Problem:
Score : 500 points

Problem Statement

There are N towns on a plane. The i-th town is located at the coordinates (x_i,y_i). There may be more than one town at the same coordinates.

You can build a road between two towns at coordinates (a,b) and (c,d) for a cost of min(|a-c|,|b-d|) yen (the currency of Japan). It is not possible to build other types of roads.

Your objective is to build roads so that it will be possible to travel between every pair of towns by traversing roads. At least how much money is necessary to achieve this?

Constraints

2 ≤ N ≤ 10^5

0 ≤ x_i,y_i ≤ 10^9

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
:
x_N y_N

Output

Print the minimum necessary amount of money in order to build roads so that it will be possible to travel between every pair of towns by traversing roads.

Sample Input 1

3
1 5
3 9
7 8

Sample Output 1

3

Build a road between Towns 1 and 2, and another between Towns 2 and 3. The total cost is 2+1=3 yen.

Sample Input 2

6
8 3
4 9
12 19
18 1
13 5
7 6

Sample Output 2

8

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

#include <bits/stdc++.h>

#define TYPE(c)   remove_reference_t<decltype(c)>
#define REP(i, n) for(TYPE(n) i = 0; i < n; i++)
#define FOR(v, c) for(TYPE(c.begin()) v = c.begin(); v != c.end(); v++)
#define ALL(c)    c.begin(), c.end()
#define SORT(c)   sort(ALL(c))
#define RSORT(c)  sort(ALL(c), greater<TYPE(c)::value_type>())
#define UNIQUE(c) c.erase(unique(ALL(c)), c.end())

using namespace std;
using ll = long long;

constexpr int MOD  = (int)1e9 + 7;
constexpr int INF  = (int)1e9 + 1;
constexpr ll  LINF = (ll)1e18 + 1;
template<typename S, typename T> ostream& operator<<(ostream &os, const pair<S, T>& p)
{ return os << "(" << p.first << "," << p.second << ")"; }
template<typename T> auto operator<<(ostream& os, const T& t)
    -> typename conditional<true, ostream&, decltype(*t.begin())>::type
{ os << "["; FOR(v, t) { if(v != t.begin()) os << ","; os << *v; } os << "]"; return os; }

class Graph {
    using ll = long long;
public:
    struct Edge {
        struct Info {
            ll   cost{1};
            bool validity{true};
            Info() = default;
            explicit Info(const ll& _cost) : cost(_cost) { }
            Info(const ll& _cost, const bool& _validity) : cost(_cost), validity(_validity) { }
            friend std::ostream& operator <<(std::ostream& os, const std::shared_ptr<Info>& i) {
                os << "cost : " << i->cost << ", " << "validity : " << i->validity;
                return os;
            }
        };

        enum class Type { Directed, Undirected };

        int src, dst;
        std::shared_ptr<Info> info;

        Edge(const int& _src, const int& _dst, const std::shared_ptr<Info>& _info) : src(_src), dst(_dst), info(_info) { }
        friend std::ostream& operator <<(std::ostream& os, const Edge& e) {
            os << "src : " << e.src << ", " << "dst : " << e.dst << ", " << "info : [" << e.info << "]";
            return os;
        }
    };

    struct Vertex {
        ll cost;
        explicit Vertex(const ll& _cost) : cost(_cost) { }
        friend std::ostream& operator <<(std::ostream& os, const Vertex& v) {
            os << "cost : " << v.cost;
            return os;
        }
    };

    const Edge::Type edge_type;

    std::vector<Vertex>            vertices;
    std::vector<std::vector<Edge>> edges;

    Graph(const int&        num_vertex,
          const Edge::Type& edge_type,
          const ll&         vertex_init_cost = (ll)1e9 + 1) :
        edge_type(edge_type),
        vertices(num_vertex, Vertex(vertex_init_cost)),
        edges(num_vertex) { }

    void setCostToAllVertex(const ll& val) {
        std::fill(vertices.begin(), vertices.end(), Vertex(val));
    };

    void addEdge(const int& src, const int& dst, const ll& cost = 1) {
        auto info = std::make_shared<Edge::Info>(cost);
        edges[src].emplace_back(src, dst, info);
        if(edge_type == Edge::Type::Undirected) edges[dst].emplace_back(dst, src, info);
    }

    void invalidateEdge(const int& src, const int& dst) {
        for(auto& edge : edges[src]) if(edge.dst == dst) { edge.info->validity = false; return; }
    }

    friend std::ostream& operator <<(std::ostream& os, const Graph& g) {
        os << std::endl
           << "--- Vertex Information ---" << std::endl;
        for(size_t i = 0; i < g.vertices.size(); i++) {  os << "[[" << i << "] " << g.vertices[i] << "]" << std::endl; }
        os << "---- Edge Information ----" << std::endl;
        for(const auto& v : g.edges) { for(const auto& e : v) { os << "[" << e << "]" << std::endl; } }
        os << "--------------------------";
        return os;
    }
};

