pre=i2node(0),pren=i2node(0),suf=i2node(0),sufn=i2node(0);
   for(int i=n;i>=1;--i) {
      temp=i2node(a[i]);
      merge1(suf,temp,suf);
      temp=i2node(a[i+n]);
      merge1(sufn,temp,sufn);
      if(i%B==0) {
         sufb[i/B]=suf;
         sufnb[i/B]=sufn;
      }
   }
   for(int i=1;i<=n;++i) {
      temp=i2node(a[i]);
      merge1(pre,pre,temp);
      temp=i2node(a[i+n]);
      merge1(pren,pren,temp);
      if(i%B==0||i==1) {
         tsuff[B]=sufb[i/B+1];
         tsufnf[B]=sufnb[i/B+1];
         for(int j=(B-1);j>=1;--j) {
            temp=i2node(i-(i==1)+j>n?0:a[i-(i==1)+j]);
            merge1(tsuff[j],temp,tsuff[j+1]);
            temp=i2node(i-(i==1)+j+n>2*n?0:a[i-(i==1)+j+n]);
            merge1(tsufnf[j],temp,tsufnf[j+1]);
         }
      }
      merge1(temp,pren,tsuff[i%B+1]);
      merge1(temp,temp,pre);
      merge1(temp,temp,tsufnf[i%B+1]);
      s=max(s,temp.ans);
   }
   pre=pren=suf=sufn=i2node(0);
   for(int i=n;i>=1;--i) {
      temp=i2node(a[i]);
      merge2(suf,temp,suf);
      temp=i2node(a[i+n]);
      merge2(sufn,temp,sufn);
      if(i%B==0) {
         sufb[i/B]=suf;
         sufnb[i/B]=sufn;
      }
   }
   for(int i=1;i<=n;++i) {
      temp=i2node(a[i]);
      merge2(pre,pre,temp);
      temp=i2node(a[i+n]);
      merge2(pren,pren,temp);
      if(i%B==0||i==1) {
         tsuff[B]=sufb[i/B+1];
         tsufnf[B]=sufnb[i/B+1];
         for(int j=(B-1);j>=1;--j) {
            temp=i2node(i-(i==1)+j>n?0:a[i-(i==1)+j]);
            merge2(tsuff[j],temp,tsuff[j+1]);
            temp=i2node(i-(i==1)+j+n>2*n?0:a[i-(i==1)+j+n]);
            merge2(tsufnf[j],temp,tsufnf[j+1]);
         }
      }
      merge2(temp,pren,tsuff[i%B+1]);
      merge2(temp,temp,pre);
      merge2(temp,temp,tsufnf[i%B+1]);
      s=max(s,sum-temp.ans);
   }
   cout << (flag?mx:s) << '\n';
}
signed main() {
   ios_base::sync_with_stdio(0);
   cin.tie(0);cout.tie(0);
   int T;cin >> T;
   while(T--) Delta();
}

#include<bits/stdc++.h>
using namespace std;
#pragma GCC optimize("Ofast","inline","-ffast-math")
#pragma GCC target("avx,sse2,sse3,sse4,mmx")
 
#define int long long
 
const int MOD=998244353;
int a[200001],b[200001],vis[200001];
vector<int> Q[200001];
void dfs(int x) {
   if(vis[x]) return;
   vis[x]=true;
   for(int i:Q[x]) dfs(i);
}
void Delta() {
   int cnt=0,blk=0,n; // count,block number
   cin >> n;
   for(int i=1;i<=n;++i) {
      cin >> a[i];
      b[i]=a[i];
      Q[i].clear();
      vis[i]=0;
   }
   sort(b+1,b+1+n);
   for(int i=1;i<=n;++i) 
      if(a[i]!=b[i]) {
         Q[a[i]].push_back(b[i]);
         Q[b[i]].push_back(a[i]);
      } else cnt++;
   for(int i=1;i<=n;++i)
      if(!vis[a[i]]&&a[i]!=b[i]) {
         blk++;
         dfs(a[i]);
      }
   if(n==2) {
      if(cnt==2) cout << 0 << endl;
      else cout << -1 << endl;
      return;
   }
   if(cnt!=0) cout << blk+(n-cnt) << endl;
   else if(blk==1) cout << n+2 << endl;
   else cout << n+blk << endl;
}
signed main() {
   ios_base::sync_with_stdio(0);
   cin.tie(0);cout.tie(0);
   int T;cin >> T;
   while(T--) Delta();
}

Compare the original array (say $$$a$$$) to the sorted one (say $$$b$$$). Since we can move only one token at once, in one move we can place at max one token to its correct place. So let's try to minimise the number of moves that don't do this.

