brute force dp math

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define fast                       \
 ios_base::sync_with_stdio(0);      \
 cin.tie(0);                         \
 cout.tie(0);
 
int main(){
    fast;
    ll t;
    cin>>t;
    while(t--){
        int n;
        cin>>n;
        int r=(n%5);
        if(r==0) cout<<"2\n";
        else if(r==1) cout<<"1\n";
        else cout<<"0\n";
    }
}

constructive algorithm

~~~~~ ~~~~~

number theory math

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define fast                       \
 ios_base::sync_with_stdio(0);      \
 cin.tie(0);                         \
 cout.tie(0);
 
vector<ll> get(ll n){
    vector<ll> v;
    for(ll i=1;i*i<=n;i++){
        if(n%i==0){
            v.push_back(i);
            ll j=(n/i);
            if(j!=i) v.push_back(j);
        }
    }
    return v;
}
 
int main(){
    fast;
    ll t;
    cin>>t;
    while(t--){
        ll n;
        cin>>n;
        vector<ll> a(n);
        ll ans=0;
        for(ll i=0;i<n;i++){
            cin>>a[i];
            ans=__gcd(ans,a[i]);
        }
        sort(a.begin(),a.end());
        vector<ll> v=get(a[0]);
        for(ll i=0;i<(ll)v.size();i++){
            ll cnt=0;
            for(ll j=0;j<n;j++) if(a[j]%v[i]==0) cnt++;
            if(cnt>=(n-2)) ans=max(ans,v[i]);
        }
        vector<ll> p,s;
        for(ll i=1;i<n;i++) p.push_back(a[i]);
        s=p;
        for(ll i=1;i<(ll)p.size();i++) p[i]=__gcd(p[i],p[i-1]);
        for(ll i=(ll)s.size()-2;i>=0;i--) s[i]=__gcd(s[i],s[i+1]);
        for(ll i=0;i<(ll)p.size();i++){
            ll val=0;
            if(i-1>=0) val=__gcd(val,p[i-1]);
            if(i+1<(ll)s.size()) val=__gcd(val,s[i+1]);
            ans=max(ans,val);
        }
        ll val=0;
        for(ll i=2;i<n;i++) val=__gcd(val,a[i]);
        ans=max(ans,val);
        cout<<ans<<"\n";
    }
}

data structure greedy

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define fast                       \
 ios_base::sync_with_stdio(0);      \
 cin.tie(0);                         \
 cout.tie(0);

vector<ll> tree,lazy,v1;

void build(ll v, ll tl, ll tr){
    if(tl==tr) tree[v]=v1[tl];
    else{
        ll m=(tl+tr)/2;
        build(2*v+1,tl,m);
        build(2*v+2,m+1,tr);
        tree[v]=min(tree[2*v+1],tree[2*v+2]);
    }
}

void push(ll v){
    tree[2*v+1]+=lazy[v];
    tree[2*v+2]+=lazy[v];
    lazy[2*v+1]+=lazy[v];
    lazy[2*v+2]+=lazy[v];
    lazy[v]=0;
}

ll get(ll v, ll tl, ll tr, ll l, ll r){
    if(l>r) return 1e18;
    if(l==tl && r==tr) return tree[v];
    push(v);
    ll m=(tl+tr)/2;
    ll val1=get(2*v+1,tl,m,l,min(r,m));
    ll val2=get(2*v+2,m+1,tr,max(l,m+1),r);
    return min(val1,val2);
}

void update(ll v, ll tl, ll tr, ll l, ll r, ll x){
    if(l>r) return;
    if(l==tl && r==tr){
        tree[v]+=x;
        lazy[v]+=x;
    }
    else{
        push(v);
        ll m=(tl+tr)/2;
        update(2*v+1,tl,m,l,min(r,m),x);
        update(2*v+2,m+1,tr,max(l,m+1),r,x);
        tree[v]=min(tree[2*v+1],tree[2*v+2]);
    }
}
 
int main(){
    fast;
    ll t;
    cin>>t;
    while(t--){
        ll n,q;
        cin>>n>>q;
        vector<ll> a(n);
        for(ll i=0;i<n;i++) cin>>a[i];
        v1.clear();
        v1=a;
        for(ll i=1;i<n;i++) v1[i]+=v1[i-1];
        tree.clear();
        tree.resize(4*n);
        lazy.clear();
        lazy.resize(4*n);
        build(0,0,n-1);
