Problem:
Score : 600 points

Problem Statement

There are N dots in a two-dimensional plane. The coordinates of the i-th dot are (x_i, y_i).

We will repeat the following operation as long as possible:

Choose four integers a, b, c, d (a \neq c, b \neq d) such that there are dots at exactly three of the positions (a, b), (a, d), (c, b) and (c, d), and add a dot at the remaining position.

We can prove that we can only do this operation a finite number of times. Find the maximum number of times we can do the operation.

Constraints

1 \leq N \leq 10^5

1 \leq x_i, y_i \leq 10^5

If i \neq j, x_i \neq x_j or y_i \neq y_j.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the maximum number of times we can do the operation.

Sample Input 1

3
1 1
5 1
5 5

Sample Output 1

1

By choosing a = 1, b = 1, c = 5, d = 5, we can add a dot at (1, 5). We cannot do the operation any more, so the maximum number of operations is 1.

Sample Input 2

2
10 10
20 20

Sample Output 2

0

There are only two dots, so we cannot do the operation at all.

Sample Input 3

9
1 1
2 1
3 1
4 1
5 1
1 2
1 3
1 4
1 5

Sample Output 3

16

We can do the operation for all choices of the form a = 1, b = 1, c = i, d = j (2 \leq i,j \leq 5), and no more. Thus, the maximum number of operations is 16.

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

using System;
using System.Collections.Generic;
using System.IO;

namespace Sandbox
{
    internal class Program
    {
        private static readonly int MaxNumber = 100000;

        private static void Main(string[] args)
        {
            var scanner = new Scanner();
            var n = scanner.GetNextAsInt();

            var unionFind = new DisjointSet(MaxNumber * 2);
            var counterX = new int[MaxNumber * 2];
            var counterY = new int[MaxNumber * 2];

            for (var i = 0; i < n; i++)
            {
                var xi = scanner.GetNextAsInt();
                var yi = scanner.GetNextAsInt();
                unionFind.TryUnite(xi - 1, yi - 1 + MaxNumber);
            }

            for (var i = 0; i < MaxNumber; i++)
            {
                var rootForX = unionFind.GetRoot(i);
                var rootForY = unionFind.GetRoot(i + MaxNumber);
                counterX[rootForX]++;
                counterY[rootForY]++;
            }

            var result = 0L;

            for (var i = 0; i < MaxNumber * 2; i++)
            {
                result += (long)counterX[i] * (long)counterY[i];
            }

            Console.WriteLine(result - n);
        }

    #region Disjoint Set

    /// <summary>
    /// いくつかの要素を素集合に分割して保持するコレクションです．(Union Find)
    /// </summary>
    public class DisjointSet
    {
        // 正: Parent / 負: Size
        private readonly int[] _ParentOrSize;
        private readonly byte[] _Rank;
        private readonly HashSet<int> _Items = new HashSet<int>();

        /// <summary>
        /// 0 から <paramref name="n"/>-1 までの番号がついた <paramref name="n"/> 個の要素からなる集合を作成します．
        /// </summary>
        /// <param name="n">要素数</param>
        /// <remarks>このコンストラクタは O(N) で実行されます．</remarks>
        public DisjointSet(int n)
        {
            this._ParentOrSize = new int[n];
            this._Rank = new byte[n];
            this.Count = n;
            for (var i = 0; i < n; i++)
            {
                this._ParentOrSize[i] = -1;
            }
        }

        /// <summary>
        /// 指定した要素が属する集合の代表値を取得します．
        /// </summary>
        /// <param name="id">調べたい要素の 0-indexed での番号</param>
        /// <returns>指定した要素が属する集合の代表値</returns>
        /// <remarks>最悪計算量 O(α(N)) で実行されます．</remarks>
        public int GetRoot(int id)
        {
            if (this._ParentOrSize[id] < 0)
            {
                return id;
            }

            // 経路圧縮
            return this._ParentOrSize[id] = this.GetRoot(this._ParentOrSize[id]);
        }

