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 Rust (1.15.1) (judge verdict: Accepted)

use std::io;
use std::str::FromStr;
use std::collections::BinaryHeap;
use std::cmp::Ordering;

type Pair = (usize, usize);

fn input<T:FromStr>() -> T {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    line.trim().parse().ok().unwrap()
}

fn input_tuple() -> (usize, usize) {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    let vec: Vec<usize> = line.split_whitespace().map(|s| s.parse().unwrap()).collect();
    (vec[0], vec[1])
}

#[derive(Eq, Clone, Debug)]
struct Edge {
    from: usize,
    to: usize,
    cost: i64,
}

impl Ord for Edge {
    fn cmp(&self, other: &Self) -> Ordering {
        self.cost.cmp(&other.cost)
    }
}

impl PartialOrd for Edge {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl PartialEq for Edge {
    fn eq(&self, other: &Self) -> bool {
        self.cost == other.cost
    }
}

#[derive(Debug, PartialEq, Clone)]
struct Node {
    parent: Option<usize>, 
    size: usize,
}

#[derive(Debug, PartialEq)]
pub struct UnionFind {
    tree: Vec<Node>,
}

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        UnionFind {tree: vec![Node {parent: None, size: 1}; n]}
    }

    pub fn unite(&mut self, i: usize, j: usize) {
        let root_i = self.find(i);
        let root_j = self.find(j);
        if root_i == root_j {
            return
        }

        if self.tree[root_i].size < self.tree[root_j].size {
            self.tree[root_i].size += self.tree[root_j].size;
            self.tree[root_j].parent = Some(root_i);
        } else {
            self.tree[root_j].size += self.tree[root_i].size;
            self.tree[root_i].parent = Some(root_j);
        }

    }

    pub fn find(&mut self, i: usize) -> usize {
        match self.tree[i].parent {
            Some(p) => {
                let root = self.find(p);
                self.tree[i].parent = Some(root);
                root
            },
            None => i
        }
    }

    pub fn equiv(&mut self, i: usize, j: usize) -> bool {
        self.find(i) == self.find(j)
    }

    pub fn size(&mut self, i: usize) -> usize {
        let root = self.find(i);
        self.tree[root].size
    }
}

fn kruskal(n: usize, x_dist: &mut BinaryHeap<Edge>, y_dist: &mut BinaryHeap<Edge>) -> i64 {
    let mut union_find = UnionFind::new(n);
    let mut cost = 0;
    while x_dist.len() > 0 || y_dist.len() > 0 {
        let edge: Edge;
        if x_dist.is_empty() {
            edge = y_dist.pop().unwrap();
        } else if y_dist.is_empty() {
            edge = x_dist.pop().unwrap();
        } else {
            if x_dist.peek().unwrap().cost < y_dist.peek().unwrap().cost {
                edge = y_dist.pop().unwrap()
            } else {
                edge = x_dist.pop().unwrap();
            }
        }
        // println!("{:?}", edge);
        
        if !union_find.equiv(edge.from, edge.to) {
            cost += edge.cost;
            union_find.unite(edge.from, edge.to);
        }
    }
    cost
}

fn main() {
    let n: usize = input();
    let mut x_pos: Vec<Pair> = Vec::new();
    let mut y_pos: Vec<Pair> = Vec::new();
    let mut x_dist: BinaryHeap<Edge> = BinaryHeap::new();
    let mut y_dist: BinaryHeap<Edge> = BinaryHeap::new();
    for i in 0..n {
        let (x, y) = input_tuple();
        x_pos.push((i, x)); y_pos.push((i, y));
    }
    x_pos.sort_by(|a, b| a.1.cmp(&b.1));
    y_pos.sort_by(|a, b| a.1.cmp(&b.1));

    for i in 0..(n-1) {
        x_dist.push(Edge{from: x_pos[i].0, to: x_pos[i+1].0, cost: x_pos[i].1 as i64 - x_pos[i+1].1 as i64});
        y_dist.push(Edge{from: y_pos[i].0, to: y_pos[i+1].0, cost: y_pos[i].1 as i64 - y_pos[i+1].1 as i64});
    }

    // println!("{:?}", x_dist);
    // println!("{:?}", y_dist);

    println!("{}", -kruskal(n, &mut x_dist, &mut y_dist));

}