Problem:
Score : 1600 points

Problem Statement

A balancing network is an abstract device built up of N wires, thought of as running from left to right,
and M balancers that connect pairs of wires. The wires are numbered from 1 to N from top to bottom,
and the balancers are numbered from 1 to M from left to right.
Balancer i connects wires x_i and y_i (x_i < y_i).

Each balancer must be in one of two states: up or down.

Consider a token that starts moving to the right along some wire at a point to the left of all balancers.
During the process, the token passes through each balancer exactly once.
Whenever the token encounters balancer i, the following happens:

If the token is moving along wire x_i and balancer i is in the down state, the token moves down to wire y_i and continues moving to the right.

If the token is moving along wire y_i and balancer i is in the up state, the token moves up to wire x_i and continues moving to the right.

Otherwise, the token doesn't change the wire it's moving along.

Let a state of the balancing network be a string of length M, describing the states of all balancers.
The i-th character is ^ if balancer i is in the up state, and v if balancer i is in the down state.

A state of the balancing network is called uniforming if a wire w exists such that, regardless of the starting wire,
the token will always end up at wire w and run to infinity along it. Any other state is called non-uniforming.

You are given an integer T (1 \le T \le 2). Answer the following question:

If T = 1, find any uniforming state of the network or determine that it doesn't exist.

If T = 2, find any non-uniforming state of the network or determine that it doesn't exist.

Note that if you answer just one kind of questions correctly, you can get partial score.

Constraints

2 \leq N \leq 50000

1 \leq M \leq 100000

1 \leq T \leq 2

1 \leq x_i < y_i \leq N

All input values are integers.

Partial Score

800 points will be awarded for passing the testset satisfying T = 1.

800 points will be awarded for passing the testset satisfying T = 2.

Input

Input is given from Standard Input in the following format:

N M T
x_1 y_1
x_2 y_2
:
x_M y_M

Output

Print any uniforming state of the given network if T = 1, or any non-uniforming state if T = 2. If the required state doesn't exist, output -1.

Sample Input 1

4 5 1
1 3
2 4
1 2
3 4
2 3

Sample Output 1

^^^^^

This state is uniforming: regardless of the starting wire, the token ends up at wire 1.

Sample Input 2

4 5 2
1 3
2 4
1 2
3 4
2 3

Sample Output 2

v^^^^

This state is non-uniforming: depending on the starting wire, the token might end up at wire 1 or wire 2.

Sample Input 3

3 1 1
1 2

Sample Output 3

-1

A uniforming state doesn't exist.

Sample Input 4

2 1 2
1 2

Sample Output 4

-1

A non-uniforming state doesn't exist.

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

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

using E = System.Linq.Enumerable;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var m = ni();
        var t = ni();
        var x = new int[m];
        var y = new int[m];
        for (int i = 0; i < m; i++) {
            x[i] = ni() - 1;
            y[i] = ni() - 1;
        }

        if (t == 1) cout.WriteLine(Solve1(n, x, y));
    }

    class History {
        public int Index;
        public BitSet Before, After;
    }

    private string Solve1(int n, int[] x, int[] y) {
        var dp = E.Range(0, n).Select(_ => new BitSet(n)).ToArray();

        for (int i = 0; i < n; i++) {
            dp[i][i] = true;
        }

        for (int i = 0; i < x.Length; i++) {
            dp[x[i]] = dp[y[i]] = dp[x[i]] | dp[y[i]];
        }

        int goal = -1;

        for (int i = 0; i < n; i++) {
            if (dp[i].CountBit() == n) {
                goal = i;
                break;
            }
        }

        if (goal == -1) return "-1";

        var ans = new char[x.Length];

        var visited = new bool[n];
        visited[goal] = true;

        for (int i = x.Length - 1; i >= 0; i--) {
            var a = x[i];
            var b = y[i];
            if (visited[a] && !visited[b]) {
                ans[i] = '^';
            } else {
                ans[i] = 'v';
            }
            visited[a] = visited[b] = visited[a] | visited[b];
        }


        return new string(ans);
    }
}

public class BitSet {
    public readonly ulong[] _array;

    public BitSet(int size) {
        _array = new ulong[(size - 1) / 64 + 1];
        BitLength = size;
    }

    public int BitLength { get; private set; }

    public int CountBit() {
        int count = 0;
        foreach (ulong a in _array) {
            count += PopCount(a);
        }
        return count;
    }

    public ulong[] LongArray { get { return _array; } }

    public bool Any() {
        return Array.FindIndex(_array, a => a != 0UL) >= 0;
    }