/**
 *  最小全域木を求める(Kruskal法)
 *  NOTE: 計算量O(|E|log|V|), Dependencies: comp-Graph,
 *  @Return: 最小全域木を返す グラフが非連結の場合、連結成分ごとに最小全域木を構成する
 */
// 使用例
// auto mst = kruskal(g);
// std::cout << mst.getTotalCost() << std::endl;
Graph kruskal(const Graph& g) {
    if(g.edge_type != Graph::Edge::Type::Undirected) {
        throw std::logic_error("Graph type is invalid");
    }
    Graph mst(g.vertices.size(), Graph::Edge::Type::Undirected);
    std::vector<Graph::Edge> mst_e;
    mst_e.reserve(g.edges.size());
    for(const auto& ee : g.edges) for(const auto& e : ee) mst_e.push_back(e);
    auto sort_rule = [](const Graph::Edge& l, const Graph::Edge& r) { return l.info->cost < r.info->cost; };
    sort(mst_e.begin(), mst_e.end(), sort_rule);
    struct UF {
        struct N{ll p,r;N(const ll&_p=0,const ll&_r=0):p(_p),r(_r){}};std::vector<N>n;
        explicit UF(const ll&nn){n.reserve(nn);for(ll i=0;i<nn;i++){n.emplace_back(i);}}
        ll r(const ll&x){return(n[x].p==x)?x:n[x].p=r(n[x].p);}
        void u(const ll&x,const ll&y){ll w=r(x);ll z=r(y);
            if(w!=z){if(n[w].r<n[z].r)std::swap(w,z);n[z].p=w;if(n[w].r==n[z].r)n[w].r++;}}
        bool s(const ll&x,const ll&y){return r(x)==r(y);}
    };
    UF uf(g.vertices.size());
    for(const auto& e : mst_e) {
        if(!uf.s(e.src, e.dst)) {
            mst.addEdge(e.src, e.dst, e.info->cost);
            uf.u(e.src, e.dst);
        }
    }
    return mst;
}

void DFS(Graph&     g,
         const int& index,
         ll&       ans) {
    for(const auto& e : g.edges[index]) {
        if(g.vertices[e.dst].cost == 0) {
            g.vertices[e.dst].cost = 1;
            ans += e.info->cost;
            DFS(g, e.dst, ans);
        }
    }
}

int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);

    int N;
    std::cin >> N;

    vector<pair<pair<int, int>, int>> city_prior_x;
    vector<pair<pair<int, int>, int>> city_prior_y;
    REP(i, N) {
        int x, y;
        std::cin >> x >> y;
        city_prior_x.emplace_back(make_pair(x, y), i);
        city_prior_y.emplace_back(make_pair(y, x), i);
    }

    Graph g(N, Graph::Edge::Type::Undirected);

    sort(ALL(city_prior_x));
    sort(ALL(city_prior_y));
    for(int i = 0; i < N - 1; i++) {
        g.addEdge(city_prior_x[i].second,
                  city_prior_x[i + 1].second,
                  min(abs(city_prior_x[i].first.first  - city_prior_x[i + 1].first.first),
                      abs(city_prior_x[i].first.second - city_prior_x[i + 1].first.second)));
        g.addEdge(city_prior_y[i].second,
                  city_prior_y[i + 1].second,
                  min(abs(city_prior_y[i].first.first  - city_prior_y[i + 1].first.first),
                      abs(city_prior_y[i].first.second - city_prior_y[i + 1].first.second)));
    }

    auto mst = kruskal(g);

    mst.setCostToAllVertex(0);
    mst.vertices[0].cost = 1;
    ll ans = 0;
    DFS(mst, 0, ans);

    std::cout << ans << std::endl;

    return 0;
}