        while(q--){
            ll type;
            cin>>type;
            if(type==1){
                ll l,r;
                cin>>l>>r;
                l--;
                r--;
                ll sum=get(0,0,n-1,r,r);
                if(l-1>=0) sum-=get(0,0,n-1,l-1,l-1);
                if(sum<=0) cout<<"-1\n";
                else{
                    ll val=get(0,0,n-1,l,r);
                    if(l-1>=0) val-=get(0,0,n-1,l-1,l-1);
                    if(val>0) cout<<"0\n";
                    else{
                        val=-val;
                        cout<<(val+2)/2<<"\n";
                    }
                }
            }
            else{
                ll pos;
                cin>>pos;
                pos--;
                if(a[pos]==1) update(0,0,n-1,pos,n-1,-2);
                else update(0,0,n-1,pos,n-1,2);
                a[pos]*=-1;
            }
        }
    }
}

constructive algorithm tree

~~~~~

~~~~~

dp tree

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define fast                       \
 ios_base::sync_with_stdio(0);      \
 cin.tie(0);                         \
 cout.tie(0);

ll mod=1e9+7;

long long power(long long a,long long b){
  long long x=1,y=a;
  while(b>0){
    if(b&1ll){
      x=(x*y)%mod;
    }
    y=(y*y)%mod;
    b>>=1;
  }
  return x%mod;
}
 
long long modular_inverse(long long n){
  return power(n,mod-2);
}

#define MAXN 2000005
 
long long factorial[MAXN];
long long invfact[MAXN];
 
void cfact(){
  long long i;
  factorial[0]=1;
  factorial[1]=1;
  for(i=2;i<MAXN;i++){
    factorial[i]=factorial[i-1]*i;
    factorial[i]%=mod;
  }
  invfact[MAXN-1]=modular_inverse(factorial[MAXN-1]);
  for(i=MAXN-2;i>=0;i--){
    invfact[i]=invfact[i+1]*(i+1);
    invfact[i]%=mod;
  }
}
 
long long calcnCr(long long n,long long k){
  if(k<0 || n<k){return 0;}
  return (factorial[n]*((invfact[k]*invfact[n-k])%mod))%mod;
}

vector<vector<ll>> adj,dp;
vector<ll> sz,dp1,v1;
ll xx,yy;

void dfs(ll x, ll p){
    dp[x][0]=1;
    for(auto i:adj[x]){
        if(i!=p){
            dfs(i,x);
            for(ll j=0;j<=sz[x]+sz[i];j++) dp1[j]=0;
            for(ll j=0;j<=sz[x];j++){
                for(ll k=1;k<=sz[i];k++){
                    dp1[j+k]=(dp1[j+k]+(((dp[x][j]*dp[i][k])%mod)*calcnCr(j+k-v1[x]-v1[i],j-v1[x]))%mod)%mod;
                }
            }
            sz[x]+=sz[i];
            v1[x]+=v1[i];
            for(ll j=0;j<=sz[x];j++){
                dp[x][j]=dp1[j];
            }
        }
    }
    sz[x]++;
    for(ll j=0;j<=sz[x];j++) dp1[j]=0;
    for(ll j=0;j<sz[x];j++){
        dp1[j+1]=(dp1[j+1]+(dp[x][j]*(j+1-v1[x]))%mod)%mod;
        dp1[j]=(dp1[j]+(dp[x][j+1]*j)%mod)%mod;
        dp1[j]=(dp1[j]+(dp[x][j]*(2*j-v1[x]))%mod)%mod;
    }
    for(ll j=0;j<=sz[x];j++){
        dp[x][j]=dp1[j];
    }
}
 
int main(){
    fast;
    cfact();
    ll t;
    cin>>t;
    while(t--){
        ll n;
        cin>>n;
        adj.clear();
        adj.resize(n);
        for(ll i=0;i<n-1;i++){
            ll u,v;
            cin>>u>>v;
            u--;
            v--;
            adj[u].push_back(v);
            adj[v].push_back(u);
        }
        dp.clear();
        dp.resize(n+2);
        for(ll i=0;i<n+2;i++) dp[i].resize(n+2);
        dp1.clear();
        dp1.resize(n+2);
        sz.clear();
        sz.resize(n+2);
        v1.clear();
        v1.resize(n+2);
        dfs(0,-1);
