Problem:
Score : 400 points

Problem Statement

A maze is composed of a grid of H \times W squares - H vertical, W horizontal.

The square at the i-th row from the top and the j-th column from the left - (i,j) - is a wall if S_{ij} is # and a road if S_{ij} is ..

There is a magician in (C_h,C_w). He can do the following two kinds of moves:

Move A: Walk to a road square that is vertically or horizontally adjacent to the square he is currently in.

Move B: Use magic to warp himself to a road square in the 5\times 5 area centered at the square he is currently in.

In either case, he cannot go out of the maze.

At least how many times does he need to use the magic to reach (D_h, D_w)?

Constraints

1 \leq H,W \leq 10^3

1 \leq C_h,D_h \leq H

1 \leq C_w,D_w \leq W

S_{ij} is # or ..

S_{C_h C_w} and S_{D_h D_w} are ..

(C_h,C_w) \neq (D_h,D_w)

Input

Input is given from Standard Input in the following format:

H W
C_h C_w
D_h D_w
S_{11}\ldots S_{1W}
\vdots
S_{H1}\ldots S_{HW}

Output

Print the minimum number of times the magician needs to use the magic. If he cannot reach (D_h,D_w), print -1 instead.

Sample Input 1

4 4
1 1
4 4
..#.
..#.
.#..
.#..

Sample Output 1

1

For example, by walking to (2,2) and then using the magic to travel to (4,4), just one use of magic is enough.

Note that he cannot walk diagonally.

Sample Input 2

4 4
1 4
4 1
.##.
####
####
.##.

Sample Output 2

-1

He cannot move from there.

Sample Input 3

4 4
2 2
3 3
....
....
....
....

Sample Output 3

0

No use of magic is needed.

Sample Input 4

4 5
1 2
2 5
#.###
####.
#..##
#..##

Sample Output 4

2

Solution in C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using static System.Math;

namespace CSharpSpiral {
	class Program {
		public static void Main(string[] args) {
			var sw = new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush =
#if DEBUG
			true
#else
			false
#endif
			};
			Console.SetOut(sw);
			new Program().Solve(new ConsoleInput(
#if DEBUG
			new StreamReader("../../../inputData.txt")
#else
			Console.In
#endif
			, ' '));
			Console.Out.Flush();
		}

		public void Solve(ConsoleInput cin) {
			var H = cin.ReadInt;var W = cin.ReadInt;
			var C = cin.ReadIntArray(2);
			var D = cin.ReadIntArray(2);
			var S = new bool[H, W];
			for(int i = 0; i < H; i++) {
				var line = cin.ReadLine;
				for(int j = 0; j < W; j++) {
					S[i, j] = line[j] == '#';
				}
			}
			var G = new Graph<int>(H*W);
			for(int i = 0; i < H; i++) {
				for(int j = 0; j < W; j++) {
					if(!S[i,j]) {
						if(j<W-1&&!S[i,j+1]) {
							G.AddEdge(i * W + j, i * W + j + 1, 1);
							G.AddEdge(i * W + j + 1, i * W + j, 1);
						}
						if(i<H-1&&!S[i+1,j]) {
							G.AddEdge(i * W + j, (i + 1) * W + j, 1);
							G.AddEdge((i + 1) * W + j, i * W + j, 1);
						}
					}
				}
			}
			var groups = new int[H, W];
			for(int i = 0; i < H; i++) {
				for(int j = 0; j < W; j++) {
					groups[i, j] = -2;
				}
			}
			var count = 0;
			for(int i = 0; i < H; i++) {
				for(int j = 0; j < W; j++) {
					if(S[i, j]) {
						groups[i, j] = -1;
						continue;
					}else{
						if(groups[i,j]<0) {
							G.PointDFS(i * W + j, k => groups[k / W, k % W] = count, null);
							count++;
						}
					}
				}
			}

			var worlds = new Dijkstra(count);
			var Done = new List<int>[count];
			for(int i = 0; i < count; i++) {
				Done[i] = new List<int>();
			}
			for(int i = 0; i < H; i++) {
				for(int j = 0; j < W; j++) {
					if(S[i, j]) continue;

					var belong = groups[i, j];
					for(int ii = -2; ii <= 2; ii++) {
						if(i + ii < 0 || i + ii >= H) continue;
						for(int jj = -2; jj <= 2; jj++) {
							if(j + jj < 0 || j + jj >= W) continue;
							if(S[i + ii, j + jj]) continue;
							var nextW = groups[i + ii, j + jj];
							if(nextW != belong && !Done[belong].Contains(nextW) && !Done[nextW].Contains(belong)) {
								worlds.AddPath(nextW, belong, 1);
								Done[belong].Add(nextW);
								Done[nextW].Add(belong);
							}
						}
					}
				}
			}

			var startW = groups[C[0] - 1, C[1] - 1];
			var goalW = groups[D[0] - 1, D[1] - 1];
			var M = worlds.GetMinCost(startW, goalW);
			Console.WriteLine(M > 11111111L ? -1 : M);
		}

