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>
using namespace std;
#ifndef ONLINE_JUDGE
	#include <boost/multiprecision/cpp_int.hpp>
	#include <boost/multiprecision/cpp_dec_float.hpp>
	using bll = boost::multiprecision::cpp_int;
	using bdouble = boost::multiprecision::cpp_dec_float_100;
#endif
#ifdef LOCAL_DEV
	void debug_impl() { std::cerr << std::endl; }
	template<typename Head, typename... Tail> void debug_impl(Head head, Tail... tail) { std::cerr << " " << head << (sizeof...(tail) ? "," : ""); debug_impl(tail...); }
	#define debug(...) { std::cerr << std::boolalpha << "(" << #__VA_ARGS__ << ") ="; debug_impl(__VA_ARGS__); std::cerr << std::noboolalpha; }
#else
	#define debug(...) {}
#endif
#ifdef LOCAL_TEST
	#define BOOST_STACKTRACE_USE_ADDR2LINE
	#define BOOST_STACKTRACE_ADDR2LINE_LOCATION /usr/local/opt/binutils/bin/addr2line
	#define _GNU_SOURCE
	#include <boost/stacktrace.hpp>
	namespace std {
		template<typename T> class dvector : public std::vector<T> {
		public:
			dvector() : std::vector<T>() {}
			explicit dvector(size_t n, const T& value = T()) : std::vector<T>(n, value) {}
			dvector(const std::vector<T>& v) : std::vector<T>(v) {}
			dvector(const std::initializer_list<T> il) : std::vector<T>(il) {}
			dvector(const typename std::vector<T>::iterator first, const typename std::vector<T>::iterator last) : std::vector<T>(first, last) {}
			dvector(const std::string::iterator first, const std::string::iterator last) : std::vector<T>(first, last) {}
			T& operator[](size_t n) {
				try { return this->at(n); } catch (const std::exception& e) {
					std::cerr << boost::stacktrace::stacktrace() << std::endl; return this->at(n);
				}
			}
			const T& operator[](size_t n) const {
				try { return this->at(n); } catch (const std::exception& e) {
					std::cerr << boost::stacktrace::stacktrace() << std::endl; return this->at(n);
				}
			}
		};
	}
	class dbool {
	private:
		bool boolvalue;
	public:
		dbool() : boolvalue(false) {}
		dbool(bool b) : boolvalue(b) {}
		dbool(const dbool &b) : boolvalue(b.boolvalue) {}
		operator bool&() { return boolvalue; }
		operator const bool&() const { return boolvalue; }
	};
	template<typename T> std::ostream& operator<<(std::ostream& s, const dvector<T>& v) {
		for (int i = 0, len = v.size(); i < len; ++i){ s << v[i]; if (i < len - 1) s << "\t"; } return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const dvector< dvector<T> >& vv) {
		for (int i = 0, len = vv.size(); i < len; ++i){ s << vv[i] << std::endl; } return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::set<T>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::multiset<T>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template<typename T1, typename T2> std::ostream& operator<<(std::ostream& s, const std::map<T1, T2>& m) {
		s << "{" << std::endl; for (auto itr = m.begin(); itr != m.end(); ++itr){ s << "\t" << (*itr).first << " : " << (*itr).second << std::endl; } s << "}"; return s; }
	template<typename T1, typename T2> std::ostream& operator<<(std::ostream& s, const std::pair<T1, T2>& p) { 
		return s << "(" << p.first << ", " << p.second << ")"; }
	#define vector dvector
	#define bool dbool
	class SIGFPE_exception : std::exception {};
	class SIGSEGV_exception : std::exception {};
	void catch_SIGFPE(int e) { std::cerr << boost::stacktrace::stacktrace() << std::endl; throw SIGFPE_exception();	}
	void catch_SIGSEGV(int e) { std::cerr << boost::stacktrace::stacktrace() << std::endl; throw SIGSEGV_exception(); }
	signed convertedmain();
	signed main() { signal(SIGFPE, catch_SIGFPE); signal(SIGSEGV, catch_SIGSEGV); return convertedmain(); }
	#define main() convertedmain()
#endif
//#define int long long
using ll = long long;
//constexpr int INF = 1e9;//INT_MAX=(1<<31)-1=2147483647
constexpr ll INF = (ll)1e18;//(1LL<<63)-1=9223372036854775807
constexpr ll MOD = (ll)1e9 + 7;
constexpr double EPS = 1e-9;
constexpr ll dx[4] = {1LL, 0LL, -1LL, 0LL};
constexpr ll dy[4] = {0LL, 1LL, 0LL, -1LL};
constexpr ll dx8[8] = {1LL, 0LL, -1LL, 0LL, 1LL, 1LL, -1LL, -1LL};
constexpr ll dy8[8] = {0LL, 1LL, 0LL, -1LL, 1LL, -1LL, 1LL, -1LL};
#define rep(i, n)   for(ll i=0, i##_length=(n); i< i##_length; ++i)
#define repeq(i, n) for(ll i=1, i##_length=(n); i<=i##_length; ++i)
#define rrep(i, n)   for(ll i=(n)-1; i>=0; --i)
#define rrepeq(i, n) for(ll i=(n)  ; i>=1; --i)
#define all(v) (v).begin(), (v).end()
#define rall(v) (v).rbegin(), (v).rend()
void p() { std::cout << '\n'; }
template<typename Head, typename... Tail> void p(Head head, Tail... tail) { std::cout << head << (sizeof...(tail) ? " " : ""); p(tail...); }
template<typename T> inline void pv(std::vector<T> &v) { for(ll i=0, N=v.size(); i<N; i++) std::cout << v[i] << " \n"[i==N-1]; }
template<typename T> inline T gcd(T a, T b) { return b ? gcd(b,a%b) : a; }
template<typename T> inline T lcm(T a, T b) { return a / gcd(a,  b) * b; }
template<typename T> inline bool chmax(T &a, T b) { return a < b && (a = b, true); }
template<typename T> inline bool chmin(T &a, T b) { return a > b && (a = b, true); }
template<typename T> inline void uniq(std::vector<T> &v) { v.erase(unique(v.begin(), v.end()), v.end()); }

