Problem:
Score : 400 points

Problem Statement

Kenkoooo is planning a trip in Republic of Snuke.
In this country, there are n cities and m trains running.
The cities are numbered 1 through n, and the i-th train connects City u_i and v_i bidirectionally.
Any city can be reached from any city by changing trains.

Two currencies are used in the country: yen and snuuk.
Any train fare can be paid by both yen and snuuk.
The fare of the i-th train is a_i yen if paid in yen, and b_i snuuk if paid in snuuk.

In a city with a money exchange office, you can change 1 yen into 1 snuuk.
However, when you do a money exchange, you have to change all your yen into snuuk.
That is, if Kenkoooo does a money exchange when he has X yen, he will then have X snuuk.
Currently, there is a money exchange office in every city, but the office in City i will shut down in i years and can never be used in and after that year.

Kenkoooo is planning to depart City s with 10^{15} yen in his pocket and head for City t, and change his yen into snuuk in some city while traveling.
It is acceptable to do the exchange in City s or City t.

Kenkoooo would like to have as much snuuk as possible when he reaches City t by making the optimal choices for the route to travel and the city to do the exchange.
For each i=0,...,n-1, find the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i years.
You can assume that the trip finishes within the year.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq s,t \leq n

s \neq t

1 \leq u_i < v_i \leq n

1 \leq a_i,b_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

Any city can be reached from any city by changing trains.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m s t
u_1 v_1 a_1 b_1
:
u_m v_m a_m b_m

Output

Print n lines.
In the i-th line, print the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i-1 years.

Sample Input 1

4 3 2 3
1 4 1 100
1 2 1 10
1 3 20 1

Sample Output 1

999999999999998
999999999999989
999999999999979
999999999999897

After 0 years, it is optimal to do the exchange in City 1.

After 1 years, it is optimal to do the exchange in City 2.

Note that City 1 can still be visited even after the exchange office is closed.
Also note that, if it was allowed to keep 1 yen when do the exchange in City 2 and change the remaining yen into snuuk, we could reach City 3 with 999999999999998 snuuk, but this is NOT allowed.

After 2 years, it is optimal to do the exchange in City 3.

After 3 years, it is optimal to do the exchange in City 4.
Note that the same train can be used multiple times.

Sample Input 2

8 12 3 8
2 8 685087149 857180777
6 7 298270585 209942236
2 4 346080035 234079976
2 5 131857300 22507157
4 8 30723332 173476334
2 6 480845267 448565596
1 4 181424400 548830121
4 5 57429995 195056405
7 8 160277628 479932440
1 6 475692952 203530153
3 5 336869679 160714712
2 7 389775999 199123879

Sample Output 2

999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
//using System.Numerics;  //comment out if AOJ
using System.Text;

using Problem = Tmp.Problem;
using MyIO;

#pragma warning disable   //for AOJ

namespace Tmp
{
    class Problem : IDisposable
    {
        bool IsGCJ;
        int Repeat;
        Scanner sc;
        Printer pr;
        public Problem(bool isGCJ, Scanner scanner, Printer printer)
        {
            sc = scanner;
            pr = printer;
            IsGCJ = isGCJ;
            if (isGCJ) Repeat = sc.nextInt();
            else Read();
        }
        public Problem(bool isGCJ) : this(isGCJ, new Scanner(), new Printer()) { }
        public Problem(bool isGCJ, Scanner scanner) : this(isGCJ, scanner, new Printer()) { }
        public Problem(bool isGCJ, Printer printer) : this(isGCJ, new Scanner(), printer) { }
        public void Solve()
        {
            if (IsGCJ) for (var i = 0; i < Repeat; i++) { Read(); pr.Write("Case #" + (i + 1) + ": "); SolveOne(); }
            else SolveOne();
        }
        public void Dispose()
        {
            sc.Dispose();
            pr.Dispose();
        }
        public int Size { get { return 1; } }
        public const long Mod = 1000000007;

        // 使用する変数をここに書く
        // string S;
        // int a;
        int n, m, s, t;
        int[] u, v;
        long[] a, b;
        /// <summary>
        /// 読み込み処理をここに書く
        /// </summary>
        void Read()
        {

        }
        /// <summary>
        /// メイン処理をここに書く
        /// </summary>
        void SolveOne()
        {
            n = sc.nextInt();
            m = sc.nextInt();
            s = sc.nextInt() - 1;
            t = sc.nextInt() - 1;

            u = new int[m];
            v = new int[m];
            a = new long[m];
            b = new long[m];

            for (int i = 0; i < m; i++)
            {
                u[i] = sc.nextInt() - 1;
                v[i] = sc.nextInt() - 1;
                a[i] = sc.nextLong();
                b[i] = sc.nextLong();
            }

            List<KeyValuePair<int, long>>[] kvp1 = new List<KeyValuePair<int, long>>[n];
            List<KeyValuePair<int, long>>[] kvp2 = new List<KeyValuePair<int, long>>[n];
            for (int i = 0; i < n; i++)
            {
                kvp1[i] = new List<KeyValuePair<int, long>>();
                kvp2[i] = new List<KeyValuePair<int, long>>();
            }

            for (int i = 0; i < m; i++)
            {
                kvp1[u[i]].Add(new KeyValuePair<int, long>(v[i], a[i]));
                kvp1[v[i]].Add(new KeyValuePair<int, long>(u[i], a[i]));
                kvp2[u[i]].Add(new KeyValuePair<int, long>(v[i], b[i]));
                kvp2[v[i]].Add(new KeyValuePair<int, long>(u[i], b[i]));
            }


            var p1 = Graph.Dijkstra(kvp1, s);
            var p2 = Graph.Dijkstra(kvp2, t);

            long[] ans = new long[n];
            long min = long.MaxValue;
            for (int i = n - 1; i >= 0; i--)
            {
                long tmp = p1[i] + p2[i];
                min = Math.Min(min, tmp);
                ans[i] = min;
            }
            for (int i = 0; i < n; i++)
            {
                Console.WriteLine((long)1e15 - ans[i]);
            }
        }
    }


    static partial class Graph
    {
        //O(|V+E|log|V|)
        static public long[] Dijkstra(this List<KeyValuePair<int, long>>[] G, int s)
        {
            var n = G.Length;
            var ans = Enumerable.Repeat(1L << 60, n).ToArray();
            ans[s] = 0;
            var pq = new PriorityQueue<KeyValuePair<int, long>>((l, r) => l.Value.CompareTo(r.Value));
            pq.Enqueue(new KeyValuePair<int, long>(s, 0));
            while (pq.Count > 0)
            {
                var p = pq.Dequeue();
                if (p.Value > ans[p.Key]) continue;
                foreach (var to in G[p.Key])
                    if (p.Value + to.Value < ans[to.Key])
                    {
                        ans[to.Key] = p.Value + to.Value;
                        pq.Enqueue(new KeyValuePair<int, long>(to.Key, ans[to.Key]));
                    }
            }

            return ans;
        }
    }

}

class Program
{
    //public static RandomSFMT rand = new RandomSFMT();
    public static bool IsJudgeMode = true;
    public static bool IsGCJMode = false;
    public static bool IsSolveCreated = true;
    static void Main()
    {
        if (IsJudgeMode)
            if (IsGCJMode) using (var problem = new Problem(true, new Scanner("C-large-practice.in.txt"), new Printer("output.txt"))) problem.Solve();
            else using (var problem = new Problem(false, new Printer())) problem.Solve();
        else
        {
            var num = 1;
            int size = 0;
            decimal time = 0;
            for (var tmp = 0; tmp < num; tmp++)
            {
                using (var P = IsSolveCreated ? new Problem(false, new Scanner("input.txt"), new Printer()) : new Problem(false))
                {
                    size = P.Size;
                    //time += Func.MeasureTime(() => P.Solve());
                }
            }
            Console.WriteLine("{0}, {1}ms", size, time / num);
        }
    }
}

/// <summary>
/// カスタマイズしたIO
/// </summary>
namespace MyIO
{
    class Printer : IDisposable
    {
        bool isConsole;
        TextWriter file;
        public Printer() { file = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; isConsole = true; }
        public Printer(string path) { file = new StreamWriter(path, false) { AutoFlush = false }; isConsole = false; }
        public void Write<T>(T value) { file.Write(value); }
        public void Write(bool b) { file.Write(b ? "YES" : "NO"); }
        public void Write(string str, params object[] args) { file.Write(str, args); }
        public void WriteLine() { file.WriteLine(); }
        public void WriteLine<T>(T value) { file.WriteLine(value); }
        public void WriteLine(bool b) { file.WriteLine(b ? "YES" : "NO"); }
        public void WriteLine<T>(IEnumerable<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(List<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(T[] list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine(string str, params object[] args) { file.WriteLine(str, args); }
        public void Dispose() { file.Flush(); if (!isConsole) file.Dispose(); }
    }
    class Scanner : IDisposable
    {
        bool isConsole;
        TextReader file;
        public Scanner() { file = Console.In; }
        public Scanner(string path) { file = new StreamReader(path); isConsole = false; }
        public void Dispose() { if (!isConsole) file.Dispose(); }

        #region next読み込み
        string[] nextBuffer = new string[0];
        int BufferCnt = 0;

        char[] cs = new char[] { ' ' };

        public string next()
        {
            while (BufferCnt >= nextBuffer.Length)
            {
                string st = file.ReadLine();
                while (st == "") st = file.ReadLine();
                nextBuffer = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
                BufferCnt = 0;
            }
            return nextBuffer[BufferCnt++];
        }

        public int nextInt()
        {
            return int.Parse(next());
        }

        public long nextLong()
        {
            return long.Parse(next());
        }

        public double nextDouble()
        {
            return double.Parse(next());
        }

        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public string[] next(int n) { return enumerate(n, next); }
        public double[] nextDouble(int n) { return enumerate(n, nextDouble); }
        public int[] nextInt(int n) { return enumerate(n, nextInt); }
        public long[] nextLong(int n) { return enumerate(n, nextLong); }
        #endregion
    }
}


/// <summary>
/// 共通機能
/// </summary>
static partial class Func
{
    public const int Inf = 1073741789;  // 2 * Inf < int.MaxValue, and Inf is a prime number
    public const long InfL = 4011686018427387913L;  // 2 * InfL < long.MaxValue, and InfL is a prime number
    public static Comparison<T> DefaultComparison<T>() { return (x, y) => Comparer<T>.Default.Compare(x, y); }
    public static Comparison<T> ToComparison<T>(this IComparer<T> comp) { return comp == null ? DefaultComparison<T>() : (x, y) => comp.Compare(x, y); }
    /// <summary>
    /// Find the first number x such that pred(x) is true
    /// if pred(x) is false for all min&lt;=x&lt;max, then return max
    /// in other words, pred(max) is assumed to be true
    /// </summary>
    /// <param name="min">inclusive lower limit</param>
    /// <param name="max">exclusive upper limit</param>
    /// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
    /// <returns>first number such that satisfy pred</returns>
    public static long FirstBinary(long min, long max, Predicate<long> pred)
    {
        while (min < max)
        {
            var mid = (min + max) / 2;
            if (pred(mid)) max = mid;
            else min = mid + 1;
        }
        return min;
    }
    /// <summary>
    /// Find the first number x such that pred(x) is true
    /// if pred(x) is false for all min&lt;=x&lt;max, then return max
    /// in other words, pred(max) is assumed to be true
    /// </summary>
    /// <param name="min">inclusive lower limit</param>
    /// <param name="max">exclusive upper limit</param>
    /// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
    /// <returns>first number such that satisfy pred</returns>
    public static int FirstBinary(int min, int max, Predicate<int> pred)
    {
        while (min < max)
        {
            var mid = (min + max) / 2;
            if (pred(mid)) max = mid;
            else min = mid + 1;
        }
        return min;
    }
    public static void Swap<T>(this IList<T> array, int i, int j) { var tmp = array[i]; array[i] = array[j]; array[j] = tmp; }
    public static void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
}



