Problem:
Score : 400 points

Problem Statement

There are N pieces of sushi. Each piece has two parameters: "kind of topping" t_i and "deliciousness" d_i.
You are choosing K among these N pieces to eat.
Your "satisfaction" here will be calculated as follows:

The satisfaction is the sum of the "base total deliciousness" and the "variety bonus".

The base total deliciousness is the sum of the deliciousness of the pieces you eat.

The variety bonus is x*x, where x is the number of different kinds of toppings of the pieces you eat.

You want to have as much satisfaction as possible.
Find this maximum satisfaction.

Constraints

1 \leq K \leq N \leq 10^5

1 \leq t_i \leq N

1 \leq d_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
t_1 d_1
t_2 d_2
.
.
.
t_N d_N

Output

Print the maximum satisfaction that you can obtain.

Sample Input 1

5 3
1 9
1 7
2 6
2 5
3 1

Sample Output 1

26

If you eat Sushi 1,2 and 3:

The base total deliciousness is 9+7+6=22.

The variety bonus is 2*2=4.

Thus, your satisfaction will be 26, which is optimal.

Sample Input 2

7 4
1 1
2 1
3 1
4 6
4 5
4 5
4 5

Sample Output 2

25

It is optimal to eat Sushi 1,2,3 and 4.

Sample Input 3

6 5
5 1000000000
2 990000000
3 980000000
6 970000000
6 960000000
4 950000000

Sample Output 3

4900000016

Note that the output may not fit into a 32-bit integer type.

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

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using static System.Math;
using static AtCoderTemplate.MyConstants;
using static AtCoderTemplate.MyInputOutputs;
using static AtCoderTemplate.MyNumericFunctions;
using static AtCoderTemplate.MyAlgorithm;
using static AtCoderTemplate.MyDataStructure;
using static AtCoderTemplate.MyExtensions;
using static AtCoderTemplate.MyEnumerable;

namespace AtCoderTemplate {
    public class Program {
        public static void Main (string[] args) {
            var NK = ReadInts ();
            var N = NK[0];
            var K = NK[1];
            var td = ReadColumns (N, 2);
            var t = td[0].ToInts ();
            var d = td[1].ToLongs ();

            var sorted = Interval (0, N).OrderByDescending (i => d[i]);

            var G1 = sorted.Take (K).GroupBy (i => t[i]);
            var S = new PriorityQueue<int> ((s1, s2) => -(t[s1] - t[s2]));
            foreach (var i in G1.Where (gg => gg.Count () != 1).SelectMany (gg => gg.OrderBy (i => d[i]).Take (gg.Count () - 1))) {
                // Print (i);
                S.Enqueue (i);
            }
            // Print (d[S.Peek ()]);
            var selected = new bool[N + 1];
            foreach (var i in G1.Select (gg => gg.Key)) {
                selected[i] = true;
            }
            var G2 = sorted.Skip (K).Where (i => !selected[t[i]]).GroupBy (i => t[i]);
            var nonS = new PriorityQueue<int> ((s1, s2) => t[s1] - t[s2]);
            foreach (var s in G2.Select (gg => gg.OrderByDescending (i => d[i]).First ())) {
                nonS.Enqueue (s);
            }
            // Print (d[nonS.Peek ()]);
            var f = Enumerable.Repeat (-1L, N + 1).ToList ();
            var x = G1.Count ();
            f[x] = G1.SelectMany (gg => gg.Select (gi => d[gi])).Sum ();
            foreach (var a in x + 1 <= N + 1 ? Interval (x + 1, N + 1) : Enumerable.Empty<int> ()) {
                if (S.Size () == 0 || nonS.Size () == 0) break;
                var minS = S.Dequeue ();
                var maxNonS = nonS.Dequeue ();
                f[a] = f[a - 1] - d[minS] + d[maxNonS];
            }

            var g = Enumerable.Repeat (-1L, N + 1).ToList ();
            foreach (var i in Interval (1, N + 1)) {
                if (f[i] == -1) continue;
                g[i] = f[i] + (long) i * i;
            }
            // Print ("#");

