Problem:
Score : 500 points

Problem Statement

Given is a connected undirected graph with N vertices and M edges. The vertices are numbered 1 to N, and the edges are described by a grid of characters S.
If S_{i,j} is 1, there is an edge connecting Vertex i and j; otherwise, there is no such edge.

Determine whether it is possible to divide the vertices into non-empty sets V_1, \dots, V_k such that the following condition is satisfied. If the answer is yes, find the maximum possible number of sets, k, in such a division.

Every edge connects two vertices belonging to two "adjacent" sets. More formally, for every edge (i,j), there exists 1\leq t\leq k-1 such that i\in V_t,j\in V_{t+1} or i\in V_{t+1},j\in V_t holds.

Constraints

2 \leq N \leq 200

S_{i,j} is 0 or 1.

S_{i,i} is 0.

S_{i,j}=S_{j,i}

The given graph is connected.

N is an integer.

Input

Input is given from Standard Input in the following format:

N
S_{1,1}...S_{1,N}
:
S_{N,1}...S_{N,N}

Output

If it is impossible to divide the vertices into sets so that the condition is satisfied, print -1.
Otherwise, print the maximum possible number of sets, k, in a division that satisfies the condition.

Sample Input 1

2
01
10

Sample Output 1

2

We can put Vertex 1 in V_1 and Vertex 2 in V_2.

Sample Input 2

3
011
101
110

Sample Output 2

-1

Sample Input 3

6
010110
101001
010100
101000
100000
010000

Sample Output 3

4

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.ComponentModel.DataAnnotations;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Reflection.Metadata;
using System.Reflection.Metadata.Ecma335;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Runtime.Intrinsics.X86;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;
using Microsoft.VisualBasic.CompilerServices;

namespace AtCoder
{
	public class Program
	{
		static void Main()
		{
			using var cin = new Scanner();
			int n = cin.Int();
			var s = new string[n];
			for (int i = 0; i < n; i++) {
				s[i] = cin.Next();
			}

			int inf = 100000;
			var dist = new int[n, n];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < n; j++) {
					if (i != j) {
						dist[i, j] = s[i][j] == '0' ?  inf : 1;
					}
				}
			}

			var c = new int[n];
			c[0] = 1;
			var que = new Queue<int>();
			que.Enqueue(0);
			bool ok = true;
			while (que.Count > 0 && ok) {
				var cur = que.Dequeue();

				int nc = c[cur] == 1 ? -1 : 1;
				for (int i = 0; i < n; i++) {
					if (i == cur || s[i][cur] == '0') {
						continue;
					}

					if (c[i] == 0) {
						c[i] = nc;
						que.Enqueue(i);
					} else if (c[i] != nc) {
						ok = false;
						break;
					}
				}
			}

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

			for (int k = 0; k < n; k++) {
				for (int i = 0; i < n; i++) {
					for (int j = 0; j < n; j++) {
						dist[i, j] = Math.Min(
							dist[i, j],
							dist[i, k] + dist[k, j]);
					}
				}
			}

			int ans = 0;
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < n; j++) {
					if (dist[i, j] != inf) {
						ans = Math.Max(ans, dist[i, j]);
					}
				}
			}

			Console.WriteLine(ans + 1);
		}
	}

	public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
	{
		private readonly Func<TKey, TValue> initialzier_;
		public HashMap(Func<TKey, TValue> initialzier)
			: base()
		{
			initialzier_ = initialzier;
		}

		public HashMap(Func<TKey, TValue> initialzier, int capacity)
			: base(capacity)
		{
			initialzier_ = initialzier;
		}

		new public TValue this[TKey key]
		{
			get
			{
				if (ContainsKey(key) == false) {
					base[key] = initialzier_(key);
				}

				return base[key];
			}

			set { base[key] = value; }
		}
	}

	public struct ModInt
	{
		//public const long P = 1000000007;
		public const long P = 998244353;
		public const long ROOT = 3;