/// <summary>
/// C++のSTLを再実装
/// </summary>
#region STL
class PriorityQueue<T> : IEnumerable<T>, ICollection, IEnumerable, ICloneable
{
    Comparison<T> comp;
    List<T> list;
    public int Count { get; private set; }
    public bool IsEmpty { get { return Count == 0; } }
    public PriorityQueue(IEnumerable<T> source) : this((Comparison<T>)null, 0, source) { }
    public PriorityQueue(int capacity = 0, IEnumerable<T> source = null) : this((Comparison<T>)null, capacity, source) { }
    public PriorityQueue(IComparer<T> comp, IEnumerable<T> source) : this(comp.ToComparison(), source) { }
    public PriorityQueue(IComparer<T> comp, int capacity = 0, IEnumerable<T> source = null) : this(comp.ToComparison(), source) { list.Capacity = capacity; }
    public PriorityQueue(Comparison<T> comp, IEnumerable<T> source) : this(comp, 0, source) { }
    public PriorityQueue(Comparison<T> comp, int capacity = 0, IEnumerable<T> source = null) { this.comp = comp == null ? Func.DefaultComparison<T>() : comp; list = new List<T>(capacity); if (source != null) foreach (var x in source) Enqueue(x); }
    /// <summary>
    /// add an item
    /// this is an O(log n) operation
    /// </summary>
    /// <param name="x">item</param>
    public void Enqueue(T x)
    {
        var pos = Count++;
        list.Add(x);
        while (pos > 0)
        {
            var p = (pos - 1) / 2;
            if (comp(list[p], x) <= 0) break;
            list[pos] = list[p];
            pos = p;
        }
        list[pos] = x;
    }
    /// <summary>
    /// return the minimum element and remove it
    /// this is an O(log n) operation
    /// </summary>
    /// <returns>the minimum</returns>
    public T Dequeue()
    {
        var value = list[0];
        var x = list[--Count];
        list.RemoveAt(Count);
        if (Count == 0) return value;
        var pos = 0;
        while (pos * 2 + 1 < Count)
        {
            var a = 2 * pos + 1;
            var b = 2 * pos + 2;
            if (b < Count && comp(list[b], list[a]) < 0) a = b;
            if (comp(list[a], x) >= 0) break;
            list[pos] = list[a];
            pos = a;
        }
        list[pos] = x;
        return value;
    }
    /// <summary>
    /// look at the minimum element
    /// this is an O(1) operation
    /// </summary>
    /// <returns>the minimum</returns>
    public T Peek() { return list[0]; }
    public IEnumerator<T> GetEnumerator() { var x = (PriorityQueue<T>)Clone(); while (x.Count > 0) yield return x.Dequeue(); }
    void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
    public object Clone() { var x = new PriorityQueue<T>(comp, Count); x.list.AddRange(list); return x; }
    public void Clear() { list = new List<T>(); Count = 0; }
    public void TrimExcess() { list.TrimExcess(); }
    /// <summary>
    /// check whether item is in this queue
    /// this is an O(n) operation
    /// </summary>
    public bool Contains(T item) { return list.Contains(item); }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    void ICollection.CopyTo(Array array, int index) { CopyTo(array, index); }
    bool ICollection.IsSynchronized { get { return false; } }
    object ICollection.SyncRoot { get { return this; } }
}
class Deque<T>
{
    T[] array;
    int offset, capacity;
    public int Count { get; protected set; }
    public Deque(int capacity) { array = new T[this.capacity = capacity]; Count = 0; offset = 0; }
    public Deque() : this(16) { }
    public T this[int index] { get { return array[GetIndex(index)]; } set { array[GetIndex(index)] = value; } }
    int GetIndex(int index) { var tmp = index + offset; return tmp >= capacity ? tmp - capacity : tmp; }
    public T PeekFront() { return array[offset]; }
    public T PeekBack() { return array[GetIndex(Count - 1)]; }
    public void PushFront(T item)
    {
        if (Count == capacity) Extend();
        if (--offset < 0) offset += array.Length;
        array[offset] = item;
        Count++;
    }
    public T PopFront()
    {
        Count--;
        var tmp = array[offset++];
        if (offset >= capacity) offset -= capacity;
        return tmp;
    }
    public void PushBack(T item)
    {
        if (Count == capacity) Extend();
        var id = (Count++) + offset;
        if (id >= capacity) id -= capacity;
        array[id] = item;
    }
    public T PopBack() { return array[GetIndex(--Count)]; }
    public void Insert(int index, T item)
    {
        PushFront(item);
        for (var i = 0; i < index; i++) this[i] = this[i + 1];
        this[index] = item;
    }
    public T RemoveAt(int index)
    {
        var tmp = this[index];
        for (var i = index; i > 0; i--) this[i] = this[i - 1];
        PopFront();
        return tmp;
    }
    void Extend()
    {
        var newArray = new T[capacity << 1];
        if (offset > capacity - Count)
        {
            var length = array.Length - offset;
            Array.Copy(array, offset, newArray, 0, length);
            Array.Copy(array, 0, newArray, length, Count - length);
        }
        else Array.Copy(array, offset, newArray, 0, Count);
        array = newArray;
        offset = 0;
        capacity <<= 1;
    }
}
class PairComparer<S, T> : IComparer<Pair<S, T>>
    where S : IComparable<S>
    where T : IComparable<T>
{
    public PairComparer() { }
    public int Compare(Pair<S, T> x, Pair<S, T> y)
    {
        var p = x.First.CompareTo(y.First);
        if (p != 0) return p;
        else return x.Second.CompareTo(y.Second);
    }
}
class Pair<S, T>
{
    public S First;
    public T Second;
    public Pair() { First = default(S); Second = default(T); }
    public Pair(S s, T t) { First = s; Second = t; }
    public override string ToString() { return string.Format("({0}, {1})", First, Second); }
    public override int GetHashCode() { return First.GetHashCode() ^ Second.GetHashCode(); }
    public override bool Equals(object obj)
    {
        if (ReferenceEquals(this, obj)) return true;
        else if (obj == null) return false;
        var tmp = obj as Pair<S, T>;
        return (object)tmp != null && First.Equals(tmp.First) && Second.Equals(tmp.Second);
    }
}
#endregion //STL

#region グラフ
interface ISegmentTree
{
    void Add(int from, int to, long value);
    long Min(int from, int to);
}
class SegmentTree2 : ISegmentTree
{
    int N;
    long[] a;
    public SegmentTree2(int N) : this(new long[N]) { }
    public SegmentTree2(long[] a) { N = a.Length; this.a = a.ToArray(); }
    public void Add(int from, int to, long value) { for (var i = from; i < to; i++) a[i] += value; }
    public long Min(int from, int to) { var s = Func.InfL; for (var i = from; i < to; i++) s = Math.Min(s, a[i]); return s; }
}
class SegmentTree3 : ISegmentTree
{
    public const long Unit = Func.InfL;
    public readonly Func<long, long, long> Operator = Math.Min;
    int N2;
    long[] seg, unif;
    public SegmentTree3(int N) : this(new long[N]) { }
    public SegmentTree3(long[] a)
    {
        N2 = 1;
        while (N2 < a.Length) N2 <<= 1;
        seg = new long[2 * N2 - 1];
        unif = new long[2 * N2 - 1];
        for (var i = a.Length + N2 - 1; i < 2 * N2 - 1; i++) seg[i] = Unit;
        for (var i = 0; i < a.Length; i++) seg[i + N2 - 1] = a[i];
        for (var i = N2 - 2; i >= 0; i--) Update(i);
    }
    void LazyEvaluate(int node)
    {
        if (unif[node] != 0)
        {
            seg[node] += unif[node];
            if (node < N2 - 1) { unif[2 * node + 1] += unif[node]; unif[2 * node + 2] += unif[node]; }
            unif[node] = 0;
        }
    }
    void Update(int node) { seg[node] = Operator(seg[2 * node + 1], seg[2 * node + 2]); }
    public void Add(int from, int to, long value) { Add(from, to, value, 0, 0, N2); }
    void Add(int from, int to, long value, int node, int l, int r)
    {
        if (from <= l && r <= to) unif[node] += value;
        else if (l < to && from < r)
        {
            Add(from, to, value, 2 * node + 1, l, (l + r) >> 1);
            Add(from, to, value, 2 * node + 2, (l + r) >> 1, r);
            Update(node);
        }
        LazyEvaluate(node);
    }
    public long this[int n] { get { return Min(n, n + 1); } set { Add(n, n + 1, value - this[n]); } }
    public long Min(int from, int to) { return Min(from, to, 0, 0, N2); }
    long Min(int from, int to, int node, int l, int r)
    {
        LazyEvaluate(node);
        if (to <= l || r <= from) return Unit;
        else if (from <= l && r <= to) return seg[node];
        else return Operator(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r));
    }
}
class SegmentTree : ISegmentTree
{
    int N2;
    long[] seg, unif;
    public SegmentTree(int N) : this(new long[N]) { }
    public SegmentTree(long[] a)
    {
        N2 = 1;
        while (N2 < a.Length) N2 <<= 1;
        seg = new long[2 * N2 - 1];
        unif = new long[2 * N2 - 1];
        for (var i = a.Length + N2 - 1; i < 2 * N2 - 1; i++) seg[i] = Func.InfL;
        for (var i = 0; i < a.Length; i++) seg[i + N2 - 1] = a[i];
        for (var i = N2 - 2; i >= 0; i--) seg[i] = Math.Min(seg[2 * i + 1], seg[2 * i + 2]);
    }
    public void Add(int from, int to, long value) { Add(from, to, value, 0, 0, N2); }
    void Add(int from, int to, long value, int node, int l, int r)
    {
        if (to <= l || r <= from) return;
        else if (from <= l && r <= to) unif[node] += value;
        else
        {
            Add(from, to, value, 2 * node + 1, l, (l + r) >> 1);
            Add(from, to, value, 2 * node + 2, (l + r) >> 1, r);
            seg[node] = Math.Min(seg[2 * node + 1] + unif[2 * node + 1], seg[2 * node + 2] + unif[2 * node + 2]);
        }
    }
    public long this[int n] { get { return Min(n, n + 1); } set { Add(n, n + 1, value - this[n]); } }
    public long Min(int from, int to) { return Min(from, to, 0, 0, N2); }
    long Min(int from, int to, int node, int l, int r)
    {
        if (to <= l || r <= from) return Func.InfL;
        else if (from <= l && r <= to) return seg[node] + unif[node];
        else return Math.Min(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r)) + unif[node];
    }
}
class Eq : IEqualityComparer<List<int>>
{
    public bool Equals(List<int> x, List<int> y)
    {
        if (x == null || y == null) return x == y;
        if (x.Count != y.Count) return false;
        for (var i = 0; i < x.Count; i++) if (x[i] != y[i]) return false;
        return true;
    }
    public int GetHashCode(List<int> obj)
    {
        var x = obj.Count.GetHashCode();
        foreach (var i in obj) x ^= i.GetHashCode();
        return x;
    }
}