            // foreach (var g1 in G1) {
            //     PrintRow (g1.Select (i => d[i]));
            // }
            // Print ("#");
            // foreach (var g2 in G2) {
            //     PrintRow (g2.Select (i => d[i]));
            // }
            // // Print ($"{sorted.GroupBy (gi => t[gi]).Count ()} {G1.Count()} {G2.Count()}");
            // PrintRow (f);
            // PrintRow (g);
            var ans = g.Max ();
            Print (ans);
        }
    }

    public static class MyInputOutputs {
        /* Input & Output*/
        public static string Read () {
            return Console.ReadLine ();
        }

        public static List<string> Reads () {
            return Console.ReadLine ().Split (' ').ToList ();
        }

        public static List<List<string>> ReadRows (int rowNum) {
            /*
            入力例
            A1 B1 C1 ... Z1
            A2 B2 C2 ... Z2
            ...
            An Bn Cn ... Zn
           

            出力例
            [[A1, B1, C1, ... Z1], [A2, B2, C2, ... Z2], ... [An, Bn, Cn, ... Zn]]
            */
            return Enumerable.Range (0, rowNum).Select (i => Reads ()).ToList ();
        }

        public static List<List<string>> ReadColumns (int rowNum, int colNum) {
            /*
            入力例
            A1 B1 C1 ... Z1
            A2 B2 C2 ... Z2
            ...
            An Bn Cn ... Zn
           

            出力例
            [[A1, A2, A3, ... An], [B1, B2, B3, ... Bn], ... [Z1, Z2, Z3, ... Zn]]
            */
            var rows = ReadRows (rowNum);
            return Enumerable.Range (0, colNum).Select (i => rows.Select (items => items[i].ToString ()).ToList ()).ToList ();
        }

        public static List<List<string>> ReadGridGraph (int height, int width) {
            /*
            入力例
            A1B1C1...Z1
            A2B2C2...Z2
            ...
            AnBnCn...Zn
           

            出力例
            [[A1, B1, C1, ... Z1], [A2, B2, C2, ... Z2], ... [An, Bn, Cn, ... Zn]]
            */
            return Enumerable.Range (0, height)
                .Select (i =>
                    Read ()
                    .Select (c => c.ToString ())
                    .ToList ()
                ).ToList ();
        }

        public static int ToInt (this string str) {
            return int.Parse (str);
        }

        public static long ToLong (this string str) {
            return long.Parse (str);
        }

        public static List<int> ToInts (this List<string> strs) {
            return strs.Select (str => str.ToInt ()).ToList ();
        }

        public static List<long> ToLongs (this List<string> strs) {
            return strs.Select (str => str.ToLong ()).ToList ();
        }

        public static int ReadInt () {
            return Read ().ToInt ();
        }
        public static long ReadLong () {
            return Read ().ToLong ();
        }

        public static List<int> ReadInts () {
            return Reads ().ToInts ();
        }
        public static List<long> ReadLongs () {
            return Reads ().ToLongs ();
        }

        public static void Print<T> (T item) {
            Console.WriteLine (item);
        }
        public static void PrintIf<T1, T2> (bool condition, T1 trueResult, T2 falseResult) {
            if (condition) {
                Console.WriteLine (trueResult);
            } else {
                Console.WriteLine (falseResult);
            }
        }

        public static void PrintRow<T> (IEnumerable<T> list) {
            /* 横ベクトルで表示
            A B C D ...
            */
            if (!list.IsEmpty ()) {
                Console.Write (list.First ());
                foreach (var item in list.Skip (1)) {
                    Console.Write ($" {item}");
                }
            }
            Console.Write ("\n");
        }
        public static void PrintColomn<T> (IEnumerable<T> list) {
            /* 縦ベクトルで表示
            A
            B
            C
            D
            ...
            */
            foreach (var item in list) {
                Console.WriteLine (item);
            }
        }
        public static void PrintRows<T> (IEnumerable<IEnumerable<T>> sources) {
            foreach (var row in sources) {
                PrintRow (row);
            }
        }

        public static void PrintGridGraph<T> (IEnumerable<IEnumerable<T>> sources) {
            foreach (var row in sources) {
                Print (String.Concat (row));
            }
        }
    }

    public static class MyConstants {
        public static IEnumerable<char> lowerAlphabets = Enumerable.Range ('a', 'z' - 'a' + 1).Select (i => (char) i);
        public static IEnumerable<char> upperAlphabets = Enumerable.Range ('A', 'Z' - 'A' + 1).Select (i => (char) i);

        public static int p1000000007 = (int) Pow (10, 9) + 7;
    }

    public static class MyNumericFunctions {
        public static bool IsEven (long a) {
            return a % 2 == 0;
        }
        public static bool IsOdd (long a) {
            return !IsEven (a);
        }