		// (924844033, 5)
		// (998244353, 3)
		// (1012924417, 5)
		// (167772161, 3)
		// (469762049, 3)
		// (1224736769, 3)

		private long value_;

		public ModInt(long value)
			=> value_ = value;
		public ModInt(long value, bool mods)
		{
			if (mods) {
				value %= P;
				if (value < 0) {
					value += P;
				}
			}

			value_ = value;
		}

		public static ModInt operator +(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (lhs.value_ + rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator +(long lhs, ModInt rhs)
		{
			rhs.value_ = (lhs + rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator +(ModInt lhs, long rhs)
		{
			lhs.value_ = (lhs.value_ + rhs) % P;
			return lhs;
		}

		public static ModInt operator -(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator -(long lhs, ModInt rhs)
		{
			rhs.value_ = (P + lhs - rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator -(ModInt lhs, long rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs) % P;
			return lhs;
		}

		public static ModInt operator *(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = lhs.value_ * rhs.value_ % P;
			return lhs;
		}
		public static ModInt operator *(long lhs, ModInt rhs)
		{
			rhs.value_ = lhs * rhs.value_ % P;
			return rhs;
		}
		public static ModInt operator *(ModInt lhs, long rhs)
		{
			lhs.value_ = lhs.value_ * rhs % P;
			return lhs;
		}

		public static ModInt operator /(ModInt lhs, ModInt rhs)
		{
			long exp = P - 2;
			while (exp > 0) {
				if (exp % 2 > 0) {
					lhs *= rhs;
				}

				rhs *= rhs;
				exp /= 2;
			}

			return lhs;
		}

		public static implicit operator ModInt(long n) => new ModInt(n, true);

		public static ModInt Inverse(ModInt value) => Pow(value, P - 2);
		public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);
		public static ModInt Pow(long value, long k)
		{
			long ret = 1;
			for (k %= P - 1; k > 0; k >>= 1, value = value * value % P) {
				if ((k & 1) == 1) {
					ret = ret * value % P;
				}
			}
			return new ModInt(ret);
		}


		public static Span<ModInt> NumberTheoreticTransform(
			Span<int> values, bool inverses = false)
		{
			var mods = new ModInt[values.Length];
			for (int i = 0; i < mods.Length; i++) {
				mods[i] = new ModInt(values[i]);
			}

			return NumberTheoreticTransform(mods, inverses);
		}

		public static Span<ModInt> NumberTheoreticTransform(
			Span<long> values, bool inverses = false)
		{
			var mods = new ModInt[values.Length];
			for (int i = 0; i < mods.Length; i++) {
				mods[i] = new ModInt(values[i]);
			}

			return NumberTheoreticTransform(mods, inverses);
		}

		public static Span<ModInt> NumberTheoreticTransform(
			Span<ModInt> a, bool inverses = false)
		{
			int n = a.Length;
			if (n == 1) {
				return a;
			}

			var b = new ModInt[n].AsSpan();
			int r = inverses
				? (int)(P - 1 - (P - 1) / n)
				: (int)((P - 1) / n);
			ModInt s = Pow(ROOT, r);
			var kp = new ModInt[n / 2 + 1];
			kp.AsSpan().Fill(1);

			for (int i = 0; i < n / 2; ++i) {
				kp[i + 1] = kp[i] * s;
			}

			int l = n / 2;
			for (int i = 1; i < n; i <<= 1, l >>= 1) {
				r = 0;
				for (int j = 0; j < l; ++j, r += i) {
					s = kp[i * j];
					for (int k = 0; k < i; ++k) {
						var p = a[k + r];
						var q = a[k + r + n / 2];
						b[k + 2 * r] = p + q;
						b[k + 2 * r + i] = (p - q) * s;
					}
				}

				var temp = a;
				a = b;
				b = temp;
			}

			if (inverses) {
				s = Inverse(n);
				for (int i = 0; i < n; i++) {
					a[i] = a[i] * s;
				}
			}

			return a;
		}

		public static ModInt[,] NumberTheoreticTransform2D(ModInt[,] a, bool inverses = false)
		{
			int h = a.GetLength(0);
			int w = a.GetLength(1);
			if (h == 1 && w == 1) {
				return a;
			}

			var b = new ModInt[h, w];

