Problem:
Score : 1900 points

Problem Statement

Ringo has a tree with N vertices.
The i-th of the N-1 edges in this tree connects Vertex A_i and Vertex B_i and has a weight of C_i.
Additionally, Vertex i has a weight of X_i.

Here, we define f(u,v) as the distance between Vertex u and Vertex v, plus X_u + X_v.

We will consider a complete graph G with N vertices.
The cost of its edge that connects Vertex u and Vertex v is f(u,v).
Find the minimum spanning tree of G.

Constraints

2 \leq N \leq 200,000

1 \leq X_i \leq 10^9

1 \leq A_i,B_i \leq N

1 \leq C_i \leq 10^9

The given graph is a tree.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 X_2 ... X_N
A_1 B_1 C_1
A_2 B_2 C_2
:
A_{N-1} B_{N-1} C_{N-1}

Output

Print the cost of the minimum spanning tree of G.

Sample Input 1

4
1 3 5 1
1 2 1
2 3 2
3 4 3

Sample Output 1

22

We connect the following pairs: Vertex 1 and 2, Vertex 1 and 4, Vertex 3 and 4. The costs are 5, 8 and 9, respectively, for a total of 22.

Sample Input 2

6
44 23 31 29 32 15
1 2 10
1 3 12
1 4 16
4 5 8
4 6 15

Sample Output 2

359

Sample Input 3

2
1000000000 1000000000
2 1 1000000000

Sample Output 3

3000000000

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
#define sqr(x) ((x)*(x))
#define ll long long
#define ull unsigned long long
#define ui unsigned int
#define ld long double
#define vi vector<int>
#define HEAP(...) priority_queue<__VA_ARGS__ >
#define heap(...) priority_queue<__VA_ARGS__,vector<__VA_ARGS__ >,greater<__VA_ARGS__ > > 
#define pii pair<int,int> 
#define pb push_back
#define mp make_pair
#define debuge cerr<<"isok"<<endl
#define debug(x) cerr<<#x<<"="<<x<<endl
#define dprintf(...) fprintf(stderr,__VA_ARGS__)
#define SS second
#define FF first
#define ls (k<<1)
#define rs (k<<1|1)
#define clr(a,x) memset(a,x,sizeof(a))
#define cpy(a,x) memcpy(a,x,sizeof(a))
#define file(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout)
#define SZ(x) ((int)x.size())
using namespace std;
template<class T> inline void gmin(T &x,const T &y){if(x>y) x=y;}
template<class T> inline void gmax(T &x,const T &y){if(x<y) x=y;}
const int BufferSize=1<<16;
char buffer[BufferSize],*Bufferhead,*Buffertail;
bool Terminal;
inline char Getchar(){
	if(Bufferhead==Buffertail){
		int l=fread(buffer,1,BufferSize,stdin);
		if(!l){Terminal=1;return 0;}
		Buffertail=(Bufferhead=buffer)+l;
	}
	return *Bufferhead++;
}
template<class T>inline bool read(T &x){
	x=0;char c=Getchar(),rev=0;
	while(c<'0'||c>'9'){rev|=c=='-';c=Getchar();if(Terminal)return 0;}
	while(c>='0'&&c<='9') x=x*10+c-'0',c=Getchar();
	if(c=='.'){
		c=Getchar();double t=0.1;
		while(c>='0'&&c<='9') x=x+(c-'0')*t,c=Getchar(),t=t/10;
	}
	x=rev?-x:x;
	return 1;
}
template<class T1,class T2> inline bool read(T1 &x,T2 &y){return read(x)&read(y);}
template<class T1,class T2,class T3> inline bool read(T1 &x,T2 &y,T3 &z){return read(x)&read(y)&read(z);}
template<class T1,class T2,class T3,class T4> inline bool read(T1 &x,T2 &y,T3 &z,T4 &w){return read(x)&read(y)&read(z)&read(w);}
inline bool reads(char *x){
	char c=Getchar();
	while(c<33||c>126){c=Getchar();if(Terminal)return 0;}
	while(c>=33&&c<=126) (*x++)=c,c=Getchar();
	*x=0;return 1;
}
template<class T>inline void print(T x,const char c='\n'){
	if(!x){putchar('0');putchar(c);return;}
	if(x<0) putchar('-'),x=-x;
	int m=0,a[20];
	while(x) a[m++]=x%10,x/=10;
	while(m--) putchar(a[m]+'0');
	putchar(c);
}
//--------------------------------head---------------------------------------------