        /// <summary>
        /// 冪を得る
        /// </summary>
        /// <param name="b">底</param>
        /// <param name="n">冪指数</param>
        /// <param name="divisor">返り値がintを超えないようにdivisorで割ったあまりを得る</param>
        /// <returns>bのn乗(をdivisorで割ったあまり)</returns>
        public static int PowRem (long b, int n, int divisor) {
            return Enumerable.Repeat (b, n)
                .Aggregate (1, (accm, i) => (int) ((accm * i) % divisor));
        }

        /// <summary>
        /// 順列の総数を得る
        /// O(N-K)
        /// </summary>
        /// <param name="n">全体の数</param>
        /// <param name="k">並べる数</param>
        /// <param name="divisor">返り値がintを超えないようにdivisorで割った余りを得る</param>
        /// <returns>nPk (をdivisorで割った余り)</returns>
        public static int nPk (int n, int k, int divisor) {
            if (k > n) {
                return 0;
            } else {
                return Enumerable.Range (n - k + 1, k).Aggregate (1, ((i, m) => (i * m) % divisor));
            }
        }

        /// <summary>
        /// 階乗を得る
        /// O(N)
        /// </summary>
        /// <param name="n"></param>
        /// <param name="divisor">返り値がintを超えないようにdivisorで割った余りを得る</param>
        /// <returns>n! (をdivisorで割った余り)</returns>
        public static int Fact (int n, int divisor) {
            return nPk (n, n, divisor);
        }

        /// <summary>
        /// 階乗のテーブルを得る
        /// O(N)
        /// </summary>
        /// <param name="nmax"></param>
        /// <param name="divisor">返り値がintを超えないようにdivisorで割った余りを得る</param>
        /// <returns></returns>
        public static int[] FactTable (int nmax, int divisor) {
            return Enumerable.Range (1, nmax)
                .Scanl (1, (accm, i) => (int) ((long) accm * i % divisor))
                .ToArray ();
        }

        public static int[, ] PascalsTriangle (int nmax, int kmax, int divisor) {
            var comb = new int[2000 + 1, 2000 + 1];
            foreach (var n in MyEnumerable.Interval (0, 2000 + 1)) {
                foreach (var k in MyEnumerable.Interval (0, 2000 + 1)) {
                    if (n < k) continue;

                    if (k == 0) {
                        comb[n, k] = 1;
                    } else {
                        comb[n, k] = (int) (((long) comb[n - 1, k - 1] + comb[n - 1, k]) % divisor);
                    }
                }
            }

            return comb;
        }

        /// <summary>
        /// 最大公約数を得る 
        /// O(log N)
        /// </summary>
        /// <param name="m">自然数</param>
        /// <param name="n">自然数</param>
        /// <returns></returns>
        public static long GCD (long m, long n) {
            // GCD(m,n) = GCD(n, m%n)を利用
            // m%n = 0のとき、mはnで割り切れるので、nが最大公約数
            if (m <= 0L || n <= 0L) throw new ArgumentOutOfRangeException ();

            if (m < n) return GCD (n, m);
            while (m % n != 0L) {
                var n2 = m % n;
                m = n;
                n = n2;
            }
            return n;
        }

        /// <summary>
        /// 最小公倍数を得る
        /// O(log N)
        /// </summary>
        /// <param name="m"></param>
        /// <param name="n"></param>
        /// <returns></returns>
        public static long LCM (long m, long n) {
            var ans = checked ((long) (BigInteger.Multiply (m, n) / GCD (m, n)));
            return ans;
        }

        /// <summary>
        /// 約数列挙(非順序)
        /// O(√N)
        /// </summary>
        /// <param name="m">m > 0</param>
        /// <returns></returns>
        public static IEnumerable<long> Divisor (long m) {
            if (!(m > 0)) throw new ArgumentOutOfRangeException ();

            var front = Enumerable.Range (1, (int) Sqrt (m))
                .Select (i => (long) i)
                .Where (d => m % d == 0);
            return front.Concat (front.Where (x => x * x != m).Select (x => m / x));
        }

        public static IEnumerable<int> Divisor (int m) {
            if (!(m > 0)) throw new ArgumentOutOfRangeException ();

            var front = Enumerable.Range (1, (int) Sqrt (m))
                .Where (d => m % d == 0);
            return front.Concat (front.Where (x => x * x != m).Select (x => m / x));
        }