        cout<<dp[0][1]<<"\n";
    }
}

data structure

#include<iostream>
#include<vector>
#include<tuple>
#include<unordered_map>
#include<unordered_set>
#include<functional>
#include<assert.h>
using namespace::std;
using lint=long long;
#define rep(i,n) for(lint i=0;(i)<lint(n);(i)++)
template<typename T>
class dynamic_connectivity{
	class euler_tour_tree{
		public:
		struct node;
		using np=node*;
		using lint=long long;
		struct node{
			np ch[2]={nullptr,nullptr};
			np p=nullptr;
			int l,r,sz,sz2;
			T val=et,sum=et;
			bool exact=1;
			bool child_exact;
			bool edge_connected=0;
			node(){}
			node(int l,int r):l(l),r(r),sz(l==r),sz2(0),child_exact(l<r){}
			bool is_root() {
				return !p;
			}
		};
		vector<unordered_map<int,np>>ptr;
		np get_node(int l,int r){
			if(ptr[l].find(r)==ptr[l].end())ptr[l][r]=new node(l,r);
			return ptr[l][r];
		}
		np root(np t){
			if(!t)return t;
			while(t->p)t=t->p;
			return t;
		}
		bool same(np s,np t){
			if(s)splay(s);
			if(t)splay(t);
			return root(s)==root(t);
		}
		np reroot(np t){
			auto s=split(t);
			return merge(s.second,s.first);
		}
		pair<np,np> split(np s){
			splay(s);
			np t=s->ch[0];
			if(t)t->p=nullptr;
			s->ch[0]=nullptr;
			return {t,update(s)};
		}
		pair<np,np> split2(np s){
			splay(s);
			np t=s->ch[0];
			np u=s->ch[1];
			if(t)t->p=nullptr;
			s->ch[0]=nullptr;
			if(u)u->p=nullptr;
			s->ch[1]=nullptr;
			return {t,u};
		}
		tuple<np,np,np> split(np s,np t){
			auto u=split2(s);
			if(same(u.first,t)){
				auto r=split2(t);
				return make_tuple(r.first,r.second,u.second);
			}else{
				auto r=split2(t);
				return make_tuple(u.first,r.first,r.second);
			}
		}
		template<typename First, typename... Rest>
		np merge(First s,Rest... t){
			return merge(s,merge(t...));
		}
		np merge(np s,np t){
			if(!s)return t;
			if(!t)return s;
			while(s->ch[1])s=s->ch[1];
			splay(s);
			s->ch[1]=t;
			if(t)t->p=s;
			return update(s);
		}
		int size(np t){return t?t->sz:0;}
		int size2(np t){return t?t->sz2:0;}
		np update(np t){
			t->sum=et;
			if(t->ch[0])t->sum=fn(t->sum,t->ch[0]->sum);
			if(t->l==t->r)t->sum=fn(t->sum,t->val);
			if(t->ch[1])t->sum=fn(t->sum,t->ch[1]->sum);
			t->sz=size(t->ch[0])+(t->l==t->r)+size(t->ch[1]);
			t->sz2=size2(t->ch[0])+(t->edge_connected)+size2(t->ch[1]);
			t->child_exact=(t->ch[0]?t->ch[0]->child_exact:0)|(t->l<t->r&&t->exact)|(t->ch[1]?t->ch[1]->child_exact:0);
			return t;
		}
		// void push(np t){
		// 	//遅延評価予定
		// }
		void rot(np t,bool b){
			np x=t->p,y=x->p;
			if((x->ch[1-b]=t->ch[b]))t->ch[b]->p=x;
			t->ch[b]=x,x->p=t;
			update(x);update(t);
			if((t->p=y)){
				if(y->ch[0]==x)y->ch[0]=t;
				if(y->ch[1]==x)y->ch[1]=t;
				update(y);
			}
		}
		void splay(np t){
			//push(t);
			while(!t->is_root()){
				np q=t->p;
				if(q->is_root()){
					//push(q), push(t);
					rot(t,q->ch[0]==t);
				}else{
					np r=q->p;
					//push(r), push(q), push(t);
					bool b=r->ch[0]==q;
					if(q->ch[1-b]==t)rot(q,b),rot(t,b);
					else rot(t,1-b),rot(t,b);
				}
			}
		}
		void debug(np t){
			if(!t)return;
			debug(t->ch[0]);
			cerr<<t->l<<"-"<<t->r<<" ";
			debug(t->ch[1]);
		}
		constexpr static T et=T();
		constexpr static T fn(T s,T t){
			return s+t;
		}
		public:
		euler_tour_tree(){}
		euler_tour_tree(int sz){
			ptr.resize(sz);
			for(int i=0;i<sz;i++)ptr[i][i]=new node(i,i);
		}
		int size(int s){
			np t=get_node(s,s);