	}

	static class ExtendClass {
		public static string Join<T>(this IEnumerable<T> X, string separator = " ")
			=> string.Join(separator, X);
	}
	class ConsoleInput {
		private readonly TextReader _stream;
		private readonly char _separator = ' ';
		private readonly Queue<string> inputStream;
		public ConsoleInput(TextReader stream, char separator = ' ') {
			this._separator = separator;
			this._stream = stream;
			inputStream = new Queue<string>();
		}
		public string Read {
			get {
				if(inputStream.Count != 0) return inputStream.Dequeue();
				string[] tmp = _stream.ReadLine().Split(_separator);
				for(int i = 0; i < tmp.Length; ++i)
					inputStream.Enqueue(tmp[i]);
				return inputStream.Dequeue();
			}
		}
		public string ReadLine { get { return _stream.ReadLine(); } }
		public int ReadInt { get { return int.Parse(Read); } }
		public long ReadLong { get { return long.Parse(Read); } }
		public double ReadDouble { get { return double.Parse(Read); } }
		public string[] ReadStrArray(long N) {
			var ret = new string[N];
			for(long i = 0; i < N; ++i)
				ret[i] = Read;
			return ret;
		}
		public int[] ReadIntArray(long N) { var ret = new int[N]; for(long i = 0; i < N; ++i) ret[i] = ReadInt; return ret; }
		public long[] ReadLongArray(long N) { var ret = new long[N]; for(long i = 0; i < N; ++i) ret[i] = ReadLong; return ret; }
		public double[] ReadDoubleArray(long N) { var ret = new double[N]; for(long i = 0; i < N; ++i) ret[i] = ReadDouble; return ret; }
		public bool isEOF {
			get { return _stream.Peek() == -1; }
		}
	}
	/// <summary>
	/// Get min cost between two points
	/// </summary>
	public class Dijkstra {
		private int maxIndex = -1;
		private const int INF = Int32.MaxValue;

		private int[,] _edgeArray;

		/// <summary>
		/// Dijkstra, get min cost between two points
		/// should not contain negatie cost path
		/// </summary>
		/// <param name="size">max index of vertices</param>
		public Dijkstra(int size) {
			maxIndex = size + 1;
			_edgeArray = new int[maxIndex, maxIndex];

			for(int i = 0; i < _edgeArray.GetLength(0); i++) {
				for(int j = 0; j < _edgeArray.GetLength(1); j++) {
					_edgeArray[i, j] = INF;
					if(i == j) _edgeArray[i, j] = 0;
				}
			}
		}

		// Add a path(no direction) with its cost
		public void AddPath(int s, int t, int cost) {
			_edgeArray[s, t] = Math.Min(_edgeArray[s, t], cost);
			_edgeArray[t, s] = _edgeArray[s, t];
		}

		//Get the min cost between s and t
		//return Int32.MaxValue if no path
		public int GetMinCost(int s, int t) {
			int[] cost = new int[maxIndex];
			for(int i = 0; i < cost.Length; i++) cost[i] = INF;
			cost[s] = 0;

			var priorityQueue = new PriorityQueue<ComparablePair<int, int>>(maxIndex);
			priorityQueue.Push(new ComparablePair<int, int>(0, s));

			while(priorityQueue.Count() > 0) {
				var costDestinationPair = priorityQueue.Pop();
				if(cost[costDestinationPair.Item2] < costDestinationPair.Item1) continue;

				for(int i = 0; i < maxIndex; i++) {
					int newCostToi = _edgeArray[costDestinationPair.Item2, i] == INF ? INF :
							costDestinationPair.Item1 + _edgeArray[costDestinationPair.Item2, i];
					if(newCostToi < cost[i]) {
						cost[i] = newCostToi;
						priorityQueue.Push(new ComparablePair<int, int>(newCostToi, i));
					}
				}
			}

			return cost[t];
		}
	}

	public class ComparablePair<T, U>: IComparable where T : IComparable<T> {
		public readonly T Item1;
		public readonly U Item2;

		public int CompareTo(object obj) {
			ComparablePair<T, U> castedObj = (ComparablePair<T, U>)obj;
			return this.Item1.CompareTo(castedObj.Item1);
		}

		public ComparablePair(T t, U u) {
			Item1 = t;
			Item2 = u;
		}
	}
	public class PriorityQueue<T> where T : IComparable {
		private IComparer<T> _comparer = null;
		private int _type = 0;

		private T[] _heap;
		private int _sz = 0;

		private int _count = 0;