        /// <summary>
        /// 公約数列挙(非順序)
        /// O(√N)
        /// </summary>
        /// <param name="m">m > 0</param>
        /// <param name="n">n > 0 </param>
        /// <returns></returns>
        public static IEnumerable<long> CommonDivisor (long m, long n) {
            if (!(m > 0)) throw new ArgumentOutOfRangeException ();

            if (m < n) return CommonDivisor (n, m);
            return Divisor (m).Where (md => n % md == 0);
        }

        public static IEnumerable<int> CommonDivisor (int m, int n) {
            if (!(m > 0)) throw new ArgumentOutOfRangeException ();

            if (m < n) return CommonDivisor (n, m);
            return Divisor (m).Where (md => n % md == 0);
        }

        /// <summary>
        /// nまでの素数を得る
        /// </summary>
        /// <param name="n">n > 1</param>
        /// <returns></returns>
        public static IEnumerable<int> Prime (int n) {
            if (!(n > 1)) throw new ArgumentOutOfRangeException ();

            var primes = new List<int> ();
            foreach (var i in Enumerable.Range (2, n - 1)) {
                if (primes.All (p => i % p != 0)) {
                    primes.Add (i);
                }
            }
            return primes;
        }

        /// <summary>
        /// 順列を得る
        /// O(N!)
        /// </summary>
        public static IEnumerable<IEnumerable<T>> Perm<T> (IEnumerable<T> source, int n) {
            if (n == 0 || source.IsEmpty () || source.Count () < n) {
                return Enumerable.Empty<IEnumerable<T>> ();
            } else if (n == 1) {
                return source.Select (i => new List<T> { i });
            } else {
                var nexts = source.Select ((x, i) =>
                    new { next = source.Take (i).Concat (source.Skip (i + 1)), selected = source.Take (i + 1).Last () });
                return nexts.SelectMany (next => Perm (next.next, n - 1).Select (item => item.Prepend (next.selected)));
            }
        }
    }

    public static class MyAlgorithm {
        /// <summary>
        /// めぐる式二分探索法
        /// O(log N)
        /// </summary>
        /// <param name="list">探索するリスト</param>
        /// <param name="predicate">条件の述語関数</param>
        /// <param name="ng">条件を満たさない既知のindex</param>
        /// <param name="ok">条件を満たす既知のindex</param>
        /// <returns>条件を満たすindexの内、隣がfalseとなるtrueのindexを返す</returns>
        public static int BinarySearch<T> (List<T> list, Func<T, bool> predicate, int ng, int ok) {
            while (Abs (ok - ng) > 1) {
                int mid = (ok + ng) / 2;
                if (predicate (list[mid])) {
                    ok = mid;
                } else {
                    ng = mid;
                }
            }
            return ok;
        }

        /// <summary>
        /// 左二分探索法
        /// </summary>
        /// <param name="list">条件の境界(falseとtrueの変わるところ)で、trueが左にあるリスト</param>
        /// <param name="predicate">条件の述語関数</param>
        /// <returns>右隣がfalseになるtrueのindexを返す</returns>
        public static int LeftBinarySearch<T> (List<T> list, Func<T, bool> predicate) {
            return BinarySearch (list, predicate, list.Count, -1);
        }

        /// <summary>
        /// 右二分探索法
        /// </summary>
        /// <param name="list">条件の境界(falseとtrueの変わるところ)で、trueが右にあるリスト</param>
        /// <param name="predicate">条件の述語関数</param>
        /// <returns>左隣がfalseになるtrueのindexを返す</returns>
        public static int RightBinarySearch<T> (List<T> list, Func<T, bool> predicate) {
            return BinarySearch (list, predicate, -1, list.Count);
        }