    public bool All() {
        return CountBit() == BitLength;
    }

    public BitSet Slice(int begin, int end) {
        var bitset = new BitSet(end - begin);
        int jump = begin >> 6;
        int move = begin & 63;
        int arrayLength = bitset._array.Length;
        int lastIndex = 0;
        for (int i = 0; i + jump < _array.Length && i < bitset._array.Length; i++) {
            bitset._array[i] = unchecked(
                (_array[i + jump] >> move) |
                ((move > 0 && i + jump + 1 < _array.Length) ? (_array[i + jump + 1] << (64 - move)) : 0)
            );
            lastIndex = i;
        }
        int lastBit = bitset.BitLength & 63;
        if (lastBit > 0) {
            bitset._array[lastIndex] &= (1UL << lastBit) - 1;
        }
        return bitset;
    }

    public static BitSet operator <<(BitSet b, int shift) {
        if (shift == 0) {
            return b;
        }

        var bitset = new BitSet(b.BitLength);
        int jump = shift >> 6;
        int move = shift & 63;
        int arrayLength = bitset._array.Length;
        for (int i = jump; i < arrayLength; i++) {
            bitset._array[i] = unchecked(
                (b._array[i - jump] << move) |
                (move > 0 && i - jump - 1 >= 0 ? (b._array[i - jump - 1] >> (64 - move)) : 0)
            );
        }
        return bitset;
    }

    public static BitSet operator >>(BitSet b, int shift) {
        if (shift == 0) {
            return b;
        }

        var bitset = new BitSet(b.BitLength);
        int jump = shift >> 6;
        int move = shift & 63;
        int arrayLength = bitset._array.Length;
        for (int i = 0; i + jump < arrayLength; i++) {
            bitset._array[i] = unchecked(
                (b._array[i + jump] >> move) |
                (move > 0 && i + jump + 1 < arrayLength ? (b._array[i + jump + 1] << (64 - move)) : 0)
            );
        }
        int lastBit = bitset.BitLength & 63;
        if (lastBit > 0) {
            bitset._array[bitset._array.Length - 1] &= (1UL << lastBit) - 1;
        }
        return bitset;
    }


    public static BitSet operator &(BitSet b1, BitSet b2) {
        var bitset = new BitSet(b1.BitLength);
        for (int i = 0; i < bitset._array.Length; i++) {
            bitset._array[i] = b1._array[i] & b2._array[i];
        }
        return bitset;
    }

    public static BitSet operator |(BitSet b1, BitSet b2) {
        var bitset = new BitSet(b1.BitLength);
        for (int i = 0; i < bitset._array.Length; i++) {
            bitset._array[i] = b1._array[i] | b2._array[i];
        }
        return bitset;
    }

    public static BitSet operator ^(BitSet b1, BitSet b2) {
        var bitset = new BitSet(b1.BitLength);
        for (int i = 0; i < bitset._array.Length; i++) {
            bitset._array[i] = b1._array[i] ^ b2._array[i];
        }
        return bitset;
    }

    private static int PopCount(ulong x) {
        x = (x & 0x5555555555555555UL) + ((x >> 1) & 0x5555555555555555UL);
        x = (x & 0x3333333333333333UL) + ((x >> 2) & 0x3333333333333333UL);
        x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0fUL;
        x += x >> 8;
        x += x >> 16;
        x += x >> 32;
        return (int)(x & 0x7FLU);
    }

    public static bool operator ==(BitSet b1, BitSet b2) {
        return b1.Equals(b2);
    }

    public static bool operator !=(BitSet b1, BitSet b2) {
        return !(b1 == b2);
    }

    public override bool Equals(object obj) {
        if (obj is BitSet) {
            return Equals((BitSet)obj);
        }
        return false;
    }

    public bool Equals(BitSet b) {
        if (BitLength != b.BitLength) {
            return false;
        }

        for (int i = 0; i < _array.Length; i++) {
            if (_array[i] != b._array[i]) {
                return false;
            }
        }
        return true;
    }

    public override int GetHashCode() {
        int hash = BitLength.GetHashCode();
        foreach (ulong a in _array) {
            hash = unchecked(hash * 31) ^ a.GetHashCode();
        }
        return hash;
    }

    public bool this[int index] {
        get {
            return (_array[index >> 6] & (1UL << (index & 63))) != 0UL;
        }
        set {
            if (value) {
                _array[index >> 6] |= 1UL << (index & 63);
            } else {
                _array[index >> 6] &= ~(1UL << (index & 63));
            }
        }
    }

    public override string ToString() {
        var b = new StringBuilder();
        for (int i = BitLength - 1; i >= 0; i--) {
            b.Append(this[i] ? '1' : '0');
        }
        return b.ToString();
    }
}
// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#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 });
        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) {
        cin = reader;
        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
}

internal static class LinqPadExtension {
    public static 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) {
        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) {
        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 (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            string line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            string[] tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (string token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}