To make optimal moves, model a directed multi-edged graph. Start with only vertices ($$$1$$$ to $$$n$$$), no edges, and add a directed edge from $$$a[i]$$$ to $$$b[i]$$$ when $$$a[i]!=b[i]$$$, for each $$$i$$$ (meaning that $$$b[i]$$$ should be placed in place of $$$a[i]$$$).

We can follow an optimal scheme like this: For now let us assume there is at least one position where correct token is placed (let it be $$$p$$$). Choose an incorrectly placed token, place it on top of $$$p$$$. This creates a gap. Place the correct token for this place. This moves the gap somewhere else. Again choose a corect token for the gap and so on. In the end, place the token stacked on top of $$$p$$$ into the gap, close the gap, and all the tokens we touched are placed correctly.

In the graph, placing a token from $$$b[i]$$$ to gap at $$$a[i]$$$ would mean moving from $$$a[i]$$$ to $$$b[i]$$$ in the graph along an edge and removing the edge. Thus the problem reduces to removing all edges of the graph in minimum number of moves.

The graph is split into disjoint connected componnts. We can thus independently resolve each component optimally for minimum number of moves.

Next, we prove that a connected component is always an euler circuit.

Claim $$$1$$$: The component would have a cycle. If it doesn't, then following some path $$$a \rightarrow b \rightarrow ...$$$, we will reach a dead end, which means that the last element has no place to go, which is not true since it belongs to the array and must have a correct position to be placed at. By induction, we can keep removing cycles (by moving along them and removing edges), and so the component is basically a set of directed cycles intersecting at vertices.

Claim $$$2$$$: It will hence form an euler circuit. Let us go along a cycle. It starts at $$$v$$$, ends at $$$v$$$. Now if there is a cyce intersecting at some vertex $$$v'$$$ in the cycle, on reaching $$$v'$$$ resolve that cycle, return at $$$v'$$$, continue on the older cycle. By this inductive argument, we can cover all edges exactly once and return to $$$v$$$, making an euler circuit.

To find the precise number of moves to resolve a component involving $$$s$$$ tokens, we need to create a gap in $$$1$$$ move, place $$$s-1$$$ tokens correctly, and place the last token we moved initially. Total $$$s+1$$$ moves. To create a gap, we need to place a token from our position to some position not in the component.

Now, proper casework:

Let the number of tokens not placed correctly be $$$co$$$. Let the number of disjoint connected components be $$$x$$$.

If $$$co<n$$$: There is always a correctly placed position to stack our token on and create a gap. So, each component of size s will be resolved in $$$s+1$$$. So total $$$co+x$$$ moves.

If $$$x>1$$$: Since there are $$$>1$$$ components, we can use a position from the other component to create gap for one component. Again $$$co+x$$$.

$$$co==n$$$ and $$$x==1$$$: It may be a single cycle, or $$$>1$$$ joint cycles. If there are $$$>1$$$ joint cycles, resolve one of the cycles, then it reduces to the cases above. Totals to $$$n+2$$$ moves. For exactly one cycle, graph is $$$1 \rightarrow 2 \rightarrow ...n \rightarrow 1$$$. So stack $$$1$$$ on $$$2$$$, resolve rest $$$n-2$$$, total $$$n-1$$$. Then resolve $$$1$$$ and $$$2$$$ which takes $$$3$$$ moves. So total again is $$$n+2$$$. Since $$$n+1$$$ is not possible in this case, and we construct a solution in $$$n+2$$$, it is optimal.

Final solution:

1. $$$(co < n)$$$ or $$$(x>1)$$$ : $$$co+x$$$

2. $$$(co==n)$$$ and $$$(x==1)$$$ : $$$n+2$$$