/// <summary>
/// 使わないで
/// </summary>
/// <typeparam name="T"></typeparam>
class MultiSortedSet<T> : IEnumerable<T>, ICollection<T>
{
    public IComparer<T> Comparer { get; private set; }
    private SortedSet<T> keys;
    private Dictionary<T, int> mult;
    public int Multiplicity(T item) { return mult[item]; }
    public int Count { get; private set; }
    public MultiSortedSet(IComparer<T> comp)
    {
        keys = new SortedSet<T>(Comparer = comp);
        mult = new Dictionary<T, int>();
    }
    public MultiSortedSet(Comparison<T> comp) : this(Comparer<T>.Create(comp)) { }
    public MultiSortedSet() : this(Func.DefaultComparison<T>()) { }
    public void Add(T item) { Add(item, 1); }
    private void Add(T item, int num)
    {
        Count += num;
        if (!keys.Contains(item)) { keys.Add(item); mult.Add(item, num); }
        else mult[item] += num;
    }
    public void AddRange(IEnumerable<T> list) { foreach (var x in list) Add(x); }
    public bool Remove(T item)
    {
        if (!keys.Contains(item)) return false;
        Count--;
        if (mult[item] == 1) { keys.Remove(item); mult.Remove(item); }
        else mult[item]--;
        return true;
    }
    public bool Overlaps(IEnumerable<T> other) { return keys.Overlaps(other); }
    public bool IsSupersetOf(IEnumerable<T> other) { return keys.IsSupersetOf(other); }
    public bool IsSubsetOf(IEnumerable<T> other) { return keys.IsSubsetOf(other); }
    public bool IsProperSubsetOf(IEnumerable<T> other) { return keys.IsProperSubsetOf(other); }
    public bool IsProperSupersetOf(IEnumerable<T> other) { return keys.IsProperSupersetOf(other); }
    public void ExceptWith(IEnumerable<T> other) { foreach (var x in other) if (Contains(x)) Remove(x); }
    public void IntersectWith(IEnumerable<T> other)
    {
        var next = new MultiSortedSet<T>(Comparer);
        foreach (var x in other) if (Contains(x) && !next.Contains(x)) next.Add(x, mult[x]);
        keys = next.keys; mult = next.mult;
    }
    public void CopyTo(T[] array) { CopyTo(array, 0); }
    public void CopyTo(T[] array, int index) { foreach (var item in array) array[index++] = item; }
    public void CopyTo(T[] array, int index, int count) { var i = 0; foreach (var item in array) { if (i++ >= count) return; array[index++] = item; } }
    public bool Contains(T item) { return keys.Contains(item); }
    public void Clear() { keys.Clear(); mult.Clear(); Count = 0; }
    public IEnumerator<T> Reverse() { foreach (var x in keys.Reverse()) for (var i = 0; i < mult[x]; i++) yield return x; }
    public IEnumerator<T> GetEnumerator() { foreach (var x in keys) for (var i = 0; i < mult[x]; i++) yield return x; }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    public T Max { get { return keys.Max; } }
    public T Min { get { return keys.Min; } }
    public bool IsReadOnly { get { return false; } }
}
class SkewHeap<T> : IEnumerable<T>
{
    class Node : IEnumerable<T>
    {
        public Node l, r;
        public T val;
        public Node(T x) { l = r = null; val = x; }
        public IEnumerator<T> GetEnumerator()
        {
            if (l != null) foreach (var x in l) yield return x;
            yield return val;
            if (r != null) foreach (var x in r) yield return x;
        }
        IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    }
    public int Count { get; private set; }
    Node head;
    Comparison<T> comp;
    public bool IsEmpty { get { return head != null; } }
    public SkewHeap(Comparison<T> c) { comp = c; Count = 0; }
    public SkewHeap() : this(Func.DefaultComparison<T>()) { }
    public SkewHeap(IComparer<T> c) : this(Func.ToComparison(c)) { }
    private SkewHeap(Comparison<T> c, Node h) : this(c) { head = h; }
    public void Push(T x) { var n = new Node(x); head = Meld(head, n); Count++; }
    public T Peek() { return head.val; }
    public T Pop() { var x = head.val; head = Meld(head.l, head.r); Count--; return x; }
    // a.comp must be equivalent to b.comp
    // a, b will be destroyed
    public static SkewHeap<T> Meld(SkewHeap<T> a, SkewHeap<T> b) { return new SkewHeap<T>(a.comp, a.Meld(a.head, b.head)); }
    public void MeldWith(SkewHeap<T> a) { head = Meld(head, a.head); }
    Node Meld(Node a, Node b)
    {
        if (a == null) return b;
        else if (b == null) return a;
        if (comp(a.val, b.val) > 0) Func.Swap(ref a, ref b);
        a.r = Meld(a.r, b);
        Func.Swap(ref a.l, ref a.r);
        return a;
    }
    public IEnumerator<T> GetEnumerator() { return head.GetEnumerator(); }
    IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
// [0, Size) の整数の集合を表す
class BITSet : BinaryIndexedTree
{
    public BITSet(int size) : base(size) { }
    public void Add(int item) { Add(item, 1); }
    public bool Contains(int item) { return Sum(item, item + 1) > 0; }
    public int Count(int item) { return Sum(item, item + 1); }
    // 順位 = item が小さい方から何番目か(0-indexed)
    public int GetRank(int item) { return Sum(0, item); }
    public void Remove(int item) { Add(item, -1); }
    public void RemoveAll(int item) { Add(item, -Count(item)); }
    // 0-indexed で順位が rank のものを求める
    // ない場合は Size が返る
    public int GetValue(int rank) { return Func.FirstBinary(0, Size, t => Sum(0, t + 1) >= rank + 1); }
}
class RangeBIT
{
    public int N { get; private set; }
    long[,] bit;
    public RangeBIT(int N) { bit = new long[2, this.N = N + 1]; }
    public RangeBIT(int[] array) : this(array.Length)
    {
        for (var i = 1; i < N; i++) bit[0, i] = array[i - 1];
        for (var i = 1; i < N - 1; i++) if (i + (i & (-i)) < N) bit[0, i + (i & (-i))] += bit[0, i];
    }
    public RangeBIT(long[] array) : this(array.Length)
    {
        for (var i = 1; i < N; i++) bit[0, i] = array[i - 1];
        for (var i = 1; i < N - 1; i++) if (i + (i & (-i)) < N) bit[0, i + (i & (-i))] += bit[0, i];
    }
    public void Add(int from, int to, long value)
    {
        Add2(0, from + 1, -value * from);
        Add2(1, from + 1, value);
        Add2(0, to + 1, value * to);
        Add2(1, to + 1, -value);
    }
    void Add2(int which, int i, long value) { while (i < N) { bit[which, i] += value; i += i & (-i); } }
    long Sum(int to) { return Sum2(0, to) + Sum2(1, to) * to; }
    public long Sum(int from, int to) { return Sum(to) - Sum(from); }
    long Sum2(int which, int i) { var sum = 0L; while (i > 0) { sum += bit[which, i]; i -= i & (-i); } return sum; }
}
class RMQ
{
    int N2;
    int[] segtree;
    public RMQ(int N) : this(new int[N]) { }
    public RMQ(int[] array)
    {
        N2 = 1;
        while (N2 < array.Length) N2 <<= 1;
        segtree = new int[2 * N2 - 1];
        for (var i = 0; i < 2 * N2 - 1; i++) segtree[i] = Func.Inf;
        for (var i = 0; i < array.Length; i++) segtree[i + N2 - 1] = array[i];
        for (var i = N2 - 2; i >= 0; i--) segtree[i] = Math.Min(segtree[2 * i + 1], segtree[2 * i + 2]);
    }
    public void Update(int index, int value)
    {
        index += N2 - 1;
        segtree[index] = value;
        while (index > 0)
        {
            index = (index - 1) / 2;
            segtree[index] = Math.Min(segtree[index * 2 + 1], segtree[index * 2 + 2]);
        }
    }
    public int this[int n] { get { return Min(n, n + 1); } set { Update(n, value); } }
    public int Min(int from, int to) { return Min(from, to, 0, 0, N2); }
    int Min(int from, int to, int node, int l, int r)
    {
        if (to <= l || r <= from) return Func.Inf;
        else if (from <= l && r <= to) return segtree[node];
        else return Math.Min(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r));
    }
}

class BinaryIndexedTree3D
{
    public int X { get; private set; }
    public int Y { get; private set; }
    public int Z { get; private set; }
    int[,,] bit;
    public BinaryIndexedTree3D(int X, int Y, int Z)
    {
        this.X = X; this.Y = Y; this.Z = Z;
        bit = new int[X + 1, Y + 1, Z + 1];
    }
    public BinaryIndexedTree3D(int[,,] array)
        : this(array.GetLength(0), array.GetLength(1), array.GetLength(2))
    {
        for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) for (var z = 0; z < Z; z++) Add(x, y, z, array[x, y, z]);
    }
    public void Add(int x, int y, int z, int value)
    {
        for (var i = x + 1; i <= X; i += i & (-i)) for (var j = y + 1; j <= Y; j += j & (-j)) for (var k = z + 1; k <= Z; k += k & (-k)) bit[i, j, k] += value;
    }
    public int Sum(int x0, int y0, int z0, int x1, int y1, int z1)
    { return Sum(x1, y1, z1) - Sum(x0, y1, z1) - Sum(x1, y0, z1) - Sum(x1, y1, z0) + Sum(x1, y0, z0) + Sum(x0, y1, z0) + Sum(x0, y0, z1) - Sum(x0, y0, z0); }
    int Sum(int x, int y, int z)
    {
        var sum = 0;
        for (var i = x; i > 0; i -= i & (-i)) for (var j = y; j > 0; j -= j & (-j)) for (var k = y; k > 0; k -= k & (-k)) sum += bit[i, j, k];
        return sum;
    }
}
class BinaryIndexedTree2D
{
    public int X { get; private set; }
    public int Y { get; private set; }
    int[,] bit;
    public BinaryIndexedTree2D(int X, int Y)
    {
        this.X = X; this.Y = Y;
        bit = new int[X + 1, Y + 1];
    }
    public BinaryIndexedTree2D(int[,] array)
        : this(array.GetLength(0), array.GetLength(1))
    {
        for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) Add(x, y, array[x, y]);
    }
    public void Add(int x, int y, int value) { for (var i = x + 1; i <= X; i += i & (-i)) for (var j = y + 1; j <= Y; j += j & (-j)) bit[i, j] += value; }
    public int Sum(int x0, int y0, int x1, int y1) { return Sum(x0, y0) + Sum(x1, y1) - Sum(x0, y1) - Sum(x1, y0); }
    int Sum(int x, int y) { var sum = 0; for (var i = x; i > 0; i -= i & (-i)) for (var j = y; j > 0; j -= j & (-j)) sum += bit[i, j]; return sum; }
}
class BinaryIndexedTree
{
    public int Size { get; private set; }
    int[] bit;
    public BinaryIndexedTree(int size)
    {
        Size = size;
        bit = new int[size + 1];
    }
    public BinaryIndexedTree(int[] array) : this(array.Length)
    {
        for (var i = 0; i < Size; i++) bit[i + 1] = array[i];
        for (var i = 1; i < Size; i++) if (i + (i & (-i)) <= Size) bit[i + (i & (-i))] += bit[i];
    }
    // index is 0-indexed
    public void Add(int index, int value) { for (var i = index + 1; i <= Size; i += i & (-i)) bit[i] += value; }
    // from, to is 0-indexed
    // from is inclusive, to is exclusive
    public int Sum(int from, int to) { return Sum(to) - Sum(from); }
    int Sum(int to) { var sum = 0; for (var i = to; i > 0; i -= i & (-i)) sum += bit[i]; return sum; }
}