			{
				int n = w;
				int r = inverses
					? (int)(P - 1 - (P - 1) / n)
					: (int)((P - 1) / n);
				ModInt s = Pow(ROOT, r);
				var kp = new ModInt[n / 2 + 1];
				kp.AsSpan().Fill(1);

				for (int i = 0; i < n / 2; ++i) {
					kp[i + 1] = kp[i] * s;
				}

				for (int y = 0; y < h; y++) {
					int l = n / 2;
					for (int i = 1; i < n; i <<= 1, l >>= 1) {
						r = 0;
						for (int j = 0; j < l; ++j, r += i) {
							s = kp[i * j];
							for (int k = 0; k < i; ++k) {
								var p = a[y, k + r];
								var q = a[y, k + r + n / 2];
								b[y, k + 2 * r] = p + q;
								b[y, k + 2 * r + i] = (p - q) * s;
							}
						}

						var temp = a;
						a = b;
						b = temp;
					}

					if (inverses) {
						s = Inverse(n);
						for (int i = 0; i < n; i++) {
							a[y, i] = a[y, i] * s;
						}
					}
				}
			}

			for (int i = 0; i < h; i++) {
				for (int j = 0; j < w; j++) {
					b[h, w] = 0;
				}
			}

			{
				int n = h;
				int r = inverses
					? (int)(P - 1 - (P - 1) / n)
					: (int)((P - 1) / n);
				ModInt s = Pow(ROOT, r);
				var kp = new ModInt[n / 2 + 1];
				kp.AsSpan().Fill(1);

				for (int i = 0; i < n / 2; ++i) {
					kp[i + 1] = kp[i] * s;
				}

				for (int x = 0; x < w; x++) {
					int l = n / 2;
					for (int i = 1; i < n; i <<= 1, l >>= 1) {
						r = 0;
						for (int j = 0; j < l; ++j, r += i) {
							s = kp[i * j];
							for (int k = 0; k < i; ++k) {
								var p = a[k + r, x];
								var q = a[k + r + n / 2, x];
								b[k + 2 * r, x] = p + q;
								b[k + 2 * r + i, x] = (p - q) * s;
							}
						}

						var temp = a;
						a = b;
						b = temp;
					}

					if (inverses) {
						s = Inverse(n);
						for (int i = 0; i < n; i++) {
							a[i, x] = a[i, x] * s;
						}
					}
				}
			}

			return a;
		}

		public static Span<ModInt> Convolve(ReadOnlySpan<ModInt> a, ReadOnlySpan<ModInt> b)
		{
			int resultLength = a.Length + b.Length - 1;
			int nttLenght = 1;
			while (nttLenght < resultLength) {
				nttLenght <<= 1;
			}

			var aa = new ModInt[nttLenght];
			a.CopyTo(aa);
			var bb = new ModInt[nttLenght];
			b.CopyTo(bb);

			var fa = NumberTheoreticTransform(aa);
			var fb = NumberTheoreticTransform(bb);
			for (int i = 0; i < nttLenght; i++) {
				fa[i] *= fb[i];
			}

			var convolved = NumberTheoreticTransform(fa, true);
			return convolved.Slice(0, resultLength);
		}

		public long ToLong() => value_;
		public override string ToString() => value_.ToString();
	}

	public static class Helper
	{
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(ref T target, T value) where T : IComparable<T>
			=> target = target.CompareTo(value) > 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
		{
			if (target.CompareTo(value) > 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(ref T target, T value) where T : IComparable<T>
			=> target = target.CompareTo(value) < 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
		{
			if (target.CompareTo(value) < 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Array<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToArray();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static List<T> List<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToList();

