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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;
using System.Threading;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var s = new string[n];
        var c = new int[n];
        for (int i = 0; i < n; i++) {
            s[i] = ns();
            c[i] = ni();
        }
        var isPalindrome = new bool[n, 22, 22];
        for (int i = 0; i < n; i++) {
            var ss = s[i];
            for (int k = 0; k < 2; k++) {
                for (int j = 0; j + k < ss.Length; j++) {
                    int left = j;
                    int right = j + k;
                    while (0 <= left && right < ss.Length && ss[left] == ss[right]) {
                        isPalindrome[i, left, right] = true;
                        left--;
                        right++;
                    }
                }
            }
        }

        for (int _ = 0; _ < 2; _++) {

        }
        var visited = new bool[n, 22, 22, 2];
        var inf = long.MaxValue;

        var pq = new PriorityQueue<(long cost, int k, int left, int right, int leftToRight)>();
        long ans = inf;

        for (int i = 0; i < n; i++) {
            pq.Enqueue((c[i], i, 0, s[i].Length - 1, 1));
        }

        while (pq.Count > 0) {
            var (cost, k, left, right, leftToRight) = pq.Dequeue();
            if (left == right + 1 || isPalindrome[k, left, right]) {
                ans = cost;
                break;
            }
            if (left > right) continue;
            if (visited[k, left, right, leftToRight]) continue;
            visited[k, left, right, leftToRight] = true;
            var f = s[k];
            for (int j = 0; j < n; j++) {
                var t = s[j];
                var maxMatch = Math.Min(right - left + 1, t.Length);
                int match = 0;
                if (leftToRight == 1) {
                    for (int i = left, p = t.Length - 1; i <= right && p >= 0; i++, p--) {
                        if (f[i] == t[p]) {
                            match++;
                        } else {
                            break;
                        }
                    }
                    if (match == maxMatch) {
                        if (left + match <= right) {
                            pq.Enqueue((cost + c[j], k, left + match, right, 1));
                        } else {
                            pq.Enqueue((cost + c[j], j, 0, t.Length - 1 - match, 0));
                        }
                    }
                } else {
                    for (int i = right, p = 0; i >= left && p < t.Length; i--, p++) {
                        if (f[i] == t[p]) {
                            match++;
                        } else {
                            break;
                        }
                    }
                    if (match == maxMatch) {
                        if (left <= right - match) {
                            pq.Enqueue((cost + c[j], k, left, right - match, 0));
                        } else {
                            pq.Enqueue((cost + c[j], j, match, t.Length - 1, 1));
                        }
                    }
                }
            }
        }

        if (ans >= inf) ans = -1;
        cout.WriteLine(ans);
    }
}

public class PriorityQueue<T> {
    private T[] _array = new T[128];
    private int _size = 0;
    private readonly Comparison<T> _compare;

    public PriorityQueue() {
        _compare = Comparer<T>.Default.Compare;
    }

    public PriorityQueue(Comparison<T> comp) {
        _compare = comp;
    }

    public int Count { get { return _size; } }

    private void Swap(int a, int b) {
        var t = _array[a];
        _array[a] = _array[b];
        _array[b] = t;
    }

    private void Expand() {
        var newlist = new T[_array.Length << 1];
        Array.Copy(_array, newlist, _array.Length);
        _array = newlist;
    }

    public bool Any() {
        return _size > 0;
    }

    public void Enqueue(T newValue) {
        int pos = ++_size;
        if (_size >= _array.Length) {
            Expand();
        }
        _array[pos] = newValue;
        while (pos > 1) {
            int parent = pos >> 1;
            if (_compare(_array[parent], _array[pos]) > 0) {
                Swap(parent, pos);
                pos = parent;
            } else {
                break;
            }
        }
    }

    public T Peek() {
        return _array[1];
    }

    public T Dequeue() {
        var top = _array[1];
        int pos = 1;
        _array[pos] = _array[_size--];
        while ((pos << 1) <= _size) {
            int left = pos << 1;
            int right = left | 1;
            int next = left;
            if (right <= _size && _compare(_array[left], _array[right]) > 0) {
                next = right;
            }
            if (_compare(_array[pos], _array[next]) > 0) {
                Swap(next, pos);
                pos = next;
            } else {
                break;
            }
        }

        return top;
    }
};
// PREWRITEN CODE BEGINS FROM HERE

static public class StringExtensions {
    static public string JoinToString<T>(this IEnumerable<T> source, string separator = " ") {
        return string.Join(separator, source);
    }
}

internal partial class Solver : Scanner {
    static readonly int? StackSizeInMebiByte = null; //50;
    public static void StartAndJoin(Action action, int maxStackSize) {
        var thread = new Thread(new ThreadStart(action), maxStackSize);
        thread.Start();
        thread.Join();
    }

    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        if (StackSizeInMebiByte.HasValue) {
            StartAndJoin(() => new Solver(Console.In, Console.Out).Run(), StackSizeInMebiByte.Value * 1024 * 1024);
        } else {
            new Solver(Console.In, Console.Out).Run();
        }
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
    private readonly TextWriter cerr;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
        cerr = Console.Error;
    }

    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

#if DEBUG
internal static class LinqPadExtension {
    public static string TextDump<T>(this T obj) {
        if (obj is IEnumerable) return (obj as IEnumerable).Cast<object>().JoinToString().Dump();
        else return obj.ToString().Dump();
    }
    public static T Dump<T>(this T obj) {
        return LINQPad.Extensions.Dump(obj);
    }
}
#endif

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}