class Amoeba
{
    public const int Dimension = 2;
    public const double Alpha = 1;  // reflection
    public const double Beta = 1 + 2.0 / Dimension; // expansion
    public const double Gamma = 0.75 - 0.5 / Dimension; // contraction
    public const double Delta = 1 - 1.0 / Dimension;    // shrink
    public Pair<AmoebaState, double>[] a;
    public AmoebaState m;
    public void Initiate()
    {
        Array.Sort(a, (x, y) => x.Second.CompareTo(y.Second));
        m = new AmoebaState();
        for (var i = 0; i < Dimension; i++) m.Add(a[i].First);
        m.Multiply(1.0 / Dimension);
    }
    void PartialSort(int i, int j) { if (a[i].Second > a[j].Second) a.Swap(i, j); }
    void Accept(AmoebaState point, double value)
    {
        var tmp = Func.FirstBinary(0, Dimension, x => a[x].Second >= value);
        if (tmp != Dimension) m.Add((point - a[Dimension - 1].First) / Dimension);
        for (var i = Dimension; i > tmp; i--) a[i] = a[i - 1];
        a[tmp].First = point;
        a[tmp].Second = value;
    }
    public void Search()
    {
        var r = m + Alpha * (m - a[Dimension].First);
        var fr = r.Func();
        if (a[0].Second <= fr && fr < a[Dimension - 1].Second) { Accept(r, fr); return; }
        var diff = r - m;
        if (fr < a[0].Second)
        {
            var e = m + Beta * diff;
            var fe = e.Func();
            if (fe < fr) Accept(e, fe);
            else Accept(r, fr);
        }
        else
        {
            var tmp = Gamma * diff;
            var o = m + tmp;
            var fo = o.Func();
            var i = m - tmp;
            var fi = i.Func();
            if (fi < fo) { o = i; fo = fi; }
            if (fo < a[Dimension - 1].Second) Accept(o, fo);
            else Shrink();
        }
    }
    void Shrink()
    {
        var tmp = (1 - Delta) * a[0].First;
        for (var i = 1; i <= Dimension; i++) { a[i].First.Multiply(Delta); a[i].First.Add(tmp); a[i].Second = a[i].First.Func(); }
        Initiate();
    }
}
class AmoebaState
{
    public static int Dimension = 2;
    public double[] vec;
    public AmoebaState() { vec = new double[Dimension]; }
    public AmoebaState(params double[] elements) : this() { elements.CopyTo(vec, 0); }
    public double this[int n] { get { return vec[n]; } set { vec[n] = value; } }
    public void Multiply(double r) { for (var i = 0; i < Dimension; i++) vec[i] *= r; }
    public void Add(AmoebaState v) { for (var i = 0; i < Dimension; i++) vec[i] += v.vec[i]; }
    public static AmoebaState operator +(AmoebaState p) { return new AmoebaState(p.vec); }
    public static AmoebaState operator -(AmoebaState p) { var tmp = new AmoebaState(p.vec); tmp.Multiply(-1); return tmp; }
    public static AmoebaState operator /(AmoebaState p, double r) { var tmp = new AmoebaState(p.vec); tmp.Multiply(1 / r); return tmp; }
    public static AmoebaState operator *(double r, AmoebaState p) { var tmp = new AmoebaState(p.vec); tmp.Multiply(r); return tmp; }
    public static AmoebaState operator *(AmoebaState p, double r) { return r * p; }
    public static AmoebaState operator +(AmoebaState p, AmoebaState q) { var tmp = +p; tmp.Add(q); return tmp; }
    public static AmoebaState operator -(AmoebaState p, AmoebaState q) { var tmp = -q; tmp.Add(p); return tmp; }
    public double Func()
    {
        return 0;//P.Func(vec[0], vec[1]);
    }
    public static Problem P;
}
class BucketList<T> : ICollection<T>, IEnumerable<T>, ICollection, IEnumerable
{
    public Comparison<T> comp { get; protected set; }
    public int BucketSize = 20;
    public int Count { get { var sum = 0; var bucket = Head; while (bucket != null) { sum += bucket.Count; bucket = bucket.Next; } return sum; } }
    public int NumOfBucket { get; protected set; }
    public Bucket<T> Head { get; protected set; }
    public Bucket<T> Tail { get; protected set; }
    public BucketList(IComparer<T> comp) : this(comp.ToComparison()) { }
    public BucketList(Comparison<T> comp = null) { Head = null; Tail = null; NumOfBucket = 0; this.comp = comp == null ? Func.DefaultComparison<T>() : comp; }
    protected void AddAfter(Bucket<T> pos, Bucket<T> bucket)
    {
        Debug.Assert(bucket != null && bucket.Count > 0 && pos != null && pos.Parent == this && comp(pos.Tail.Value, bucket.Head.Value) <= 0
                    && (pos.Next == null || comp(pos.Next.Head.Value, bucket.Tail.Value) >= 0));
        bucket.Parent = this;
        bucket.Prev = pos;
        bucket.Next = pos.Next;
        if (pos != Tail) pos.Next.Prev = bucket;
        else Tail = bucket;
        pos.Next = bucket;
        NumOfBucket++;
    }
    protected void AddBefore(Bucket<T> pos, Bucket<T> bucket)
    {
        Debug.Assert(bucket != null && bucket.Count > 0 && pos != null && pos.Parent == this && comp(pos.Head.Value, bucket.Tail.Value) >= 0
                    && (pos.Prev == null || comp(pos.Prev.Tail.Value, bucket.Head.Value) <= 0));
        bucket.Parent = this;
        bucket.Prev = pos.Prev;
        bucket.Next = pos;
        if (pos != Head) pos.Prev.Next = bucket;
        else Head = bucket;
        pos.Prev = bucket;
        NumOfBucket++;
    }
    protected void AddAfter(Bucket<T> bucket, BucketNode<T> node)
    {
        Debug.Assert(node != null && bucket != null && bucket.Parent == this && node.Parent.Parent == this && comp(bucket.Tail.Value, node.Value) <= 0
                    && (bucket.Next == null || comp(bucket.Next.Head.Value, node.Value) >= 0));
        var tmp = new Bucket<T>(this, bucket, bucket.Next);
        tmp.InitiateWith(node);
        if (bucket != Tail) bucket.Next.Prev = tmp;
        else Tail = tmp;
        bucket.Next = tmp;
        NumOfBucket++;
    }
    protected void AddBefore(Bucket<T> bucket, BucketNode<T> node)
    {
        Debug.Assert(node != null && bucket != null && bucket.Parent == this && node.Parent.Parent == this && comp(bucket.Head.Value, node.Value) >= 0
                    && (bucket.Prev == null || comp(bucket.Prev.Tail.Value, node.Value) <= 0));
        var tmp = new Bucket<T>(this, bucket.Prev, bucket);
        tmp.InitiateWith(node);
        if (bucket != Head) bucket.Prev.Next = tmp;
        else Head = tmp;
        bucket.Prev = tmp;
        NumOfBucket++;
    }
    public void AddAfter(BucketNode<T> node, T item)
    {
        Debug.Assert(node != null && node.Parent.Parent == this && comp(node.Value, item) <= 0
                    && ((node.Next == null && (node.Parent.Next == null || comp(node.Parent.Next.Head.Value, item) >= 0))
                        || comp(node.Next.Value, item) >= 0));
        var bucket = node.Parent;
        var tmp = new BucketNode<T>(item, bucket, node, node.Next);
        if (!bucket.AddAfter(node, tmp))
        {
            if (node.Next == null && (bucket.Next == null || bucket.Next.Count >= BucketSize)) AddAfter(bucket, tmp);
            else if (node.Next == null) AddBefore(bucket.Next.Head, item);
            else
            {
                node.Next.Prev = tmp;
                node.Next = tmp;
                while (node.Next.Next != null) node = node.Next;
                item = node.Next.Value;
                bucket.Tail = node;
                node.Next = null;
                AddAfter(node, item);
            }
        }
    }
    public void AddBefore(BucketNode<T> node, T item)
    {
        Debug.Assert(node != null && node.Parent.Parent == this && comp(node.Value, item) >= 0
                    && ((node.Prev == null && (node.Parent.Prev == null || comp(node.Parent.Prev.Tail.Value, item) <= 0))
                        || comp(node.Prev.Value, item) <= 0));
        var bucket = node.Parent;
        var tmp = new BucketNode<T>(item, bucket, node.Prev, node);
        if (!bucket.AddBefore(node, tmp))
        {
            if (node.Prev == null && (bucket.Prev == null || bucket.Prev.Count >= BucketSize)) AddBefore(bucket, tmp);
            else if (node.Prev == null) AddAfter(bucket.Prev.Tail, item);
            else
            {
                node.Prev.Next = tmp;
                node.Prev = tmp;
                while (node.Prev.Prev != null) node = node.Prev;
                item = node.Prev.Value;
                bucket.Head = node;
                node.Prev = null;
                AddBefore(node, item);
            }
        }
    }
    // (node, index)
    // index is the position of node in node.Parent
    public Tuple<BucketNode<T>, int> UpperBound(Predicate<T> pred)
    {
        if (NumOfBucket == 0) return null;
        if (pred(Tail.Tail.Value)) return new Tuple<BucketNode<T>, int>(Tail.Tail, Tail.Count - 1);
        var bucket = Tail;
        while (bucket.Prev != null && !pred(bucket.Prev.Tail.Value)) bucket = bucket.Prev;
        var node = bucket.Tail;
        var index = bucket.Count - 1;
        while (node.Prev != null && !pred(node.Prev.Value)) { node = node.Prev; index--; }
        if (node.Prev == null) return bucket.Prev == null ? null : new Tuple<BucketNode<T>, int>(bucket.Prev.Tail, bucket.Prev.Count - 1);
        else return new Tuple<BucketNode<T>, int>(node.Prev, index - 1);
    }
    public Tuple<BucketNode<T>, int> UpperBound(T item) { return LowerBound(x => comp(x, item) <= 0); }
    // (node, index)
    // index is the position of node in node.Parent
    public Tuple<BucketNode<T>, int> LowerBound(Predicate<T> pred)
    {
        if (NumOfBucket == 0) return null;
        if (pred(Head.Head.Value)) return new Tuple<BucketNode<T>, int>(Head.Head, 0);
        var bucket = Head;
        while (bucket.Next != null && !pred(bucket.Next.Head.Value)) bucket = bucket.Next;
        var node = bucket.Head;
        var index = 0;
        while (node.Next != null && !pred(node.Next.Value)) { node = node.Next; index++; }
        if (node.Next == null) return bucket.Next == null ? null : new Tuple<BucketNode<T>, int>(bucket.Next.Head, 0);
        else return new Tuple<BucketNode<T>, int>(node.Next, index + 1);
    }
    public Tuple<BucketNode<T>, int> LowerBound(T item) { return LowerBound(x => comp(x, item) >= 0); }
    public void InitiateWith(Bucket<T> bucket)
    {
        Debug.Assert(bucket != null && bucket.Count > 0);
        RemoveAll();
        Head = Tail = bucket;
        bucket.Parent = this;
        NumOfBucket++;
    }
    public void InitiateWith(T item)
    {
        RemoveAll();
        Head = Tail = new Bucket<T>(this, null, null);
        Head.Head = Head.Tail = new BucketNode<T>(item, Head, null, null);
        Head.Count++;
        NumOfBucket++;
    }
    public void AddFirst(Bucket<T> bucket) { if (NumOfBucket == 0) InitiateWith(bucket); else AddBefore(Head, bucket); }