/*-----8<-----template-----8<-----*/

class Edge {
public:
	ll from,to,cost;
	Edge() {}
	Edge(ll from, ll to, ll cost) : from(from), to(to), cost(cost) {}
};
ostream& operator<<(ostream& s, const Edge& e) {
	s << "{ " << e.from << " -> " << e.to << ", " << e.cost << " }";
	return s;
}

class UnionFind {
public:
	vector<ll> v;
	UnionFind(size_t size) : v(size, -1) {}

	ll root(ll x) {
		return (v[x] < 0 ? x : v[x] = root(v[x]));
	}

	bool is_root(ll x) {
		return x == root(x);
	}

	bool is_same(ll x, ll y) {
		return root(x) == root(y);
	}

	bool unite(ll x, ll y) {
		x = root(x);
		y = root(y);
		if (x == y) return false;
		if (v[x] > v[y]) swap(x, y);
		v[x] += v[y];
		v[y] = x;
		return true;
	}

	ll size(ll x) {
		return -v[root(x)];
	}
};

/*
グラフ上のボロノイ図を作る 要Edge,UnionFind
引数
　g:探索するグラフ
　startv:探索するスタートノード番号の集合
戻り値
  vector<Edge>: ボロノイ図で接しているセルの情報
    e.from,e.to = 接しているセルの2ノード
    e.cost = e.from-e.to間の距離, 接点までの位置は÷2してください
*/
vector<Edge> voronoi(vector<vector<Edge>> &g, vector<ll> &startv){
	priority_queue<Edge, vector<Edge>, function<bool(Edge,Edge)>> que(
		[](const Edge &x, const Edge &y){
			return x.cost != y.cost ? (x.cost > y.cost) : 
				(x.from != y.from ? x.from < y.from : x.to < y.to);
		}
	);
	for(ll v : startv){
		que.push(Edge(v, v, 0));
	}

	vector<ll> dist(g.size(), INF);
	vector<ll> prev(g.size(), -1);
	UnionFind u(g.size());
	vector<Edge> ans;
	while(!que.empty()){
		Edge e = que.top();
		que.pop();
		if(dist[e.to] == INF){
			dist[e.to] = e.cost;
			prev[e.to] = e.from;
			for(Edge f : g[e.to]){
				que.push(Edge(e.from, f.to, e.cost + f.cost));
			}
		}else{
			if(!u.is_same(e.from, prev[e.to])){
				//prev[e.to] と e.from が出会った
				ans.emplace_back(e.from, prev[e.to], dist[e.to] + e.cost);
				u.unite(e.from, prev[e.to]);
			}
		}
	}
	return ans;
}


////


/*
計算量：O(ElogE)
引数
　g:探索するグラフ
戻り値
　return:最小全域木を構成する辺のコストの総和
　ming:最小全域木のグラフ
*/
ll kruskal(const vector<vector<Edge>> &g, vector<vector<Edge>> &ming) {
	ll n = g.size();
	ming.assign(n, vector<Edge>());
 	UnionFind uf(n);
	priority_queue<Edge, vector<Edge>, function<bool(Edge,Edge)>> que(
		[](const Edge &x, const Edge &y){
			return x.cost != y.cost ? (x.cost > y.cost) : 
				(x.from != y.from ? x.from < y.from : x.to < y.to);
		}
	);
	for(ll u=0; u<n; u++){
		for(Edge e:g[u]){
			if (u < e.to) que.push(e);
		}
	}

	ll edgecount = 0;
	ll mincost = 0;
	while (edgecount < n-1 && !que.empty()) {
		Edge e = que.top();
		que.pop();
		if (uf.unite(e.from, e.to)) {
			edgecount++;
			ming[e.from].push_back(e);
			mincost += e.cost;
		}
	}
	return mincost;
}


template< typename G >
class DoublingLowestCommonAncestor {
public:
	const G &g;
	vector< int > depth;//depth[v]=根からvまでの距離
	const int LOG;
	vector< vector< int > > table;

