Problem:
Score : 600 points

Problem Statement

There are N points in a two-dimensional plane. The initial coordinates of the i-th point are (x_i, y_i). Now, each point starts moving at a speed of 1 per second, in a direction parallel to the x- or y- axis. You are given a character d_i that represents the specific direction in which the i-th point moves, as follows:

If d_i = R, the i-th point moves in the positive x direction;

If d_i = L, the i-th point moves in the negative x direction;

If d_i = U, the i-th point moves in the positive y direction;

If d_i = D, the i-th point moves in the negative y direction.

You can stop all the points at some moment of your choice after they start moving (including the moment they start moving).
Then, let x_{max} and x_{min} be the maximum and minimum among the x-coordinates of the N points, respectively. Similarly, let y_{max} and y_{min} be the maximum and minimum among the y-coordinates of the N points, respectively.

Find the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}) and print it.

Constraints

1 \leq N \leq 10^5

-10^8 \leq x_i,\ y_i \leq 10^8

x_i and y_i are integers.

d_i is R, L, U, or D.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 d_1
x_2 y_2 d_2
.
.
.
x_N y_N d_N

Output

Print the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}).

The output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-9}.

Sample Input 1

2
0 3 D
3 0 L

Sample Output 1

0

After three seconds, the two points will meet at the origin. The value in question will be 0 at that moment.

Sample Input 2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Sample Output 2

97.5

The answer may not be an integer.

Sample Input 3

20
6 -10 R
-4 -9 U
9 6 D
-3 -2 R
0 7 D
4 5 D
10 -10 U
-1 -8 U
10 -6 D
8 -5 U
6 4 D
0 3 D
7 9 R
9 -4 R
3 10 D
1 9 U
1 -6 U
9 -8 R
6 7 D
7 -3 D

Sample Output 3

273

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text;
using E = System.Linq.Enumerable;

partial class Solver
{
    public int[] dx = new int[] { 0, 1, 0, -1 };
    public int[] dy = new int[] { 1, 0, -1, 0 };
    public string ds = "URDL";

    long[] GetFirstAndLast(long[] array)
    {
        Array.Sort(array);
        if (array.Length <= 2) return array;
        return new[] { array[0], array[array.Length - 1] };
    }

    public double GetConflictTime(long x1, long dx1, long x2, long dx2)
    {
        if (dx1 == dx2) return 0;
        if (dx1 < dx2) return GetConflictTime(x2, dx2, x1, dx1);
        var d = dx1 - dx2;
        return Math.Max(0, 1.0 * (x2 - x1) / d);
    }

    public void Run()
    {
        var N = ni();
        var x = new long[N];
        var y = new long[N];
        var d = new string[N];
        for (int i = 0; i < N; i++)
        {
            x[i] = ni();
            y[i] = ni();
            d[i] = ns();
        }

        double ans = 1e100;

        var candidates = new List<double>();

        candidates.Add(0);

        {
            var xinfo = E.Range(0, N)
                .Select(i => new { X = x[i], D = dx[ds.IndexOf(d[i][0])] })
                .GroupBy(p => p.D)
                .Select(p => new { D = p.Key, Array = p.Select(v => v.X).ToArray() })
                .SelectMany(p => GetFirstAndLast(p.Array).Select(xx => new { p.D, X = xx }))
                .ToArray();

            for (int i = 0; i < xinfo.Length; i++)
            {
                for (int j = 0; j < i; j++)
                {
                    candidates.Add(GetConflictTime(xinfo[i].X, xinfo[i].D, xinfo[j].X, xinfo[j].D));
                }
            }
        }

        {
            var yinfo = E.Range(0, N)
                .Select(i => new { Y = y[i], D = dy[ds.IndexOf(d[i][0])] })
                .GroupBy(p => p.D)
                .Select(p => new { D = p.Key, Array = p.Select(v => v.Y).ToArray() })
                .SelectMany(p => GetFirstAndLast(p.Array).Select(yy => new { p.D, Y = yy }))
                .ToArray();

            for (int i = 0; i < yinfo.Length; i++)
            {
                for (int j = 0; j < i; j++)
                {
                    candidates.Add(GetConflictTime(yinfo[i].Y, yinfo[i].D, yinfo[j].Y, yinfo[j].D));
                }
            }
        }


        ans = candidates.Distinct().Min(t => GetAreaAt(N, x, y, d, t));

        cout.WriteLine("{0:0.0000000000000}", ans);
    }

    private double GetAreaAt(int N, long[] x, long[] y, string[] d, double mid)
    {
        double xmin = 1e100;
        double ymin = 1e100;
        double xmax = -1e100;
        double ymax = -1e100;

        for (int i = 0; i < N; i++)
        {
            var dindex = ds.IndexOf(d[i][0]);
            var nx = x[i] + dx[dindex] * mid;
            var ny = y[i] + dy[dindex] * mid;
            xmin = Math.Min(xmin, nx);
            ymin = Math.Min(ymin, ny);
            xmax = Math.Max(xmax, nx);
            ymax = Math.Max(ymax, ny);
        }

        return (ymax - ymin) * (xmax - xmin);
    }
}

// PREWRITEN CODE BEGINS FROM HERE
partial class Solver : Scanner
{
    public static void Main(string[] args)
    {
#if LOCAL
        new Solver(Console.In, Console.Out).Run();
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

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

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

public static class LinqPadExtension
{
    static public T Dump<T>(this T obj)
    {
#if LOCAL
        return LINQPad.Extensions.Dump(obj);
#else
        return obj;
#endif
    }
}
public class Scanner
{
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

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

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

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size)
    {
        var array = new string[size];
        for (int i = 0; i < size; i++) array[i] = Next();
        return array;
    }
    public int[] NextIntArray(int size)
    {
        var array = new int[size];
        for (int i = 0; i < size; i++) array[i] = NextInt();
        return array;
    }

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

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

    public string Next()
    {
        if (TokenQueue.Count == 0)
        {
            if (!StockTokens()) throw new InvalidOperationException();
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext()
    {
        if (TokenQueue.Count > 0)
            return true;
        return StockTokens();
    }
    static readonly char[] _separator = new[] { ' ' };
    private bool StockTokens()
    {
        while (true)
        {
            var line = Reader.ReadLine();
            if (line == null) return false;
            var tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) continue;
            foreach (var token in tokens)
                TokenQueue.Enqueue(token);
            return true;
        }
    }
}