        // Dictionaryがcapacity近くになるとゲロ重かったので削除
        // 配列にするかも

        // /// <summary>
        // /// DynamicProgrammingの形式(貰うDP)
        // /// 項dp_(i,j,k...)の漸化式、初項、漸化式の計算順序（添字のパターン）、から漸化式を計算する
        // /// </summary>
        // /// <param name="calculationOrders">漸化式の計算順序、項dp_(i,j,k...)の添字(i,j,k...)のトポロジカル順序を持つシーケンス</param>
        // /// <param name="initialConditions">初項とその添字のペア、のシーケンス</param>
        // /// <param name="recurrenceRelation">項dp_(i,j,k...)の漸化式、以前の項の計算はDictonaryから得る</param>
        // /// <typeparam name="Indexes">項dp_(i,j,k...)の添字(i,j,k...)</typeparam>
        // /// <typeparam name="Result">項dp_(i,j,k...)の計算結果</typeparam>
        // /// <returns>項dp_(i,j,k...)の全ての計算結果をDictionaryで返す</returns>
        // public static Dictionary<Indexes, Result> DynamicProgramming<Indexes, Result> (
        //     IEnumerable<Indexes> calculationOrders //
        //     , IEnumerable<KeyValuePair<Indexes, Result>> initialConditions //
        //     , Func<Dictionary<Indexes, Result>, Indexes, Result> recurrenceRelation) {
        //     var conditions = initialConditions.ToDictionary (kv => kv.Key, kv => kv.Value);
        //     foreach (var order in calculationOrders) {
        //         conditions.Add (order, recurrenceRelation (conditions, order));
        //     }
        //     return conditions;
        // }

        // .NetFramework 4.6.2 ではタプルの記法がまだ使えない

        // /// <summary>
        // /// DynamicProgrammingの形式(貰うDP)
        // /// 項dp_(i,j,k...)の漸化式、初項、漸化式の計算順序（添字のパターン）、から漸化式を計算する
        // /// </summary>
        // /// <param name="calculationOrders">漸化式の計算順序、項dp_(i,j,k...)の添字(i,j,k...)のトポロジカル順序を持つシーケンス</param>
        // /// <param name="initialConditions">初項とその添字のペア、のシーケンス</param>
        // /// <param name="recurrenceRelation">項dp_(i,j,k...)の漸化式、以前の項の計算はDictonaryから得る</param>
        // /// <typeparam name="Indexes">項dp_(i,j,k...)の添字(i,j,k...)</typeparam>
        // /// <typeparam name="Result">項dp_(i,j,k...)の計算結果</typeparam>
        // /// <returns>項dp_(i,j,k...)の全ての計算結果をDictionaryで返す</returns>
        // public static Dictionary<Indexes, Result> DynamicProgramming<Indexes, Result> (
        //     IEnumerable<Indexes> calculationOrders //
        //     , IEnumerable < (Indexes indexes, Result result) > initialConditions //
        //     , Func<Dictionary<Indexes, Result>, Indexes, Result> recurrenceRelation) {
        //     var conditions = initialConditions.ToDictionary (kv => kv.indexes, kv => kv.result);
        //     foreach (var order in calculationOrders) {
        //         conditions.Add (order, recurrenceRelation (conditions, order));
        //     }
        //     return conditions;
        // }

        /// <summary>
        /// 右へのしゃくとり法の形式
        /// </summary>
        /// <param name="n">なめるシーケンスの長さ</param>
        /// <param name="Predicate">更新のための条件関数。indexの(left,right)をとり、条件を満たすとUpdateを行う</param>
        /// <param name="initialCondition">初期状態。</param>
        /// <param name="Update">状態更新関数。indexのleft, rightと前のconditionをとり、更新したconditionを返す</param>
        /// <typeparam name="TR"></typeparam>
        /// <returns></returns>
        public static TR TwoPointersRight<TR> (int n, Func<int, int, bool> Predicate, TR initialCondition, Func<int, int, TR, TR> Update) {
            var l = 0;
            var r = 0;
            TR condition = initialCondition;
            while (r < n) {
                while (r < n && !Predicate (l, r)) {
                    r += 1;
                }
                while (r < n && l != r && Predicate (l, r)) {
                    condition = Update (l, r, condition);
                    l += 1;
                }
            }
            return condition;
        }

        public static void TreeBFS (
            int nodeCount,
            IEnumerable<int> initialNodes,
            Func<int, IEnumerable<int>> getChilds,
            Predicate<int> continuePred,
            Action<int, int> updateItem) {

            var que = new Queue<int> ();
            var visited = new bool[nodeCount];
            foreach (var node in initialNodes) {
                que.Enqueue (node);
                visited[node] = true;
            }

            while (!que.IsEmpty ()) {
                var parent = que.Dequeue ();
                foreach (var child in getChilds (parent)) {
                    if (visited[child] || continuePred (child)) continue;
                    updateItem (parent, child);
                    que.Enqueue (child);
                    visited[child] = true;
                }
            }
        }