	DoublingLowestCommonAncestor(const G &g) : g(g), depth(g.size()), LOG(32 - __builtin_clz(g.size())) {
		table.assign(LOG, vector< int >(g.size(), -1));
	}

	void dfs(int idx, int par, int d) {
		table[0][idx] = par;
		depth[idx] = d;
		for(auto &to : g[idx]) {
			if(to != par) dfs(to, idx, d + 1);
		}
	}

	void build(int v) {
		dfs(v, -1, 0);
		for(int k = 0; k + 1 < LOG; k++) {
			for(int i = 0,tablesize = table[k].size(); i < tablesize; i++) {
				if(table[k][i] == -1) table[k + 1][i] = -1;
				else table[k + 1][i] = table[k][table[k][i]];
			}
		}
	}

	int query(int u, int v) {
		if(depth[u] > depth[v]) swap(u, v);
		for(int i = LOG - 1; i >= 0; i--) {
			if(((depth[v] - depth[u]) >> i) & 1) v = table[i][v];
		}
		if(u == v) return u;
		for(int i = LOG - 1; i >= 0; i--) {
			if(table[i][u] != table[i][v]) {
				u = table[i][u];
				v = table[i][v];
			}
		}
		return table[0][u];
	}
};

void dijkstra(const vector<vector<Edge>> &g, ll start, vector<ll> &dist, vector<ll> &prev) {
	ll gsize = g.size();
	dist.assign(gsize, INF); dist[start] = 0;
	prev.assign(gsize, -1);

	priority_queue<Edge, vector<Edge>, function<bool(Edge,Edge)>> que(
		[](const Edge &x, const Edge &y){
			return x.cost != y.cost ? (x.cost > y.cost) : 
				(x.from != y.from ? x.from < y.from : x.to < y.to);
		}
	);
	que.push(Edge(-1, start, 0));
	while (!que.empty()) {
		Edge e = que.top();
		que.pop();
		if (prev[e.to] != -1) continue;
		prev[e.to] = e.from;
		for(Edge f : g[e.to]) {
			if (dist[f.to] > e.cost + f.cost) {
				dist[f.to] = e.cost + f.cost;
				que.push(Edge(f.from, f.to, e.cost+f.cost));
			}
		}
	}
}

/*-----8<-----library-----8<-----*/

ll N;
vector<ll> A;

void solve() {
	cin>>N;
	vector<ll> X(N,0);
	rep(i,N)cin>>X[i];
	vector<ll> A(N-1,0),B(N-1,0),C(N-1,0);
	rep(i,N-1){
		cin>>A[i]>>B[i]>>C[i];
		A[i]--;B[i]--;
	}

	vector<vector<ll>> g(N);
	rep(i,N-1){
		g[A[i]].push_back(B[i]);
		g[B[i]].push_back(A[i]);
	}
	DoublingLowestCommonAncestor< vector<vector<ll>> > lca(g);
	lca.build(0);

	vector<vector<Edge>> gc(N);
	rep(i,N-1){
		Edge e(A[i],B[i],C[i]);
		Edge f(B[i],A[i],C[i]);
		gc[A[i]].push_back(e);
		gc[B[i]].push_back(f);
	}
	vector<ll> dist, prev;
	dijkstra(gc, 0, dist, prev);
	auto getdist = [&](ll x,ll y){
		ll lcav=lca.query(x,y);
		return dist[x]+dist[y]-2*dist[lcav];
	};

	vector<vector<Edge>> g1(2*N);
	rep(i,N){
		Edge e(i, N+i, X[i]);
		g1[e.from].push_back(e);
		Edge f(N+i, i, X[i]);
		g1[f.from].push_back(f);
	}
	rep(i,N-1){
		ll d=getdist(A[i],B[i]);
		Edge e(N+A[i], N+B[i], d);
		g1[e.from].push_back(e);
		Edge f(N+B[i], N+A[i], d);
		g1[f.from].push_back(f);
	}

	vector<ll> startv;
	rep(i,N)startv.push_back(i);
	vector<Edge> volg = voronoi(g1, startv);


	vector<vector<Edge>> g2(N);
	for(Edge e : volg){
		ll d=getdist(e.from,e.to);
		Edge f1(e.from, e.to, d+X[e.from]+X[e.to]);
		g2[f1.from].push_back(f1);
		Edge f2(e.to, e.from, d+X[e.from]+X[e.to]);
		g2[f2.from].push_back(f2);
	}

	vector<vector<Edge>> ming;
	ll mincost = kruskal(g2, ming);
	printf("%lld\n", mincost);
}

signed main() {
	solve();
	return 0;
}