Problem:
Score : 600 points

Problem Statement

We have N strings of lowercase English letters: S_1, S_2, \cdots, S_N.

Takahashi wants to make a string that is a palindrome by choosing one or more of these strings - the same string can be chosen more than once - and concatenating them in some order of his choice.

The cost of using the string S_i once is C_i, and the cost of using it multiple times is C_i multiplied by that number of times.

Find the minimum total cost needed to choose strings so that Takahashi can make a palindrome.

If there is no choice of strings in which he can make a palindrome, print -1.

Constraints

1 \leq N \leq 50

1 \leq |S_i| \leq 20

S_i consists of lowercase English letters.

1 \leq C_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
S_1 C_1
S_2 C_2
:
S_N C_N

Output

Print the minimum total cost needed to choose strings so that Takahashi can make a palindrome, or -1 if there is no such choice.

Sample Input 1

3
ba 3
abc 4
cbaa 5

Sample Output 1

7

We have ba, abc, and cbaa.

For example, we can use ba once and abc once for a cost of 7, then concatenate them in the order abc, ba to make a palindrome.
Also, we can use abc once and cbaa once for a cost of 9, then concatenate them in the order cbaa, abc to make a palindrome.

We cannot make a palindrome for a cost less than 7, so we should print 7.

Sample Input 2

2
abcab 5
cba 3

Sample Output 2

11

We can choose abcab once and cba twice, then concatenate them in the order abcab, cba, cba to make a palindrome for a cost of 11.

Sample Input 3

4
ab 5
cba 3
a 12
ab 10

Sample Output 3

8

We can choose a once, which is already a palindrome, but it is cheaper to concatenate ab and cba.

Sample Input 4

2
abc 1
ab 2

Sample Output 4

-1

We cannot make a palindrome, so we should print -1.

Solution in Rust (1.42.0) (judge verdict: Wrong Answer)

use itertools::Itertools;
use proconio::{input, marker::Bytes};
use std::cmp::Reverse;
use std::collections::{BinaryHeap, HashMap};

fn main() {
    input!(s: [(Bytes, u64)]);

    #[derive(Clone, Debug, PartialEq, Eq, PartialOrd, Ord, Hash)]
    enum Fix {
        Pre(Vec<u8>),
        Suf(Vec<u8>),
    };
    use Fix::*;

    let mut depth = HashMap::new();
    let mut que = BinaryHeap::new();

    for (s, c) in &s {
        let c = *c;
        depth.insert(Pre(s.clone()), c);
        depth.insert(Suf(s.clone()), c);
        que.push((Reverse(c), Pre(s.clone())));
        que.push((Reverse(c), Suf(s.clone())));
    }

    while let Some((Reverse(dv), v)) = que.pop() {
        let adj = s.iter().flat_map(|(s, c)| match v.clone() {
            Pre(v) => {
                let w = &v.iter().copied().rev().collect_vec();
                if s.ends_with(w) {
                    Some((Suf(s[..s.len() - v.len()].into()), c))
                } else if w.ends_with(s) {
                    Some((Pre(v[..s.len()].into()), c))
                } else {
                    None
                }
            }
            Suf(v) => {
                let w = &v.iter().copied().rev().collect_vec();
                if s.starts_with(w) {
                    Some((Pre(s[v.len()..].into()), c))
                } else if w.starts_with(s) {
                    Some((Suf(v[..v.len() - s.len()].into()), c))
                } else {
                    None
                }
            }
        });

        for (w, &c) in adj {
            let alt = dv + c;
            if let Some(dw) = depth.get_mut(&w) {
                if *dw > alt {
                    *dw = alt;
                    que.push((Reverse(alt), w.clone()));
                }
            } else {
                depth.insert(w.clone(), alt);
                que.push((Reverse(alt), w.clone()));
            }
        }
    }

    let ans = depth
        .iter()
        .filter(|&(s, _)| {
            let s = match s {
                Pre(s) => s,
                Suf(s) => s,
            };
            s.iter().eq(s.iter().rev())
        })
        .map(|(_, d)| d)
        .min();
    if let Some(ans) = ans {
        println!("{}", ans)
    } else {
        println!("-1")
    }
}