        /// <summary>
        /// 指定した 2 つの要素が属する集合同士を 1 つに統合することを試みます．
        /// </summary>
        /// <param name="x">最初の要素の 0-indexed での番号</param>
        /// <param name="y">2 つ目の要素の 0-indexed での番号</param>
        /// <returns>統合に成功したならば true，そうでなければ false．</returns>
        /// <remarks>最悪計算量 O(α(N)) で実行されます．</remarks>
        public bool TryUnite(int x, int y)
        {
            this._Items.Add(x);
            this._Items.Add(y);

            x = this.GetRoot(x);
            y = this.GetRoot(y);
            if (x == y)
            {
                return false;
            }

            if (this._Rank[x] < this._Rank[y])
            {
                var tmp = x;
                x = y;
                y = tmp;
            }

            this._ParentOrSize[x] += this._ParentOrSize[y];
            this._ParentOrSize[y] = x;
            if (this._Rank[x] == this._Rank[y])
            {
                this._Rank[x]++;
            }

            this.Count--;
            return true;
        }

        public IReadOnlyCollection<int> GetItems() => this._Items;

        /// <summary>
        /// 指定した要素が属する集合のサイズを取得します．
        /// </summary>
        /// <param name="x">指定する要素の 0-indexed での番号</param>
        /// <returns>集合のサイズ</returns>
        /// <remarks>最悪計算量 O(α(N)) で実行されます．</remarks>
        public int Size(int x) => - this._ParentOrSize[this.GetRoot(x)];

        /// <summary>
        /// 指定した 2 つの要素が属する集合が同じかどうかを判定する．計算量 O(α(N))．
        /// </summary>
        /// <param name="x">最初の要素の 0-indexed での番号</param>
        /// <param name="y">2 つ目の要素の 0-indexed での番号</param>
        /// <returns>同じならば true，そうでなければ false．</returns>
        /// <remarks>最悪計算量 O(α(N)) で実行されます．</remarks>
        public bool IsSame(int x, int y) => this.GetRoot(x) == this.GetRoot(y);

        /// <summary>
        /// 集合の数を取得します。
        /// </summary>
        /// <returns></returns>
        public int Count { get; private set; }
    }

    #endregion

        #region String Scanner

        public class Scanner
        {
            private const char Separator = ' ';
            private readonly StreamReader _Stream = new StreamReader(Console.OpenStandardInput());
            private int _Pos;
            private string[] _Line = new string[0];

            #region get a element

            public string GetNext()
            {
                // ReSharper disable once InvertIf
                if (this._Pos >= this._Line.Length)
                {
                    // ReSharper disable once PossibleNullReferenceException
                    this._Line = this._Stream.ReadLine().Split(Separator);
                    this._Pos = 0;
                }

                return this._Line[this._Pos++];
            }

            public int GetNextAsInt()
            {
                return int.Parse(this.GetNext());
            }

            public long GetNextAsLong()
            {
                return long.Parse(this.GetNext());
            }

            public double GetNextAsDouble()
            {
                return double.Parse(this.GetNext());
            }

            #endregion

            #region get array

            public string[] ToArrayFromNextLine()
            {
                if (this._Pos >= this._Line.Length)
                {
                    // ReSharper disable once PossibleNullReferenceException
                    this._Line = this._Stream.ReadLine().Split(Separator);
                }

                this._Pos = this._Line.Length;
                return this._Line;
            }

            public int[] ToIntArrayFromNextLine()
            {
                return ToIntArray(this.ToArrayFromNextLine());
            }

            public long[] ToLongArrayFromNextLine()
            {
                return ToLongArray(this.ToArrayFromNextLine());
            }

            public double[] ToDoubleArrayFromNextLine()
            {
                return ToDoubleArray(this.ToArrayFromNextLine());
            }

            #endregion

            #region convert array

            private static int[] ToIntArray(string[] array)
            {
                var result = new int[array.Length];
                for (var i = 0; i < array.Length; i++)
                {
                    result[i] = int.Parse(array[i]);
                }

                return result;
            }

            private static long[] ToLongArray(string[] array)
            {
                var result = new long[array.Length];
                for (var i = 0; i < array.Length; i++)
                {
                    result[i] = long.Parse(array[i]);
                }

                return result;
            }

            private static double[] ToDoubleArray(string[] array)
            {
                var result = new double[array.Length];
                for (var i = 0; i < array.Length; i++)
                {
                    result[i] = double.Parse(array[i]);
                }

                return result;
            }

            #endregion
        }

        #endregion
    }
}