			splay(t);
			return t->sz;
		}
		bool same(int s,int t){
			return same(get_node(s,s),get_node(t,t));
		}
		void set_size(int sz){
			ptr.resize(sz);
			for(int i=0;i<sz;i++)ptr[i][i]=new node(i,i);
		}
		void update(int s,T x){
			np t=get_node(s,s);
			splay(t);
			t->val=fn(t->val,x);
			update(t);
		}
		void edge_update(int s,function<void(int,int)> g){
			np t=get_node(s,s);
			splay(t);
			function<void(np)>dfs=[&](np t){
				assert(t);
				if(t->l<t->r&&t->exact){
					splay(t);
					t->exact=0;
					update(t);
					g(t->l,t->r);
					return;
				}
				if(t->ch[0]&&t->ch[0]->child_exact)dfs(t->ch[0]);
				else dfs(t->ch[1]);
			};
			while(t&&t->child_exact){
				dfs(t);
				splay(t);
			}
		}
		bool try_reconnect(int s,function<bool(int)> f){
			np t=get_node(s,s);
			splay(t);
			function<bool(np,int)>dfs=[&](np t,int idx)->bool{
				assert(t);
				if(t->edge_connected&&(size2(t->ch[0])==idx)){
					splay(t);
					return f(t->l);
				}
				if(idx<size2(t->ch[0]))return dfs(t->ch[0],idx);
				else return dfs(t->ch[1],idx-size2(t->ch[0])-(t->edge_connected));
			};
			while(size2(t)){
				if(dfs(t,0))return 1;
				splay(t);
			}
			return 0;
		}
		void edge_connected_update(int s,bool b){
			np t=get_node(s,s);
			splay(t);
			t->edge_connected=b;
			update(t);
		}
		bool link(int l,int r){
			if(same(l,r))return 0;
			merge(reroot(get_node(l,l)),get_node(l,r),reroot(get_node(r,r)),get_node(r,l));
			return 1;
		}
		bool cut(int l,int r){
			if(ptr[l].find(r)==ptr[l].end())return 0;
			np s,t,u;
			tie(s,t,u)=split(get_node(l,r),get_node(r,l));
			merge(s,u);
			np p=ptr[l][r];
			np q=ptr[r][l];
			ptr[l].erase(r);
			ptr[r].erase(l);
			delete p;delete q;
			return 1;
		}
		T get_sum(int p,int v){
			cut(p,v);
			np t=get_node(v,v);
			splay(t);
			T res=t->sum;
			link(p,v);
			return res;
		}
		T get_sum(int s){
			np t=get_node(s,s);
			splay(t);
			return t->sum;
		}
	};
	int dep=1;
	vector<euler_tour_tree> ett;
	vector<vector<unordered_set<int>>>edges;
	int sz;
	public:
	dynamic_connectivity(int sz):sz(sz){
		ett.emplace_back(sz);
		edges.emplace_back(sz);
	}
	bool link(int s,int t){
		if(s==t)return 0;
		if(ett[0].link(s,t))return 1;
		edges[0][s].insert(t);
		edges[0][t].insert(s);
		if(edges[0][s].size()==1)ett[0].edge_connected_update(s,1);
		if(edges[0][t].size()==1)ett[0].edge_connected_update(t,1);
		return 0;
	}
	bool same(int s,int t){
		return ett[0].same(s,t);
	}
	int size(int s){
		return ett[0].size(s);
	}
	vector<int>get_vertex(int s){
		return ett[0].vertex_list(s);
	}
	void update(int s,T x){
		ett[0].update(s,x);
	}
	T get_sum(int s){
		return ett[0].get_sum(s);
	}
	bool cut(int s,int t){
		if(s==t)return 0;
		for(int i=0;i<dep;i++){
			edges[i][s].erase(t);
			edges[i][t].erase(s);
			if(edges[i][s].size()==0)ett[i].edge_connected_update(s,0);
			if(edges[i][t].size()==0)ett[i].edge_connected_update(t,0);
		}
		for(int i=dep-1;i>=0;i--){
			if(ett[i].cut(s,t)){
				if(dep-1==i){
					dep++;
					ett.emplace_back(sz);
					edges.emplace_back(sz);
				}
				return !try_reconnect(s,t,i);
			}
		}
		return 0;
	}
	bool try_reconnect(int s,int t,int k){
		for(int i=0;i<k;i++){
			ett[i].cut(s,t);
		}
		for(int i=k;i>=0;i--){
			if(ett[i].size(s)>ett[i].size(t))swap(s,t);
			function<void(int,int)> g=[&](int s,int t){ett[i+1].link(s,t);};
			ett[i].edge_update(s,g);
			function<bool(int)> f=[&](int x)->bool{