    public void AddLast(Bucket<T> bucket) { if (NumOfBucket == 0) InitiateWith(bucket); else AddAfter(Tail, bucket); }
    public void AddFirst(T item) { if (NumOfBucket == 0) InitiateWith(item); else AddBefore(Head.Head, item); }
    public void AddLast(T item) { if (NumOfBucket == 0) InitiateWith(item); else AddAfter(Tail.Tail, item); }
    public void Clear() { RemoveAll(); }
    public void RemoveAll() { Head = Tail = null; NumOfBucket = 0; }
    public void RemoveFirst() { if (NumOfBucket == 0) throw new InvalidOperationException(); else Remove(Head.Head); }
    public void RemoveLast() { if (NumOfBucket == 0) throw new InvalidOperationException(); else Remove(Tail.Tail); }
    // remove item and return whether item was removed or not
    public bool Remove(T item) { var node = Find(item); if (node != null) Remove(node); return node != null; }
    public void Remove(Bucket<T> bucket)
    {
        Debug.Assert(bucket != null && bucket.Parent == this);
        NumOfBucket--;
        if (bucket == Head && bucket == Tail) { Head = Tail = null; }
        else if (bucket == Head) { Head.Next.Prev = null; Head = Head.Next; }
        else if (bucket == Tail) { Tail.Prev.Next = null; Tail = Tail.Prev; }
        else { bucket.Prev.Next = bucket.Next; bucket.Next.Prev = bucket.Prev; }
    }
    public void Remove(BucketNode<T> node) { Debug.Assert(node != null && node.Parent.Parent == this); if (!node.Parent.Remove(node)) Remove(node.Parent); }
    protected void RemoveRange(Bucket<T> from, Bucket<T> to, int indexFrom = -1, int indexTo = -1)
    {
        Debug.Assert(from != null && to != null && from.Parent == this && to.Parent == this);
        if (indexFrom < 0) indexFrom = from.Index;
        if (indexTo < 0) indexTo = to.Index;
        if (indexFrom == 0 && indexTo == NumOfBucket - 1) { Clear(); return; }
        else if (indexFrom == 0) { Head = to.Next; Head.Prev = null; }
        else if (indexTo == NumOfBucket - 1) { Tail = from.Prev; Tail.Next = null; }
        else { from.Prev.Next = to.Next; to.Next.Prev = from.Prev; }
        NumOfBucket -= indexTo - indexFrom + 1;
    }
    public void RemoveRange(BucketNode<T> from, BucketNode<T> to, int indexFrom = -1, int indexTo = -1)
    {
        Debug.Assert(from != null && to != null && from.Parent.Parent == this && to.Parent.Parent == this);
        if (indexFrom < 0) indexFrom = from.Index;
        if (indexTo < 0) indexTo = to.Index;
        var bucketFrom = from.Parent;
        var bucketTo = to.Parent;
        if (bucketFrom == bucketTo)
        {
            var bucket = bucketFrom;
            if (indexFrom == 0 && indexTo == bucket.Count - 1) Remove(bucket);
            else bucket.RemoveRange(from, to, indexFrom, indexTo);
        }
        else
        {
            var bf = bucketFrom.Index;
            var bt = bucketTo.Index;
            Debug.Assert(bf < bt);
            if (bt > bf + 1) RemoveRange(bucketFrom.Next, bucketTo.Prev, bf + 1, bt - 1);
            if (indexFrom == 0) { Remove(bucketFrom); RemoveRange(bucketTo.Head, to, 0, indexTo); }
            else if (indexTo == bucketTo.Count - 1) { Remove(bucketTo); RemoveRange(from, bucketFrom.Tail, indexFrom, bucketFrom.Count - 1); }
            else
            {
                bucketFrom.RemoveRange(from, bucketFrom.Tail, indexFrom, bucketFrom.Count - 1);
                bucketTo.RemoveRange(bucketTo.Head, to, 0, indexTo);
                if (bucketFrom.Count + bucketTo.Count < BucketSize) Adjust();
            }
        }
    }
    public void Adjust()
    {
        var array = this.ToArray();
        Clear();
        var length = array.Length;
        BucketSize = (int)Math.Sqrt(length + 1);
        var count = (length + BucketSize - 1) / BucketSize;
        for (var i = 0; i < count; i++)
        {
            var bucket = new Bucket<T>(this, null, null);
            var lim = Math.Min(BucketSize * (i + 1), length);
            for (var j = BucketSize * i; j < lim; j++) bucket.AddLast(array[j]);
            AddLast(bucket);
        }
    }
    public BucketNode<T> Find(T item) { var node = LowerBound(item); if (node == null || comp(node.Item1.Value, item) != 0) return null; else return node.Item1; }
    public BucketNode<T> FindLast(T item) { var node = UpperBound(item); if (node == null || comp(node.Item1.Value, item) != 0) return null; else return node.Item1; }
    public IEnumerator<T> GetEnumerator()
    {
        var bucket = Head;
        while (bucket != null)
        {
            var node = bucket.Head;
            while (node != null) { yield return node.Value; node = node.Next; }
            bucket = bucket.Next;
        }
    }
    public void Add(T item) { var ub = LowerBound(item); if (ub != null) AddBefore(ub.Item1, item); else AddLast(item); }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    public void CopyTo(Array array, int index) { foreach (var item in this) array.SetValue(item, index++); }
    public bool IsSynchronized { get { return false; } }
    public object SyncRoot { get { return this; } }
    public bool IsReadOnly { get { return false; } }
    public bool Contains(T item) { return Find(item) != null; }
    public void CopyTo(T[] array, int index) { foreach (var item in this) array[index++] = item; }
    public override string ToString()
    {
        var sb = new StringBuilder();
        sb.Append("<Start>\n");
        var node = Head;
        while (node != null) { sb.Append($"{node.ToString()}\n"); node = node.Next; }
        sb.Append("<end>");
        return sb.ToString();
    }
    public bool Check()
    {
        if (NumOfBucket == 0) return Head == null && Tail == null;
        if (Head.Prev != null || Tail.Next != null) return false;
        var bucket = Head;
        var c = 1;
        while (bucket.Next != null)
        {
            if (!CheckConnection(bucket) || !CheckBucket(bucket)) return false;
            bucket = bucket.Next;
            c++;
        }
        return bucket == Tail && CheckBucket(Tail) && c == NumOfBucket;
    }
    bool CheckConnection(Bucket<T> bucket)
    {
        if (bucket.Next == null) return bucket == Tail;
        else return bucket.Next.Prev == bucket && comp(bucket.Tail.Value, bucket.Next.Head.Value) <= 0;
    }
    bool CheckBucket(Bucket<T> bucket) { return bucket.Count > 0 && bucket.Count <= BucketSize && bucket.Parent == this; }
    public void Start(Func<string, T> parser, Func<T> random)
    {
        BucketNode<T> x = null, y = null;
        var help = true;
        while (true)
        {
            Console.Clear();
            Console.WriteLine($"{Count} items, {NumOfBucket} buckets(size : {BucketSize})");
            Console.WriteLine(this);
            Console.WriteLine(Check() ? "OK!" : "NG!");
            if (help)
            {
                Console.WriteLine("when val is omitted, random value will be used.");
                Console.WriteLine("a val : add val");
                Console.WriteLine("r val : remove val");
                Console.WriteLine("j : adjust");
                Console.WriteLine("c : clear");
                Console.WriteLine("h : disable/enable help message");
                Console.WriteLine("x : set x");
                Console.WriteLine("x h : set x to head");
                Console.WriteLine("x t : set x to tail");
                Console.WriteLine("x n : set x to x.next");
                Console.WriteLine("x p : set x to x.prev");
                Console.WriteLine("x f val : set x to lower bound of val");
                Console.WriteLine("y : set y");
                Console.WriteLine("x : exchange x and y");
                Console.WriteLine("d : remove from x to y");
                Console.WriteLine("q : quit");
            }
            if (x != null) Console.WriteLine($"x = {x.Value} <- {x.Parent}");
            if (y != null) Console.WriteLine($"y = {y.Value} <- {y.Parent}");
            Console.Write("enter command > ");
            var command = Console.ReadLine().Split();
            if (command[0].Length > 1 && command[0][1] == 'd')
                Console.WriteLine("debug...");
            if (command[0].StartsWith("a")) { if (command.Length > 1) Add(parser(command[1])); else Add(random()); }
            else if (command[0].StartsWith("r")) { if (command.Length > 1) Remove(parser(command[1])); else Remove(random()); }
            else if (command[0].StartsWith("c")) Clear();
            else if (command[0].StartsWith("j")) Adjust();
            else if (command[0].StartsWith("h")) help = !help;
            else if (command[0].StartsWith("x")) SetVariable(command, ref x, parser, random);
            else if (command[0].StartsWith("y")) SetVariable(command, ref y, parser, random);
            else if (command[0].StartsWith("e")) { var tmp = x; x = y; y = tmp; }
            else if (command[0].StartsWith("d")) { RemoveRange(x, y, x.Index, y.Index); x = null; y = null; }
            else if (command[0].StartsWith("q")) break;
        }
    }
    void SetVariable(string[] command, ref BucketNode<T> x, Func<string, T> parser, Func<T> random)
    {
        if (command[1].StartsWith("h")) x = Head.Head;
        else if (command[1].StartsWith("t")) x = Tail.Tail;
        else if (command[1].StartsWith("n"))
        {
            if (x.Next != null) x = x.Next;
            else if (x.Parent.Next != null) x = x.Parent.Next.Head;
            else { Console.WriteLine("x is the last element..."); Console.ReadKey(true); }
        }
        else if (command[1].StartsWith("p"))
        {
            if (x.Prev != null) x = x.Prev;
            else if (x.Parent.Prev != null) x = x.Parent.Prev.Tail;
            else { Console.WriteLine("x is the first element..."); Console.ReadKey(true); }
        }
        else if (command[1].StartsWith("f")) { if (command.Length > 2) x = LowerBound(parser(command[2])).Item1; else x = LowerBound(random()).Item1; }
    }
}
// bucket cannot be empty
class Bucket<T>
{
    public BucketList<T> Parent;
    public int Count;
    public Bucket<T> Prev;
    public Bucket<T> Next;
    public BucketNode<T> Head;
    public BucketNode<T> Tail;
    public Bucket(BucketList<T> parent, Bucket<T> prev, Bucket<T> next) { Parent = parent; Prev = prev; Next = next; Head = null; Tail = null; }
    public int Index
    {
        get
        {
            var count = 0;
            var node = Parent.Head;
            while (node != this) { node = node.Next; count++; }
            return count;
        }
    }
    public bool AddAfter(BucketNode<T> node, BucketNode<T> item) { return AddAfter(node, item.Value); }