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, T init)
			where T : struct
		{
			var array = new T[n, m];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					array[i, j] = init;
				}
			}

			return array;
		}
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, Func<int, int, T> initializer)
		{
			var array = new T[n, m];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					array[i, j] = initializer(i, j);
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,] Array3<T>(int n1, int n2, int n3, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						array[i1, i2, i3] = init;
					}
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3, n4];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						for (int i4 = 0; i4 < n4; i4++) {
							array[i1, i2, i3, i4] = init;
						}
					}
				}
			}

			return array;
		}

		private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt4(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			for (int dn = 0; dn < 4; dn++) {
				int d4i = i + delta4_[dn];
				int d4j = j + delta4_[dn + 1];
				if ((uint)d4i < (uint)imax && (uint)d4j < (uint)jmax) {
					action(d4i, d4j);
				}
			}
		}

		private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt8(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			for (int dn = 0; dn < 8; dn++) {
				int d8i = i + delta8_[dn];
				int d8j = j + delta8_[dn + 1];
				if ((uint)d8i < (uint)imax && (uint)d8j < (uint)jmax) {
					action(d8i, d8j);
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void ForEachSubBit(int bit, Action<int> action)
		{
			for (int sub = bit; sub >= 0; sub--) {
				sub &= bit;
				action(sub);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Reverse(string src)
		{
			var chars = src.ToCharArray();
			for (int i = 0, j = chars.Length - 1; i < j; i++, j--) {
				var tmp = chars[i];
				chars[i] = chars[j];
				chars[j] = tmp;
			}
			return new string(chars);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Join<T>(this IEnumerable<T> values, string separator = "")
			=> string.Join(separator, values);
	}

	public class Scanner : IDisposable
	{
		private const int BUFFER_SIZE = 1024;
		private const int ASCII_CHAR_BEGIN = 33;
		private const int ASCII_CHAR_END = 126;
		private readonly string filePath_;
		private readonly Stream stream_;
		private readonly byte[] buf_ = new byte[BUFFER_SIZE];
		private int length_ = 0;
		private int index_ = 0;
		private bool isEof_ = false;

		public Scanner(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				stream_ = Console.OpenStandardInput();
			} else {
				filePath_ = file;
				stream_ = new FileStream(file, FileMode.Open);
			}

			Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush = false
			});
		}

		public void Dispose()
		{
			Console.Out.Flush();
			stream_.Dispose();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public char Char()
		{
			byte b;
			do {
				b = Read();
			} while (b < ASCII_CHAR_BEGIN || ASCII_CHAR_END < b);

			return (char)b;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string Next()
		{
			var sb = new StringBuilder();
			for (var b = Char(); b >= ASCII_CHAR_BEGIN && b <= ASCII_CHAR_END; b = (char)Read()) {
				sb.Append(b);
			}

			return sb.ToString();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int() => (int)Long();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int(int offset) => Int() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int) Int2(int offset = 0)
			=> (Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int) Int3(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int, int) Int4(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int[] ArrayInt(int length, int offset = 0)
		{
			int[] Array = new int[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Int(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long()
		{
			long ret = 0;
			byte b;
			bool ng = false;
			do {
				b = Read();
			} while (b != '-' && (b < '0' || '9' < b));

			if (b == '-') {
				ng = true;
				b = Read();
			}

			for (; true; b = Read()) {
				if (b < '0' || '9' < b) {
					return ng ? -ret : ret;
				} else {
					ret = ret * 10 + b - '0';
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long(long offset) => Long() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long) Long2(long offset = 0)
			=> (Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long) Long3(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long, long) Long4(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long[] ArrayLong(int length, long offset = 0)
		{
			long[] Array = new long[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Long(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double() => double.Parse(Next(), CultureInfo.InvariantCulture);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double(double offset) => Double() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double) Double2(double offset = 0)
			=> (Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double) Double3(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double, double) Double4(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double[] ArrayDouble(int length, double offset = 0)
		{
			double[] Array = new double[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Double(offset);
			}
			return Array;
		}

		private byte Read()
		{
			if (isEof_) {
				throw new EndOfStreamException();
			}

			if (index_ >= length_) {
				index_ = 0;
				if ((length_ = stream_.Read(buf_, 0, BUFFER_SIZE)) <= 0) {
					isEof_ = true;
					return 0;
				}
			}

			return buf_[index_++];
		}

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

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