				for(auto itr=edges[i][x].begin();itr!=edges[i][x].end();){
					auto y=*itr;
					itr=edges[i][x].erase(itr);
					edges[i][y].erase(x);
					if(edges[i][x].size()==0)ett[i].edge_connected_update(x,0);
					if(edges[i][y].size()==0)ett[i].edge_connected_update(y,0);
					if(ett[i].same(x,y)){
						edges[i+1][x].insert(y);
						edges[i+1][y].insert(x);
						if(edges[i+1][x].size()==1)ett[i+1].edge_connected_update(x,1);
						if(edges[i+1][y].size()==1)ett[i+1].edge_connected_update(y,1);
					}else{
						for(int j=0;j<=i;j++){
							ett[j].link(x,y);
						}
						return 1;
					}
				}
				return 0;
			};
			if(ett[i].try_reconnect(s,f))return 1;
		}
		return 0;
	}
};

vector<vector<lint>> adj;
vector<bool> vis;
bool ok1,ok2;
lint comp,sz;

void dfs(lint x){
    if(x%2) ok1=true;
    else ok2=true;
    vis[x]=true;
    sz++;
    for(auto i:adj[x]){
        if(!vis[i]) dfs(i);
    }
}

int main(){
    lint t;
    cin>>t;
    while(t--){
        lint n,q;
        cin>>n>>q;
        dynamic_connectivity<lint> dc(n+1);
        vector<lint> p(n+1);
        adj.clear();
        adj.resize(n+1);
        for(lint i=1;i<=n;i++){
            cin>>p[i];
            adj[i].push_back(p[i]);
            adj[p[i]].push_back(i);
            dc.link(i,p[i]);
            if(i%2) dc.update(i,1);
        }
        lint cnt1=0,cnt2=0;
        comp=0;
        vis.clear();
        vis.resize(n+1,false);
        for(lint i=1;i<=n;i++){
            if(!vis[i]){
                ok1=false;
                ok2=false;
                comp++;
                sz=0;
                dfs(i);
                if(ok1 && !ok2 && sz>1) cnt1++;
                if(!ok1 && ok2 && sz>1) cnt2++;
            }
        }
        cout<<(n-comp+2*(max(cnt1,cnt2)))<<" ";
        while(q--){
            lint x,y;
            cin>>x>>y;
            if(dc.same(x,y)){
                comp++;
                lint sum=dc.get_sum(x);
                lint siz=dc.size(x);
                if(sum==siz && siz>1) cnt1--;
                if(sum==0 && siz>1) cnt2--;
                dc.cut(x,p[x]);
                dc.cut(y,p[y]);
                dc.link(x,p[y]);
                dc.link(y,p[x]);
                sum=dc.get_sum(x);
                siz=dc.size(x);
                if(sum==siz && siz>1) cnt1++;
                if(sum==0 && siz>1) cnt2++;
                sum=dc.get_sum(y);
                siz=dc.size(y);
                if(sum==siz && siz>1) cnt1++;
                if(sum==0 && siz>1) cnt2++;
            }
            else{
                comp--;
                lint sum=dc.get_sum(x);
                lint siz=dc.size(x);
                if(sum==siz && siz>1) cnt1--;
                if(sum==0 && siz>1) cnt2--;
                sum=dc.get_sum(y);
                siz=dc.size(y);
                if(sum==siz && siz>1) cnt1--;
                if(sum==0 && siz>1) cnt2--;
                lint sz1=dc.size(x);
                if(sz1>2) dc.cut(x,p[x]);
                sz1=dc.size(y);
                if(sz1>2) dc.cut(y,p[y]);
                dc.link(x,p[y]);
                dc.link(y,p[x]);
                sum=dc.get_sum(x);
                siz=dc.size(x);
                if(sum==siz && siz>1) cnt1++;
                if(sum==0 && siz>1) cnt2++;
            }
            swap(p[x],p[y]);
            cout<<(n-comp+2*(max(cnt1,cnt2)))<<" ";
        }
        cout<<"\n";
    }
}