        /// <summary>
        /// グラフの辺
        /// </summary>
        public struct Edge {
            public int SourceNode { get; }
            public int TargetNode { get; }
            public long Weight { get; }

            public Edge (int sourceNode, int targetNode, long weight) {
                SourceNode = sourceNode;
                TargetNode = targetNode;
                Weight = weight;
            }
        }

        public static IEnumerable<Edge> CreateEdges (List<int> sourceNodes, List<int> targetNodes, List<long> weights) {
            return Enumerable.Range (0, sourceNodes.Count)
                .Select (i => new Edge (sourceNode: sourceNodes[i], targetNode: targetNodes[i], weight: weights[i]));
        }

        public static List<Dictionary<int, long>> CreateAdjacencyListOfUndirectedGraph (IEnumerable<Edge> edges, int nodeNum) {
            var adjacencyList = Enumerable.Range (0, nodeNum).Select (_ => new Dictionary<int, long> ()).ToList ();
            foreach (var edge in edges) {
                adjacencyList[edge.SourceNode].Add (edge.TargetNode, edge.Weight);
                adjacencyList[edge.TargetNode].Add (edge.SourceNode, edge.Weight);
            }
            return adjacencyList;
        }

        public static List<List<long>> CreateAdjacencyMatrixOfUndirectedGraph (IEnumerable<Edge> edges, int nodeNum, long weightOfNoEdge) {
            var adjacencyMatrix = Enumerable.Range (0, nodeNum)
                .Select (i =>
                    Enumerable.Range (0, nodeNum)
                    .Select (k =>
                        weightOfNoEdge
                    ).ToList ()
                ).ToList ();
            foreach (var edge in edges) {
                adjacencyMatrix[edge.SourceNode][edge.TargetNode] = edge.Weight;
                adjacencyMatrix[edge.TargetNode][edge.SourceNode] = edge.Weight;
            }
            return adjacencyMatrix;
        }

        /// <summary>
        /// ワーシャルフロイド法
        /// O(N^3)
        /// </summary>
        /// <param name="adjacencyMatrix">隣接行列</param>
        /// <param name="nodeNum">ノードの数</param>
        /// <returns>各ノード間の最短距離を重みとする隣接行列</returns>
        public static List<List<long>> WarshallFloyd (List<List<long>> adjacencyMatrix) {
            var nodeNum = adjacencyMatrix.Count;
            var res = Enumerable.Range (0, nodeNum).Select (i => new List<long> (adjacencyMatrix[i])).ToList ();
            foreach (var b in Enumerable.Range (0, nodeNum)) {
                foreach (var a in Enumerable.Range (0, nodeNum)) {
                    foreach (var c in Enumerable.Range (0, nodeNum)) {
                        res[a][c] = Min (res[a][c], res[a][b] + res[b][c]);
                    }
                }
            }
            return res;
        }

    }

    public static class MyDataStructure {
        public class UnionFind {
            List<int> parent;
            List<int> size;
            public UnionFind (int N) {
                // 最初はすべて異なるグループ番号(root)が割り当てられる
                parent = Enumerable.Range (0, N).ToList ();
                size = Enumerable.Repeat (1, N).ToList ();
            }

            // 頂点uの属するグループ番号(root)を探す
            int Root (int u) {
                if (parent[u] == u) {
                    return u;
                } else {
                    var root = Root (parent[u]);
                    parent[u] = root; // 経路圧縮
                    return root;
                }
            }

            // 2つのグループを統合する(rootが異なる場合、同じrootにする)
            public void Unite (int u, int v) {
                int root_u = Root (u);
                int root_v = Root (v);
                if (root_u == root_v) return;
                parent[root_u] = root_v; // root_vをroot_uの親とする
                size[root_v] += size[root_u];
            }

            public int Size (int u) {
                return size[Root (u)];
            }

            public bool IsSame (int u, int v) {
                int root_u = Root (u);
                int root_v = Root (v);
                return root_u == root_v;
            }
        }

        /// <summary>
        /// Comparison<T>(順序を定める高階関数 t -> t -> Orderingみたいなもの)からIComparer<T>の実体へ変換するクラス
        /// </summary>
        /// <typeparam name="T">順序付けられる型</typeparam>
        public class ComparisonToComparerConverter<T> : IComparer<T> {
            Comparison<T> comparison;
            public ComparisonToComparerConverter (Comparison<T> comparison) {
                this.comparison = comparison;
            }
            public int Compare (T x, T y) {
                return comparison (x, y);
            }
        }