		/// <summary>
		/// Priority Queue with custom comparer
		/// </summary>
		public PriorityQueue(IComparer<T> comparer) {
			_heap = new T[128];
			_comparer = comparer;
		}

		/// <summary>
		/// Priority queue
		/// </summary>
		/// <param name="type">0: asc, 1:desc</param>
		public PriorityQueue(int type = 0) {
			_heap = new T[128];
			_type = type;
		}

		private int Compare(T x, T y) {
			if(_comparer != null) return _comparer.Compare(x, y);
			return _type == 0 ? x.CompareTo(y) : y.CompareTo(x);
		}

		public void Push(T x) {
			_count++;
			if(_count > _heap.Length) {
				var newheap = new T[_heap.Length * 2];
				for(int n = 0; n < _heap.Length; n++) newheap[n] = _heap[n];
				_heap = newheap;
			}

			//node number
			var i = _sz++;

			while(i > 0) {
				//parent node number
				var p = (i - 1) / 2;

				if(Compare(_heap[p], x) <= 0) break;

				_heap[i] = _heap[p];
				i = p;
			}

			_heap[i] = x;
		}

		public T Pop() {
			_count--;

			T ret = _heap[0];
			T x = _heap[--_sz];

			int i = 0;
			while(i * 2 + 1 < _sz) {
				//children
				int a = i * 2 + 1;
				int b = i * 2 + 2;

				if(b < _sz && Compare(_heap[b], _heap[a]) < 0) a = b;

				if(Compare(_heap[a], x) >= 0) break;

				_heap[i] = _heap[a];
				i = a;
			}

			_heap[i] = x;

			return ret;
		}

		public int Count() {
			return _count;
		}

		public T Peek() {
			return _heap[0];
		}

		public bool Contains(T x) {
			for(int i = 0; i < _sz; i++) if(x.Equals(_heap[i])) return true;
			return false;
		}

		public void Clear() {
			while(this.Count() > 0) this.Pop();
		}

		public IEnumerator<T> GetEnumerator() {
			var ret = new List<T>();

			while(this.Count() > 0) {
				ret.Add(this.Pop());
			}

			foreach(var r in ret) {
				this.Push(r);
				yield return r;
			}
		}

		public T[] ToArray() {
			T[] array = new T[_sz];
			int i = 0;

			foreach(var r in this) {
				array[i++] = r;
			}

			return array;
		}
	}
	class Graph<T> {
		public readonly int N;
		public T[] Nodes { get; }
		protected List<int>[] Nexts { get; }
		protected Dictionary<int, T>[] Edges { get; set; }
		public Graph(int _N) {
			N = _N;
			Nodes = new T[_N];
			Nexts = new List<int>[_N];
			Edges = new Dictionary<int, T>[_N];
			for(int i = 0; i < _N; i++) {
				Nexts[i] = new List<int>();
				Edges[i] = new Dictionary<int, T>();
			}
		}
		public void Update(int i, T value) {
			Nodes[i] = value;
		}
		public void AddEdge(int from, int to, T value) {
			Edges[from].Add(to, value);
			Nexts[from].Add(to);
		}
		public void PointDFS(int from, Action<int> onEnter, Action<int> onLeave) {
			var S = new Stack<int>();
			S.Push(from);
			var entered = new bool[N];
			var departed = new bool[N];//dep<=ent
			var childCount = new int[N];
			while(S.Any()) {
				var v = S.Peek();
				if(!entered[v]) {
					onEnter?.Invoke(v);
					entered[v] = true;
				}
				for(int i = childCount[v]; i < Nexts[v].Count; i++) {
					if(!entered[Nexts[v][i]]) {
						S.Push(Nexts[v][i]);
						childCount[v] = i + 1;
						goto NextLoop;
					}
				}
				S.Pop();
				childCount[v] = Nexts[v].Count;
				if(!departed[v]) {
					onLeave?.Invoke(v);
					departed[v] = true;
				}
			NextLoop:;
			}
		}
		public void MaxMatching() {
			var matching = new List<Tuple<int, int>>();
		}
	}
}