const ll inf=0x3f3f3f3f3f3f3f3f;
const int N=200005,M=100005,mod=1e9+7;
template<class T,class S> inline void ch(T &x,const S y){x=(x+y)%mod;}
inline int exp(int x,int y,const int mod=::mod){
	int ans=1;
	while(y){
		if(y&1) ans=(ll)ans*x%mod;
		x=(ll)x*x%mod;y>>=1;
	}return ans;
}

int n,m,val[N],c[N],fa[N],vis[N];
ll ans=0;
struct node{pair<ll,int> mn,mn2;}INF,dp[N];
pair<ll,int> f[N];
vector<pii> g[N];

inline void make(node &a,pair<ll,int> b){
	if(b.FF<a.mn.FF){
		if(b.SS!=a.mn.SS) a.mn2=a.mn;
		a.mn=b;
	}else if(b.FF<a.mn2.FF&&b.SS!=a.mn.SS) a.mn2=b;
}
inline node operator + (node a,node b){
	node tmp=INF;
	make(tmp,a.mn);make(tmp,a.mn2);
	make(tmp,b.mn);make(tmp,b.mn2);
	return tmp;
}
inline node operator * (node a,int b){
	a.mn.FF+=b;a.mn2.FF+=b;
	return a;
}

inline void dfs(int x,int fa=0){
	dp[x]=INF;
	for(auto i:g[x]){
		int y=i.FF,L=i.SS;
		if(y==fa) continue;
		dfs(y,x);
		dp[x]=dp[x]+dp[y]*L;
	}
	make(dp[x],mp(val[x],c[x]));
}
inline void dfs2(int x,int fa=0,node p=INF){
	node tmp=(p+dp[x])*val[x];
	gmin(f[c[x]],tmp.mn.SS==c[x]?tmp.mn2:tmp.mn);
	for(auto i:g[x]){
		int y=i.FF,L=i.SS;
		if(y==fa) continue;
		dfs2(y,x,(p+dp[x])*L);
	}
}

inline int find(int x){return fa[x]==x?x:(fa[x]=find(fa[x]));}
inline void merge(int x,int y,ll w){
	x=find(x);y=find(y);
	if(x!=y) ans+=w,fa[x]=y;
}

int main(){
#ifdef rqgao2014
	assert(freopen("input.txt","r",stdin));
#endif
	INF.mn=INF.mn2=mp(inf,0);
	scanf("%d",&n);
	for(int i=1;i<=n;i++) 
		scanf("%d",&val[i]);
	for(int i=1;i<n;i++){
		int x,y,z;scanf("%d%d%d",&x,&y,&z);
		g[x].pb(mp(y,z));
		g[y].pb(mp(x,z));
	}
	m=n;
	for(int i=1;i<=n;i++)
		c[i]=i;
	while(m>1){
		int cnt=0;
		for(int i=1;i<=m;i++)
			f[i]=mp(inf,0);
		dfs(1);dfs2(1);
//		dprintf("(%lld,%d) , (%lld,%d)\n",dp[1].mn.FF,dp[1].mn.SS,dp[1].mn2.FF,dp[1].mn2.SS);
		for(int i=1;i<=m;i++)
			fa[i]=i,vis[i]=0;
		for(int i=1;i<=m;i++)
			merge(i,f[i].SS,f[i].FF);
		for(int i=1;i<=m;i++)
			if(!vis[find(i)]) vis[find(i)]=++cnt;
		for(int i=1;i<=n;i++)
			c[i]=vis[find(c[i])];
		m=cnt;
	}
	printf("%lld\n",ans);
	return 0;
}