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.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			int n = cin.Int();
			int m = cin.Int();
			int t = cin.Int();
			int[] xs = new int[m];
			int[] ys = new int[m];
			for (int i = 0; i < m; i++) {
				xs[i] = cin.Int() - 1;
				ys[i] = cin.Int() - 1;
			}

			if (t == 1) {
				var bits = new Bitset[n];
				for (int i = 0; i < n; i++) {
					bits[i] = new Bitset(n);
				}

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

				for (int i = m - 1; i >= 0; i--) {
					bits[xs[i]] |= bits[ys[i]];
					bits[ys[i]] |= bits[xs[i]];
				}

				var flags = bits[0];
				for (int i = 1; i < n; i++) {
					flags &= bits[i];
				}

				int index = -1;
				for (int i = 0; i < n; i++) {
					if (flags[i]) {
						index = i;
						break;
					}
				}

				if (index == -1) {
					Console.WriteLine(-1);
					return;
				}

				char[] ds = new char[m];
				bool[] f = new bool[n];
				f[index] = true;
				for (int i = m - 1; i >= 0; i--) {
					if (f[xs[i]] && f[ys[i]] == false) {
						ds[i] = '^';
					} else if (f[xs[i]] == false && f[ys[i]]) {
						ds[i] = 'v';
					} else {
						ds[i] = '^';
					}

					f[xs[i]] |= f[ys[i]];
					f[ys[i]] |= f[xs[i]];
				}

				string ans = new string(ds);
				Console.WriteLine(ans);
			} else {
				Console.WriteLine(-1);
			}
		}
	}

	public class Bitset
	{
		private readonly int length_;
		private readonly ulong[] bits_;

		public int Size { get; }
		public Bitset(int size, bool fillTrue = false)
		{
			Size = size;
			length_ = ((size - 1) >> 6) + 1;
			bits_ = new ulong[length_];
			if (fillTrue) {
				int y = size % 64;
				if (y == 0) {
					for (int i = 0; i < length_; i++) {
						bits_[i] = ulong.MaxValue;
					}
				} else {
					for (int i = 0; i < length_ - 1; i++) {
						bits_[i] = ulong.MaxValue;
					}

					bits_[length_ - 1] = (1UL << y) - 1;
				}
			}
		}
		public Bitset(int size, long initialValue)
		{
			Size = size;
			length_ = ((size - 1) >> 6) + 1;
			bits_ = new ulong[length_];
			bits_[0] = (ulong)initialValue;
		}

		public bool this[int i]
		{
			get { return ((bits_[i >> 6] >> i) & 1) != 0; }
			set { bits_[i >> 6] = (bits_[i >> 6] & (ulong.MaxValue ^ (1ul << i))) | ((ulong)(value ? 1 : 0) << i); }
		}

		public static Bitset operator &(Bitset lhs, Bitset rhs)
		{
			int min = Math.Min(lhs.length_, rhs.length_);

			var ret = new Bitset(Math.Max(lhs.Size, rhs.Size));
			for (int i = 0; i < min; i++) {
				ret.bits_[i] = lhs.bits_[i] & rhs.bits_[i];
			}

			return ret;
		}
		public static Bitset operator |(Bitset lhs, Bitset rhs)
		{
			int max = Math.Max(lhs.length_, rhs.length_);
			int min = Math.Min(lhs.length_, rhs.length_);
			var ret = new Bitset(Math.Max(lhs.Size, rhs.Size));

			for (int i = 0; i < min; i++) {
				ret.bits_[i] = lhs.bits_[i] | rhs.bits_[i];
			}

			if (lhs.length_ > rhs.length_) {
				for (int i = min; i < max; i++) {
					ret.bits_[i] = lhs.bits_[i];
				}
			} else {
				for (int i = min; i < max; i++) {
					ret.bits_[i] = rhs.bits_[i];
				}
			}