        /// <summary>
        /// 優先度付きキュー
        /// 先頭参照・要素数がO(1)、要素の追加・先頭削除がO(log N)
        /// </summary>
        /// <typeparam name="T"></typeparam>
        public class PriorityQueue<T> where T : IComparable<T> {
            SortedDictionary<T, Queue<T>> dict;
            int size = 0;

            public PriorityQueue () {
                dict = new SortedDictionary<T, Queue<T>> ();
            }

            public PriorityQueue (IComparer<T> comparer) {
                dict = new SortedDictionary<T, Queue<T>> (comparer);
            }

            public PriorityQueue (Comparison<T> comparison) {
                dict = new SortedDictionary<T, Queue<T>> (new ComparisonToComparerConverter<T> (comparison));
            }

            public void Enqueue (T item) {
                if (dict.ContainsKey (item)) {
                    dict[item].Enqueue (item);
                } else {
                    var added = new Queue<T> ();
                    added.Enqueue (item);
                    dict.Add (item, added);
                }
                size += 1;
            }

            public T Peek () {
                return dict.First ().Value.First ();
            }

            public int Size () {
                return size;
            }

            public T Dequeue () {
                var first = dict.First ();
                if (first.Value.Count <= 1) {
                    dict.Remove (first.Key);
                }
                size -= 1;
                return first.Value.Dequeue ();

            }
        }
    }

    public static class MyExtensions {
        // AppendとPrependが、.NET Standard 1.6からの追加で、Mono 4.6.2 はそれに対応して仕様はあるが、実装がない
        public static IEnumerable<T> Append<T> (this IEnumerable<T> source, T element) {
            return source.Concat (Enumerable.Repeat (element, 1));
        }

        public static IEnumerable<T> Prepend<T> (this IEnumerable<T> source, T element) {
            return Enumerable.Repeat (element, 1).Concat (source);
        }

        // TakeLastとSkipLastが、.Net Standard 2.1からの追加で、Mono 4.6.2 はそれに対応していない
        public static IEnumerable<T> TakeLast<T> (this IEnumerable<T> source, int count) {
            return source.Skip (source.Count () - count);
        }

        public static IEnumerable<T> SkipLast<T> (this IEnumerable<T> source, int count) {
            return source.Take (source.Count () - count);
        }

        public static bool IsEmpty<T> (this IEnumerable<T> source) {
            return !source.Any ();
        }

        /// <summary>
        /// シーケンスの要素ごとに副作用を起こす（シーケンスはそのまま）
        /// </summary>
        /// <param name="source">シーケンス</param>
        /// <param name="action">副作用。要素をとるアクション</param>
        /// <typeparam name="T">シーケンスの要素の型</typeparam>
        /// <returns>元のシーケンス</returns>
        public static IEnumerable<T> Do<T> (this IEnumerable<T> source, Action<T> action) {
            foreach (var item in source) {
                action (item);
            }
            return source;
        }

        /// <summary>
        /// パイプライン演算子のようにデータを変換する（関数を適用する）
        /// </summary>
        /// <param name="arg">変換するデータ</param>
        /// <param name="func">変換</param>
        /// <typeparam name="T">変換元のデータの型</typeparam>
        /// <typeparam name="TR">変換先のデータの型</typeparam>
        /// <returns>変換されたデータ</returns>
        public static TR Apply<T, TR> (this T arg, Func<T, TR> func) {
            return func (arg);
        }

        /// <summary>
        /// データから副作用（出力や破壊的代入など）を生み、データを返す
        /// </summary>
        /// <param name="item">元のデータ</param>
        /// <param name="effect">副作用（出力や破壊的代入など）</param>
        /// <typeparam name="T">データの型</typeparam>
        /// <returns>副作用を起こした後のデータ</returns>
        public static T Effect<T> (this T item, Action<T> effect) {
            effect (item);
            return item;
        }

        /// <summary>
        /// シーケンスの隣り合う要素を2引数の関数に適用したシーケンスを得る
        /// </summary>
        /// <para>O(N)</para>
        /// <param name="source">元のシーケンス</param>
        /// <param name="func">2引数関数</param>
        /// <example>[1,2,3,4].MapAdjacent(f) => [f(1,2), f(2,3), f(3,4)]</example>
        public static IEnumerable<TR> MapAdjacent<T1, TR> (this IEnumerable<T1> source, Func<T1, T1, TR> func) {
            var list = source.ToList ();
            return Enumerable.Range (1, list.Count - 1)
                .Select (i => func (list[i - 1], list[i]));
        }

