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 (1.42.0) (judge verdict: Accepted)

#[allow(unused_macros)]
macro_rules ! read_value {($ next : expr , ($ ($ t : tt ) ,* ) ) => {($ (read_value ! ($ next , $ t ) ) ,* ) } ; ($ next : expr , [$ t : tt ; $ len : expr ] ) => {(0 ..$ len ) . map (| _ | read_value ! ($ next , $ t ) ) . collect ::< Vec < _ >> () } ; ($ next : expr , chars ) => {read_value ! ($ next , String ) . chars () . collect ::< Vec < char >> () } ; ($ next : expr , usize1 ) => {read_value ! ($ next , usize ) - 1 } ; ($ next : expr , $ t : ty ) => {$ next () . parse ::<$ t > () . expect ("Parse error" ) } ; }
#[allow(unused_macros)]
macro_rules ! input_inner {($ next : expr ) => {} ; ($ next : expr , ) => {} ; ($ next : expr , $ var : ident : $ t : tt $ ($ r : tt ) * ) => {let $ var = read_value ! ($ next , $ t ) ; input_inner ! {$ next $ ($ r ) * } } ; }
#[allow(unused_macros)]
macro_rules ! input {(source = $ s : expr , $ ($ r : tt ) * ) => {let mut iter = $ s . split_whitespace () ; let mut next = || {iter . next () . unwrap () } ; input_inner ! {next , $ ($ r ) * } } ; ($ ($ r : tt ) * ) => {let stdin = std :: io :: stdin () ; let mut bytes = std :: io :: Read :: bytes (std :: io :: BufReader :: new (stdin . lock () ) ) ; let mut next = move || -> String {bytes . by_ref () . map (| r | r . unwrap () as char ) . skip_while (| c | c . is_whitespace () ) . take_while (| c |! c . is_whitespace () ) . collect () } ; input_inner ! {next , $ ($ r ) * } } ; }

#[allow(dead_code)]
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
struct Edge {
    src: usize,
    dst: usize,
    weight: i32,
}

use std::cmp::Ordering;
impl Ord for Edge {
    fn cmp(&self, other: &Self) -> Ordering {
        if self.weight != other.weight {
            self.weight.cmp(&other.weight)
        } else if self.src != other.src {
            self.src.cmp(&other.src)
        } else {
            self.dst.cmp(&other.dst)
        }
    }
}

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

#[allow(dead_code)]
#[derive(Debug, Clone, PartialEq, Eq)]
struct Graph {
    g: Vec<Vec<Edge>>
}

#[allow(dead_code)]
impl Graph {
    fn new(n: usize) -> Self {
        Graph{ g: vec![vec![]; n] }
    }

    fn order(&self) -> usize {
        self.g.len()
    }

    fn add_edge(&mut self, src: usize, dst: usize, weight: i32) {
        self.g[src].push(Edge{ src, dst, weight });
        self.g[dst].push(Edge{ src: dst, dst: src, weight });
    }
}

use std::ops::Index;
impl Index<usize> for Graph {
    type Output = Vec<Edge>;

    fn index(&self, v: usize) -> &Self::Output {
        &self.g[v]
    }
}

use std::cmp::Reverse;

#[allow(dead_code)]
/// Prim's algorithm, O(E + V log V)
fn minimum_spanning_tree(g: &Graph, root: usize) -> (i32, Vec<Edge>) {
    let n = g.order();
    let mut total = 0;

    let mut visited = vec![false; n];
    let mut mst = vec![];
    let mut que = std::collections::BinaryHeap::new();  // max-heap
    que.push(Reverse(Edge{ src: root, dst: root, weight: 0 }));
    while let Some(Reverse(e)) = que.pop() {
        if visited[e.dst] {
            continue;
        }
        mst.push(e);
        total += e.weight;
        visited[e.dst] = true;
        for f in g[e.dst].iter() {
            if !visited[f.dst] {
                que.push(Reverse(*f));
            }
        }
    }
    (total, mst)
}

fn main() {
    input! {
        n: usize,
        xy: [(i32, i32); n],
    }

    let mut xpath = xy.iter().enumerate().map(|(i, &(x, _))| (x, i)).collect::<Vec<_>>();
    let mut ypath = xy.iter().enumerate().map(|(i, &(_, y))| (y, i)).collect::<Vec<_>>();
    xpath.sort();
    ypath.sort();

    let mut g = Graph::new(n);
    for i in 1..n {
        g.add_edge(xpath[i - 1].1, xpath[i].1, xpath[i].0 - xpath[i - 1].0);
        g.add_edge(ypath[i - 1].1, ypath[i].1, ypath[i].0 - ypath[i - 1].0);
    }
    let (ret, _) = minimum_spanning_tree(&g, 0);

    println!("{}", ret);
}