Problem:
Score : 700 points

Problem Statement

We have a directed weighted graph with N vertices.
Each vertex has two integers written on it, and the integers written on Vertex i are A_i and B_i.

In this graph, there is an edge from Vertex x to Vertex y for all pairs 1 \leq x,y \leq N, and its weight is {\rm min}(A_x,B_y).

We will consider a directed cycle in this graph that visits every vertex exactly once.
Find the minimum total weight of the edges in such a cycle.

Constraints

2 \leq N \leq 10^5

1 \leq A_i \leq 10^9

1 \leq B_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 B_1
A_2 B_2
:
A_N B_N

Output

Print the minimum total weight of the edges in such a cycle.

Sample Input 1

3
1 5
4 2
6 3

Sample Output 1

7

Consider the cycle 1→3→2→1. The weights of those edges are {\rm min}(A_1,B_3)=1, {\rm min}(A_3,B_2)=2 and {\rm min}(A_2,B_1)=4, for a total of 7.
As there is no cycle with a total weight of less than 7, the answer is 7.

Sample Input 2

4
1 5
2 6
3 7
4 8

Sample Output 2

10

Sample Input 3

6
19 92
64 64
78 48
57 33
73 6
95 73

Sample Output 3

227

Solution in Rust (1.42.0) (judge verdict: Accepted)

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());
    let n: usize = sc.read();
    let mut edges = Vec::with_capacity(2 * n);
    for i in 0..n {
        let a: u64 = sc.read();
        let b: u64 = sc.read();
        edges.push((a, i, true));
        edges.push((b, i, false));
    }
    edges.sort_by_key(|e| e.0);

    let mut candidates = vec![];
    if let Some(ans) = calc(&edges) {
        candidates.push(ans);
    }

    edges.swap(n, n - 1);
    if let Some(ans) = calc(&edges) {
        candidates.push(ans);
    }
    edges.swap(n, n - 1);

    edges.swap(n - 2, n);
    if let Some(ans) = calc(&edges) {
        candidates.push(ans);
    }
    edges.swap(n - 2, n);

    edges.swap(n - 1, n + 1);
    if let Some(ans) = calc(&edges) {
        candidates.push(ans);
    }
    edges.swap(n - 1, n + 1);

    candidates.sort();
    println!("{}", candidates[0]);
}

fn calc(edges: &[(u64, usize, bool)]) -> Option<u64> {
    let n = edges.len() / 2;
    let mut a_count = 0;
    let mut count = vec![0; n];
    let mut sum = 0;
    for i in 0..n {
        let (cost, i, is_a) = edges[i];
        count[i] += 1;
        sum += cost;
        if is_a {
            a_count += 1;
        }
    }
    if count.iter().any(|&e| e >= 2) || a_count == n || a_count == 0 {
        Some(sum)
    } else {
        None
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        Self(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}