        /// <summary>
        /// 累積項を要素にもつシーケンスを得る(初項は、first)
        /// <para>O(N)</para>
        /// </summary>
        /// <param name="source">元のシーケンス</param>
        /// <param name="func">2引数関数f</param>
        /// <param name="first">func(first, source[0])のための初項</param>
        /// <example> [1,2,3].Scanl1(0,f) => [0, f(0,1), f(f(0,1),2), f(f(f(0,1),2),3)]</example>
        public static IEnumerable<TR> Scanl<T, TR> (this IEnumerable<T> source, TR first, Func<TR, T, TR> func) {
            var list = source.ToList ();
            var result = new List<TR> { first };
            foreach (var i in Enumerable.Range (0, source.Count ())) {
                result.Add (func (result[i], list[i]));
            }
            return result;
        }
        /// <summary>
        /// 累積項を要素にもつシーケンスを得る（初項は、source.First()）
        /// <para>O(N)</para>
        /// </summary>
        /// <param name="source">要素数1以上のシーケンス</param>
        /// <param name="func">2引数関数f</param>
        /// <example> [1,2,3].Scanl1(f) => [1, f(1,2), f(f(1,2),3)]</example>
        public static IEnumerable<T> Scanl1<T> (this IEnumerable<T> source, Func<T, T, T> func) {
            if (source.IsEmpty ()) throw new ArgumentOutOfRangeException ();
            var list = source.ToList ();
            var result = new List<T> { list[0] };
            foreach (var i in Enumerable.Range (1, source.Count () - 1)) {
                result.Add (func (result[i - 1], list[i]));
            }
            return result;
        }

        /// <summary>
        /// 数列の和をdivisorで割った余りを計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <param name="divisor">割る数</param>
        /// <returns>数列の和をdivisorで割った余り</returns>
        public static int Sum (this IEnumerable<int> source, int divisor) {
            return source.Aggregate (0, (a, b) => (int) (((long) a + b) % divisor));
        }

        /// <summary>
        /// 数列の積を計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <returns>数列の積</returns>
        public static long Product (this IEnumerable<long> source) {
            return source.Aggregate (1L, (a, b) => a * b);
        }

        /// <summary>
        /// 数列の積をdivisorで割った余りを計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <param name="divisor">割る数</param>
        /// <returns>数列の積をdivisorで割った余り</returns>
        public static int Product (this IEnumerable<int> source, int divisor) {
            return source.Aggregate (1, (a, b) => (int) (((long) a * b) % divisor));
        }

        /// <summary>
        /// indexの列で文字列を分割する
        /// </summary>
        /// <param name="source">元の文字列</param>
        /// <param name="cutIndexes">分割する位置の列</param>
        /// <returns>分割された文字列</returns>
        /// <example>CutForIndexes("abcdef", [0, 2, 3, 5, 6]) => ["ab", "c", "de", "f"]</example>
        public static IEnumerable<string> CutForIndexes (string source, IEnumerable<int> cutIndexes) {
            return cutIndexes.MapAdjacent ((i0, i1) => source.Substring (i0, i1 - i0));
        }

    }

    public static class MyEnumerable {
        /// <summary>
        /// 左閉右開区間 [startIndex,endIndex) を得る
        /// </summary>
        /// <param name="startIndex">始まりのインデックス。含む</param>
        /// <param name="endIndex">終わりのインデックス。含まない</param>
        public static IEnumerable<int> Interval (int startIndex, int endIndex) {
            if (endIndex - startIndex < 0) new ArgumentException ();
            return Enumerable.Range (startIndex, endIndex - startIndex);
        }

        /// <summary>
        /// フラグから分割するindexの位置の列への変換
        /// </summary>
        /// <param name="flags">二進数によるフラグ</param>
        /// <param name="flagSize">フラグの数</param>
        /// <returns>分割するindexの位置の列</returns>
        /// <example> CutFlagToCutIndex(10110, 5) => [0, 2, 3, 5, 6]</example>
        public static IEnumerable<int> CutFlagToIndexes (int flags) {
            int flagSize = (int) Log (flags, 2);
            var indexes = new List<int> { 0 };
            foreach (var i in MyEnumerable.Interval (0, flagSize)) {
                if ((flags >> i) % 2 == 1) {
                    indexes.Add (i + 1);
                }
            }
            indexes.Add (flagSize + 1);
            return indexes;
        }
    }
}