    public bool AddBefore(BucketNode<T> node, BucketNode<T> item) { return AddBefore(node, item.Value); }
    public bool AddAfter(BucketNode<T> node, T item)
    {
        Debug.Assert(node != null && node.Parent == this && Parent.comp(node.Value, item) <= 0
                    && ((node.Next == null && (Next == null || Parent.comp(Next.Head.Value, item) >= 0))
                        || Parent.comp(node.Next.Value, item) >= 0));
        if (Count < Parent.BucketSize)
        {
            var tmp = new BucketNode<T>(item, this, node, node.Next);
            if (node.Next != null) node.Next.Prev = tmp;
            else Tail = tmp;
            node.Next = tmp;
            Count++;
            return true;
        }
        return false;
    }
    public bool AddBefore(BucketNode<T> node, T item)
    {
        Debug.Assert(node != null && node.Parent == this && Parent.comp(node.Value, item) >= 0
                    && ((node.Prev == null && (Prev == null || Parent.comp(Prev.Tail.Value, item) <= 0))
                        || Parent.comp(node.Prev.Value, item) <= 0));
        if (Count < Parent.BucketSize)
        {
            var tmp = new BucketNode<T>(item, this, node.Prev, node);
            if (node.Prev != null) node.Prev.Next = tmp;
            else Head = tmp;
            node.Prev = tmp;
            Count++;
            return true;
        }
        else return false;
    }
    public bool InitiateWith(BucketNode<T> node)
    {
        Head = Tail = node;
        node.Parent = this;
        node.Prev = node.Next = null;
        Count++;
        return true;
    }
    public bool InitiateWith(T item) { return InitiateWith(new BucketNode<T>(item, this, null, null)); }
    public void RemoveAll() { Head = Tail = null; Count = 0; }
    public bool AddFirst(T item) { if (Count == 0) return InitiateWith(item); else return AddBefore(Head, item); }
    public bool AddLast(T item) { if (Count == 0) return InitiateWith(item); else return AddAfter(Tail, item); }
    public bool Remove(BucketNode<T> node)
    {
        Debug.Assert(node != null && node.Parent == this);
        if (Count > 1)
        {
            Count--;
            if (node == Head) { Head.Next.Prev = null; Head = Head.Next; }
            else if (node == Tail) { Tail.Prev.Next = null; Tail = Tail.Prev; }
            else { node.Prev.Next = node.Next; node.Next.Prev = node.Prev; }
            return true;
        }
        else return false;
    }
    public bool RemoveRange(BucketNode<T> from, BucketNode<T> to, int indexFrom = -1, int indexTo = -1)
    {
        Debug.Assert(from != null && to != null && from.Parent == this && to.Parent == this);
        if (indexFrom < 0) indexFrom = from.Index;
        if (indexTo < 0) indexTo = to.Index;
        if (indexTo == 0 && indexFrom == Count - 1) return false;
        else if (indexFrom == 0) { Head = to.Next; Head.Prev = null; }
        else if (indexTo == Count - 1) { Tail = from.Prev; Tail.Next = null; }
        else { from.Prev.Next = to.Next; to.Next.Prev = from.Prev; }
        Count -= indexTo - indexFrom + 1;
        return true;
    }
    public override string ToString()
    {
        var sb = new StringBuilder();
        sb.Append("[");
        var node = Head;
        while (node != null) { sb.Append($"{node.ToString()}, "); node = node.Next; }
        if (sb.Length > 1) sb.Remove(sb.Length - 2, 2);
        sb.Append("]");
        return sb.ToString();
    }
    public bool Check()
    {
        if (Count == 0) return Head == null && Tail == null;
        if (Head.Prev != null || Tail.Next != null) return false;
        var node = Head;
        var c = 1;
        while (node.Next != null)
        {
            if (!CheckConnection(node) || !CheckNode(node)) return false;
            node = node.Next;
            c++;
        }
        return node == Tail && CheckNode(Tail) && c == Count;
    }
    bool CheckConnection(BucketNode<T> node)
    {
        if (node.Next == null) return node == Tail;
        else return node.Next.Prev == node && Parent.comp(node.Value, node.Next.Value) <= 0;
    }
    bool CheckNode(BucketNode<T> node) { return node.Parent == this; }
}
class BucketNode<T>
{
    public T Value;
    public Bucket<T> Parent;
    public BucketNode<T> Prev;
    public BucketNode<T> Next;
    public BucketNode(T item, Bucket<T> parent, BucketNode<T> prev, BucketNode<T> next) { Value = item; Parent = parent; Prev = prev; Next = next; }
    public int Index
    {
        get
        {
            var count = 0;
            var node = Parent.Head;
            while (node != this) { node = node.Next; count++; }
            return count;
        }
    }
    public override string ToString() { return Value.ToString(); }
}
class UndirectedGraph<V, E> : DirectedGraph<V, E>
{
    public UndirectedGraph(int V) : base(V) { }
    public UndirectedGraph(int V, IEnumerable<EdgeInfo<E>> edges) : base(V, edges) { }
    public override void AddEdge(EdgeInfo<E> edge)
    {
        edges.Add(edge);
        edges.Add(edge.Reverse());
        edgesFrom[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
        edgesFrom[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
        edgesTo[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
        edgesTo[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
    }
    public bool IsConnected
    {
        get
        {
            if (numberOfNodes == 0) return true;
            var used = new bool[numberOfNodes];
            var queue = new Queue<int>();
            queue.Enqueue(0);
            while (queue.Count > 0)
            {
                var v = queue.Dequeue();
                if (used[v]) continue;
                used[v] = true;
                foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
            }
            return used.All(x => x);
        }
    }
    public bool IsTree
    {
        get
        {
            if (numberOfNodes == 0) return true;
            var used = new bool[numberOfNodes];
            var queue = new Queue<int>();
            queue.Enqueue(0);
            while (queue.Count > 0)
            {
                var v = queue.Dequeue();
                if (used[v]) return false;
                used[v] = true;
                foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
            }
            return used.All(x => x);
        }
    }
    public UndirectedGraph<V, E> MinimumSpanningTreePrim(int start, Func<E, int> cost)
    {
        var graph = new UndirectedGraph<V, E>(numberOfNodes);
        nodes.CopyTo(graph.nodes, 0);
        var d = Enumerable.Repeat(Func.Inf, numberOfNodes).ToArray();
        var used = new bool[numberOfNodes];
        var queue = new PriorityQueue<Pair<EdgeInfo<E>, int>>((x, y) => x.Second.CompareTo(y.Second), numberOfNodes);
        d[start] = 0;
        queue.Enqueue(new Pair<EdgeInfo<E>, int>(new EdgeInfo<E>(-1, 0, default(E)), 0));
        while (queue.Count > 0)
        {
            var p = queue.Dequeue();
            var v = p.First.To;
            if (d[v] < p.Second) continue;
            used[v] = true;
            if (p.First.From >= 0) graph.AddEdge(v, p.First.From, p.First.Information);
            foreach (var w in EdgesFrom(v))
            {
                if (!used[w.End] && cost(w.Information) < d[w.End])
                {
                    d[w.End] = cost(w.Information);
                    queue.Enqueue(new Pair<EdgeInfo<E>, int>(new EdgeInfo<E>(v, w.End, w.Information), cost(w.Information)));
                }
            }
        }
        return graph;
    }
    public UndirectedGraph<V, E> MinimumSpanningTreeKruskal(Func<E, int> cost)
    {
        var graph = new UndirectedGraph<V, E>(numberOfNodes);
        nodes.CopyTo(graph.nodes, 0);
        var tree = new UnionFindTree(numberOfNodes);
        edges.Sort((x, y) => cost(x.Information).CompareTo(cost(y.Information)));
        foreach (var e in edges)
        {
            if (!tree.IsSameCategory(e.From, e.To))
            {
                tree.UniteCategory(e.From, e.To);
                graph.AddEdge(e);
            }
        }
        return graph;
    }
    public bool IsBipartite
    {
        get
        {
            var color = new int[numberOfNodes];
            foreach (var v in nodes)
            {
                if (color[v.Code] == 0)
                {
                    var queue = new Queue<Pair<int, int>>();
                    queue.Enqueue(new Pair<int, int>(v.Code, 1));
                    while (queue.Count > 0)
                    {
                        var w = queue.Dequeue();
                        if (color[w.First] != 0)
                        {
                            if (color[w.First] != w.Second) return false;
                            continue;
                        }
                        color[w.First] = w.Second;
                        foreach (var e in EdgesFrom(w.First)) queue.Enqueue(new Pair<int, int>(e.End, -w.Second));
                    }
                }
            }
            return true;
        }
    }
    public IEnumerable<NodeInfo<V>> GetArticulationPoints()
    {
        var visited = new bool[numberOfNodes];
        var parent = new int[numberOfNodes];
        var children = Enumerable.Range(0, numberOfNodes).Select(_ => new SortedSet<int>()).ToArray();
        var order = new int[numberOfNodes];
        var lowest = new int[numberOfNodes];
        var isroot = new bool[numberOfNodes];
        var count = 1;
        var isarticulation = new bool[numberOfNodes];
        Action<int, int> dfs = null;
        dfs = (u, prev) =>
        {
            order[u] = count;
            lowest[u] = count;
            count++;
            visited[u] = true;
            foreach (var e in edgesFrom[u])
            {
                var v = e.End;
                if (visited[v]) { if (v != prev) lowest[u] = Math.Min(lowest[u], order[v]); }
                else
                {
                    parent[v] = u;
                    if (isroot[u]) children[u].Add(v);
                    dfs(v, u);
                    lowest[u] = Math.Min(lowest[u], lowest[v]);
                    if (order[u] <= lowest[v]) isarticulation[u] = true;
                }
            }
        };
        for (var v = 0; v < numberOfNodes; v++)
        {
            if (visited[v]) continue;
            count = 1; dfs(v, -1);
            isroot[v] = true;
        }
        for (var v = 0; v < numberOfNodes; v++)
        {
            if (isroot[v]) { if (children[v].Count > 1) yield return nodes[v]; }