			return ret;
		}
		public static Bitset operator ^(Bitset lhs, Bitset rhs)
		{
			int max = Math.Max(lhs.length_, rhs.length_);
			int min = Math.Min(lhs.length_, rhs.length_);
			var ret = new Bitset(Math.Max(lhs.Size, rhs.Size));

			for (int i = 0; i < min; i++) {
				ret.bits_[i] = lhs.bits_[i] ^ rhs.bits_[i];
			}

			if (lhs.length_ > rhs.length_) {
				for (int i = min; i < max; i++) {
					ret.bits_[i] = lhs.bits_[i];
				}
			} else {
				for (int i = min; i < max; i++) {
					ret.bits_[i] = rhs.bits_[i];
				}
			}

			return ret;
		}
		public static Bitset operator <<(Bitset target, int shift)
		{
			var ret = new Bitset(target.Size);
			if (shift > target.length_ << 6) {
				return ret;
			}

			int minIndex = (shift + 63) >> 6;
			if (shift % 64 == 0) {
				for (int i = target.length_ - 1; i >= minIndex; i--) {
					ret.bits_[i] = target.bits_[i - minIndex];
				}
			} else {
				for (int i = target.length_ - 1; i >= minIndex; i--) {
					ret.bits_[i] = (target.bits_[i - minIndex + 1] << shift)
						| (target.bits_[i - minIndex] >> (64 - shift));
				}

				ret.bits_[minIndex - 1] = target.bits_[0] << shift;
			}

			return ret;
		}
		public static Bitset operator >>(Bitset target, int shift)
		{
			var ret = new Bitset(target.Size);
			if (shift > target.length_ << 6) {
				return ret;
			}

			int minIndex = (shift + 63) >> 6;
			if (shift % 64 == 0) {
				for (int i = 0; i + minIndex < ret.length_; i++) {
					ret.bits_[i] = target.bits_[i + minIndex];
				}
			} else {
				for (int i = 0; i + minIndex < ret.length_; i++) {
					ret.bits_[i] = (target.bits_[i + minIndex - 1] >> shift)
						| (target.bits_[i + minIndex] << (64 - shift));
				}

				ret.bits_[ret.length_ - minIndex] = target.bits_[ret.length_ - 1] >> shift;
			}

			return ret;
		}
		public static bool operator ==(Bitset lhs, Bitset rhs)
		{
			int max = Math.Max(lhs.length_, rhs.length_);
			int min = Math.Min(lhs.length_, rhs.length_);
			for (int i = 0; i < min; i++) {
				if (lhs.bits_[i] != rhs.bits_[i]) {
					return false;
				}
			}

			if (lhs.length_ > rhs.length_) {
				for (int i = min; i < max; i++) {
					if (lhs.bits_[i] != 0) {
						return false;
					}
				}
			} else {
				for (int i = min; i < max; i++) {
					if (rhs.bits_[i] != 0) {
						return false;
					}
				}
			}

			return true;
		}
		public static bool operator !=(Bitset lhs, Bitset rhs)
		{
			return !(lhs == rhs);
		}
		public override bool Equals(object obj)
		{
			return false;
		}

		public override int GetHashCode()
		{
			var hashCode = -723801931;
			hashCode = hashCode * -1521134295 + length_.GetHashCode();
			hashCode = hashCode * -1521134295 + EqualityComparer<ulong[]>.Default.GetHashCode(bits_);
			hashCode = hashCode * -1521134295 + Size.GetHashCode();
			return hashCode;
		}

		public override string ToString()
		{
			string ret = "";
			for (int i = 0; i < length_ - 1; i++) {
				string temp = Convert.ToString((long)bits_[i], 2);
				temp = (64 > temp.Length ? new string('0', 64 - temp.Length) : "")
					+ temp;
				ret = temp + ret;
			}

			{
				string temp = Convert.ToString((long)bits_[length_ - 1], 2);
				temp = ((Size % 64) > temp.Length ? new string('0', (Size % 64) - temp.Length) : "")
					+ temp;
				ret = temp + ret;
			}

			return ret;
		}
	}

	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private string[] buf_;
		private int index_;
		Func<string> reader_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}