            else { if (isarticulation[v]) yield return nodes[v]; }
        }
    }
    public string ToString(Func<NodeInfo<V>, string> vertex, Func<EdgeInfo<E>, string> edge)
    {
        var sb = new StringBuilder();
        sb.Append("graph G {\n");
        foreach (var v in nodes) sb.Append($"\tv{v.Code} [label = \"{vertex(v)}\"];\n");
        foreach (var e in edges) sb.Append($"\tv{e.From} -- v{e.To} [label=\"{edge(e)}\"];\n");
        sb.Append("}");
        return sb.ToString();
    }
    public override string ToString() { return ToString(v => v.ToString(), e => e.ToString()); }
}
class NodeInfo<V> : Pair<int, V>
{
    public int Code { get { return First; } set { First = value; } }
    public V Information { get { return Second; } set { Second = value; } }
    public NodeInfo() : base() { }
    public NodeInfo(int code, V info) : base(code, info) { }
}
class HalfEdgeInfo<E> : Pair<int, E>
{
    public int End { get { return First; } set { First = value; } }
    public E Information { get { return Second; } set { Second = value; } }
    public HalfEdgeInfo() : base() { }
    public HalfEdgeInfo(int end, E info) : base(end, info) { }
}
class EdgeInfo<E> : Pair<Pair<int, int>, E>
{
    public int From { get { return First.First; } set { First.First = value; } }
    public int To { get { return First.Second; } set { First.Second = value; } }
    public E Information { get { return Second; } set { Second = value; } }
    public EdgeInfo() : base() { }
    public EdgeInfo(int from, int to, E info) : base(new Pair<int, int>(from, to), info) { }
    public EdgeInfo<E> Reverse() { return new EdgeInfo<E>(To, From, Information); }
}
class DirectedGraph<V, E> : IEnumerable<NodeInfo<V>>
{
    protected int numberOfNodes;
    public int NumberOfNodes { get { return numberOfNodes; } }
    protected NodeInfo<V>[] nodes;
    protected List<EdgeInfo<E>> edges;
    protected List<HalfEdgeInfo<E>>[] edgesFrom;
    protected List<HalfEdgeInfo<E>>[] edgesTo;
    public IEnumerable<HalfEdgeInfo<E>> EdgesFrom(int node) { return edgesFrom[node]; }
    public int InDegree(int node) { return edgesTo[node].Count; }
    public int OutDegree(int node) { return edgesFrom[node].Count; }
    public IEnumerable<HalfEdgeInfo<E>> EdgesTo(int node) { return edgesTo[node]; }
    public V this[int node] { get { return nodes[node].Second; } set { nodes[node].Second = value; } }
    public IEnumerable<EdgeInfo<E>> Edges { get { return edges; } }
    public DirectedGraph(int V)
    {
        numberOfNodes = V;
        nodes = Enumerable.Range(0, V).Select(x => new NodeInfo<V>(x, default(V))).ToArray();
        edges = new List<EdgeInfo<E>>();
        edgesFrom = Enumerable.Range(0, V).Select(_ => new List<HalfEdgeInfo<E>>()).ToArray();
        edgesTo = Enumerable.Range(0, V).Select(_ => new List<HalfEdgeInfo<E>>()).ToArray();
    }
    public DirectedGraph(int V, IEnumerable<EdgeInfo<E>> edges) : this(V) { foreach (var e in edges) AddEdge(e.From, e.To, e.Information); }
    public virtual void AddEdge(EdgeInfo<E> edge)
    {
        edges.Add(edge);
        edgesFrom[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
        edgesTo[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
    }
    public void AddEdge(int from, int to, E information) { AddEdge(new EdgeInfo<E>(from, to, information)); }
    public void AddEdge(V from, V to, E information) { AddEdge(new EdgeInfo<E>(SearchNode(from).Code, SearchNode(to).Code, information)); }
    public NodeInfo<V> SearchNode(V node) { return nodes.FirstOrDefault(e => e.Information.Equals(node)); }
    public EdgeInfo<E> SearchEdge(E edge) { return edges.Find(e => e.Information.Equals(edge)); }
    public IEnumerator<NodeInfo<V>> GetEnumerator() { foreach (var v in nodes) yield return v; }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    public long[] ShortestPathLengthFrom(int from, Func<E, long> cost)
    {
        long[] d = Enumerable.Repeat(long.MaxValue/2, numberOfNodes).ToArray();
        d[from] = 0;
        var update = true;
        while (update)
        {
            update = false;
            foreach (var e in edges)
            {
                var tmp = d[e.From] + cost(e.Information);
                if (d[e.From] < Func.Inf && d[e.To] > tmp)
                {
                    d[e.To] = tmp;
                    update = true;
                }
            }
        }
        return d;
    }
    public int[] DijkstraFrom(int from, Func<E, int> cost)
    {
        var d = Enumerable.Repeat(Func.Inf, numberOfNodes).ToArray();
        var queue = new PriorityQueue<Pair<int, int>>((x, y) => x.Second.CompareTo(y.Second));
        d[from] = 0;
        queue.Enqueue(new Pair<int, int>(from, 0));
        while (!queue.IsEmpty)
        {
            var p = queue.Dequeue();
            var v = p.First;
            if (d[v] < p.Second) continue;
            foreach (var e in EdgesFrom(v))
            {
                var tmp = d[v] + cost(e.Information);
                if (d[e.End] > tmp) queue.Enqueue(new Pair<int, int>(e.End, d[e.End] = tmp));
            }
        }
        return d;
    }
    // cost(e)>=0
    public Pair<long, int>[] DijkstraFromL(int from, Func<E, long> cost)
    {
        var d = new Pair<long, int>[numberOfNodes];
        for (var i = 0; i < numberOfNodes; i++) d[i] = new Pair<long, int>(Func.InfL, -1);
        var queue = new PriorityQueue<Tuple<int, long, int>>((x, y) => x.Item2.CompareTo(y.Item2));
        d[from] = new Pair<long, int>(0, -1);
        queue.Enqueue(new Tuple<int, long, int>(from, 0, -1));
        while (!queue.IsEmpty)
        {
            var p = queue.Dequeue();
            var v = p.Item1;
            if (d[v].First < p.Item2) continue;
            foreach (var e in edgesFrom[v])
            {
                var tmp = d[v].First + cost(e.Information);
                if (d[e.End].First > tmp) queue.Enqueue(new Tuple<int, long, int>(e.End, d[e.End].First = tmp, d[e.End].Second = v));
            }
        }
        return d;
    }
    public int[,] ShortestPathLengthEachOther(Func<E, int> cost)
    {
        var d = new int[numberOfNodes, numberOfNodes];
        for (var v = 0; v < numberOfNodes; v++) for (var w = 0; w < numberOfNodes; w++) d[v, w] = Func.Inf;
        for (var v = 0; v < numberOfNodes; v++) d[v, v] = 0;
        foreach (var e in edges) if (e.From != e.To) d[e.From, e.To] = cost(e.Information);
        for (var k = 0; k < numberOfNodes; k++)
            for (var v = 0; v < numberOfNodes; v++)
                for (var w = 0; w < numberOfNodes; w++)
                    d[v, w] = Math.Min(d[v, w], d[v, k] + d[k, w]);
        return d;
    }
    public bool ContainsNegativeLoopWF(Func<E, int> cost)
    {
        var d = ShortestPathLengthEachOther(cost);
        for (var v = 0; v < numberOfNodes; v++) if (d[v, v] < 0) return true;
        return false;
    }
    public bool ContainsNegativeLoop(Func<E, int> cost)
    {
        var d = Enumerable.Repeat(0, numberOfNodes).ToArray();
        for (var v = 0; v < numberOfNodes; v++)
        {
            foreach (var e in edges)
            {
                var tmp = d[e.From] + cost(e.Information);
                if (d[e.To] > tmp)
                {
                    d[e.To] = tmp;
                    if (v == numberOfNodes - 1) return true;
                }
            }
        }
        return false;
    }
    public IEnumerable<int> ReachableFrom(int from)
    {
        var used = new bool[numberOfNodes];
        var queue = new Queue<int>();
        queue.Enqueue(from);
        while (queue.Count > 0)
        {
            var v = queue.Dequeue();
            if (used[v]) continue;
            used[v] = true;
            foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
        }
        for (var v = 0; v < numberOfNodes; v++) if (used[v]) yield return v;
    }
    public bool IsReachable(int from, int to)
    {
        var used = new bool[numberOfNodes];
        var queue = new Queue<int>();
        queue.Enqueue(from);
        while (queue.Count > 0)
        {
            var v = queue.Dequeue();
            if (v == to) return true;
            if (used[v]) continue;
            used[v] = true;
            foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
        }
        return false;
    }
    public Pair<DirectedGraph<HashSet<NodeInfo<V>>, object>, int[]> StronglyConnectedComponents()
    {
        var mark = new bool[numberOfNodes];
        var stack = new Stack<int>();
        Action<int> dfs = null;
        dfs = v =>
        {
            mark[v] = true;
            foreach (var w in edgesFrom[v]) if (!mark[w.End]) dfs(w.End);
            stack.Push(v);
        };
        for (var v = 0; v < numberOfNodes; v++) if (!mark[v]) dfs(v);
        var scc = new List<HashSet<NodeInfo<V>>>();
        mark = new bool[numberOfNodes];
        var which = new int[numberOfNodes];
        Action<int, HashSet<NodeInfo<V>>> rdfs = null;
        rdfs = (v, set) =>
        {
            set.Add(new NodeInfo<V>(v, nodes[v].Information));
            mark[v] = true;
            foreach (var w in edgesFrom[v]) if (!mark[w.End]) rdfs(w.End, set);
        };
        var M = 0;
        while (stack.Count > 0)
        {
            var v = stack.Pop();
            if (mark[v]) continue;
            var set = new HashSet<NodeInfo<V>>();
            rdfs(v, set);
            scc.Add(set);
            foreach (var w in set) which[w.Code] = M;
            M++;
        }
        var graph = new UndirectedGraph<HashSet<NodeInfo<V>>, object>(M);
        for (var v = 0; v < M; v++) graph[v] = scc[v];
        foreach (var e in edges) if (which[e.From] != which[e.To]) graph.AddEdge(which[e.From], which[e.To], null);
        return new Pair<DirectedGraph<HashSet<NodeInfo<V>>, object>, int[]>(graph, which);
    }
    public string ToString(Func<V, string> vertex, Func<E, string> edge)
    {
        var sb = new StringBuilder();
        sb.Append("digraph G {\n");
        foreach (var v in nodes) sb.Append($"\tv{v.Code} [label = \"{vertex(v.Information)}\"];\n");
        foreach (var e in edges) sb.Append($"\tv{e.From} -> v{e.To} [label=\"{edge(e.Information)}\"];\n");
        sb.Append("}");
        return sb.ToString();
    }
    public override string ToString() { return ToString(v => v.ToString(), e => e.ToString()); }
}
class UnionFindTree
{
    int N;
    int[] parent, rank, size;
    public UnionFindTree(int capacity)
    {
        N = capacity;
        parent = new int[N];
        rank = new int[N];
        size = new int[N];
        for (var i = 0; i < N; i++) { parent[i] = i; size[i] = 1; }
    }
    public int GetSize(int x) { return size[GetRootOf(x)]; }
    public int GetRootOf(int x) { return parent[x] == x ? x : parent[x] = GetRootOf(parent[x]); }
    public bool UniteCategory(int x, int y)
    {
        if ((x = GetRootOf(x)) == (y = GetRootOf(y))) return false;
        if (rank[x] < rank[y]) { parent[x] = y; size[y] += size[x]; }
        else
        {
            parent[y] = x; size[x] += size[y];
            if (rank[x] == rank[y]) rank[x]++;
        }
        return true;
    }
    public bool IsSameCategory(int x, int y) { return GetRootOf(x) == GetRootOf(y); }
}
class AVLTree<T> : IEnumerable<T>, ICollection<T>, ICollection, IEnumerable
{
    public class AVLNode : IEnumerable<T>
    {
        AVLTree<T> tree;
        int height;
        public int Height { get { return height; } }
        public int Bias { get { return Left.height - Right.height; } }
        public T Item;
        public AVLNode Parent;
        public AVLNode Left;
        public AVLNode Right;
        AVLNode(T x, AVLTree<T> tree) { this.tree = tree; Item = x; Left = tree.sentinel; Right = tree.sentinel; }
        public AVLNode(AVLTree<T> tree) : this(default(T), tree) { height = 0; Parent = null; }
        public AVLNode(T x, AVLNode parent, AVLTree<T> tree) : this(x, tree) { height = 1; Parent = parent; }
        public void Adjust() { height = 1 + Math.Max(Left.height, Right.height); }
        public void ResetAsSentinel() { height = 0; Left = tree.sentinel; Right = tree.sentinel; }
        public IEnumerator<T> GetEnumerator()
        {
            if (this != tree.sentinel)
            {
                foreach (var x in Left) yield return x;
                yield return Item;
                foreach (var x in Right) yield return x;
            }
        }
        IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    }
    AVLNode sentinel;
    Comparison<T> comp;
    Func<T, T, bool> equals;
    int count;
    // assumed to be comparer
    // i.e. comp(x,x)=0, and comp(x,y)>0 then comp(y,x)<0, and comp(x,y)>0 & comp(y,z)>0 then comp(x,z)>0
    public AVLTree(Comparison<T> comp)
    {
        sentinel = new AVLNode(this);
        sentinel.ResetAsSentinel();
        this.comp = comp == null ? Func.DefaultComparison<T>() : comp;
        if (typeof(T).IsValueType) equals = (x, y) => x.Equals(y);
        else equals = (x, y) => ReferenceEquals(x, y);
        count = 0;
    }
    public AVLTree(IComparer<T> comp = null) : this(comp.ToComparison()) { }
    void Replace(AVLNode u, AVLNode v)
    {
        var parent = u.Parent;
        if (parent.Left == u) parent.Left = v;
        else parent.Right = v;
        v.Parent = parent;
    }
    AVLNode RotateL(AVLNode v)
    {
        var u = v.Right;
        Replace(v, u);
        v.Right = u.Left;
        u.Left.Parent = v;
        u.Left = v;
        v.Parent = u;
        v.Adjust();
        u.Adjust();
        return u;
    }
    AVLNode RotateR(AVLNode u)
    {
        var v = u.Left;
        Replace(u, v);
        u.Left = v.Right;
        v.Right.Parent = u;
        v.Right = u;
        u.Parent = v;
        u.Adjust();
        v.Adjust();
        return v;
    }
    AVLNode RotateLR(AVLNode t) { RotateL(t.Left); return RotateR(t); }
    AVLNode RotateRL(AVLNode t) { RotateR(t.Right); return RotateL(t); }
    void Adjust(bool isInsertMode, AVLNode node)
    {
        while (node.Parent != sentinel)
        {
            var parent = node.Parent;
            var height = parent.Height;
            if ((parent.Left == node) == isInsertMode)
                if (parent.Bias == 2)
                    if (parent.Left.Bias >= 0) parent = RotateR(parent);
                    else parent = RotateLR(parent);
                else parent.Adjust();
            else
                if (parent.Bias == -2)
                if (parent.Right.Bias <= 0) parent = RotateL(parent);
                else parent = RotateRL(parent);
            else parent.Adjust();
            if (height == parent.Height) break;
            node = parent;
        }
    }
    public void Add(T item)
    {
        var parent = sentinel;
        var pos = sentinel.Left;
        var isLeft = true;
        count++;
        while (pos != sentinel)
            if (comp(item, pos.Item) < 0) { parent = pos; pos = pos.Left; isLeft = true; }
            else { parent = pos; pos = pos.Right; isLeft = false; }
        if (isLeft)
        {
            parent.Left = new AVLNode(item, parent, this);
            Adjust(true, parent.Left);
        }
        else
        {
            parent.Right = new AVLNode(item, parent, this);
            Adjust(true, parent.Right);
        }
    }
    // if equals(x,y) holds then !(comp(x,y)<0) and !(comp(x,y)>0) must hold
    // i.e. equals(x,y) -> comp(x,y)=0
    public bool Remove(T item, AVLNode start)
    {
        var pos = start;
        while (pos != sentinel)
        {
            if (comp(item, pos.Item) < 0) pos = pos.Left;
            else if (comp(item, pos.Item) > 0) pos = pos.Right;
            else if (equals(pos.Item, item))
            {
                if (pos.Left == sentinel)
                {
                    Replace(pos, pos.Right);
                    Adjust(false, pos.Right);
                }
                else
                {
                    var max = Max(pos.Left);
                    pos.Item = max.Item;
                    Replace(max, max.Left);
                    Adjust(false, max.Left);
                }
                count--;
                return true;
            }
            else return Remove(item, pos.Left) || Remove(item, pos.Right);
        }
        return false;
    }
    public bool Remove(T item) { return Remove(item, sentinel.Left); }
    AVLNode Max(AVLNode node)
    {
        while (node.Right != sentinel) node = node.Right;
        return node;
    }
    AVLNode Min(AVLNode node)
    {
        while (node.Left != sentinel) node = node.Left;
        return node;
    }
    public bool Contains(T item)
    {
        var pos = sentinel.Left;
        while (pos != sentinel)
        {
            if (comp(item, pos.Item) < 0) pos = pos.Left;
            else if (comp(item, pos.Item) > 0) pos = pos.Right;
            else return true;
        }
        return false;
    }
    public T Find(T item)
    {
        var pos = sentinel.Left;
        while (pos != sentinel)
        {
            if (comp(item, pos.Item) < 0) pos = pos.Left;
            else if (comp(item, pos.Item) > 0) pos = pos.Right;
            else return pos.Item;
        }
        return default(T);
    }
    public AVLNode LowerBound(Predicate<T> pred) { AVLNode node; LowerBound(pred, sentinel.Left, out node); return node; }
    public AVLNode UpperBound(Predicate<T> pred) { AVLNode node; UpperBound(pred, sentinel.Left, out node); return node; }
    public AVLNode LowerBound(T item) { return LowerBound(x => comp(x, item) >= 0); }
    public AVLNode UpperBound(T item) { return UpperBound(x => comp(x, item) <= 0); }
    bool UpperBound(Predicate<T> pred, AVLNode node, out AVLNode res)
    {
        if (node == sentinel) { res = null; return false; }
        if (pred(node.Item)) { if (!UpperBound(pred, node.Right, out res)) res = node; return true; }
        else return UpperBound(pred, node.Left, out res);
    }
    bool LowerBound(Predicate<T> pred, AVLNode node, out AVLNode res)
    {
        if (node == sentinel) { res = null; return false; }
        if (pred(node.Item)) { if (!LowerBound(pred, node.Left, out res)) res = node; return true; }
        else return LowerBound(pred, node.Right, out res);
    }
    public T Min() { return Min(sentinel.Left).Item; }
    public AVLNode MinNode() { return Min(sentinel.Left); }
    public T Max() { return Max(sentinel.Left).Item; }
    public AVLNode MaxNode() { return Max(sentinel.Left); }
    public bool IsEmpty { get { return sentinel.Left == sentinel; } }
    public void Clear() { sentinel.Left = sentinel; count = 0; sentinel.ResetAsSentinel(); }
    public IEnumerator<T> GetEnumerator() { return sentinel.Left.GetEnumerator(); }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    public void CopyTo(T[] array, int arrayIndex) { foreach (var x in this) array[arrayIndex++] = x; }
    public int Count { get { return count; } }
    public bool IsReadOnly { get { return true; } }
    public void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
    public bool IsSynchronized { get { return false; } }
    public object SyncRoot { get { return this; } }
    public override string ToString()
    {
        var nodes = new StringBuilder();
        var edges = new StringBuilder();
        ConcatSubTree(nodes, edges, sentinel.Left, "L");
        return $"digraph G {{\n{nodes.ToString()}{edges.ToString()}}}";
    }
    void ConcatSubTree(StringBuilder nodes, StringBuilder edges, AVLNode node, string code)
    {
        if (node == sentinel) return;
        nodes.Append($"\tv{code} [label = \"{node.Height}:{node.Item}\"];\n");
        if (node.Left != sentinel) edges.Append($"\tv{code} -> v{code}L;\n");
        if (node.Right != sentinel) edges.Append($"\tv{code} -> v{code}R;\n");
        ConcatSubTree(nodes, edges, node.Left, $"{code}L");
        ConcatSubTree(nodes, edges, node.Right, $"{code}R");
    }
    public bool IsBalanced() { return IsBalanced(sentinel.Left); }
    public bool IsValidBinarySearchTree() { return IsValidBinarySearchTree(sentinel.Left); }
    bool IsBalanced(AVLNode node) { return node == sentinel || (Math.Abs(node.Bias) < 2 && IsBalanced(node.Left) && IsBalanced(node.Right)); }
    bool IsValidBinarySearchTree(AVLNode node)
    {
        return node == sentinel || (Small(node.Item, node.Left) && Large(node.Item, node.Right)
               && IsValidBinarySearchTree(node.Left) && IsValidBinarySearchTree(node.Right));
    }
    bool Small(T item, AVLNode node) { return node == sentinel || (comp(item, node.Item) >= 0 && Small(item, node.Left) && Small(item, node.Right)); }
    bool Large(T item, AVLNode node) { return node == sentinel || (comp(item, node.Item) <= 0 && Large(item, node.Left) && Large(item, node.Right)); }
    public static void CheckAVL(Random rand, int N)
    {
        Comparison<double> comp = (x, y) => x.CompareTo(y);
        var avl = new AVLTree<double>(comp);
        var toBeLeft = new double[N];
        var toBeRemoved = new double[N];
        for (var i = 0; i < N; i++) avl.Add(toBeRemoved[i] = rand.NextDouble());
        for (var i = 0; i < N; i++) avl.Add(toBeLeft[i] = rand.NextDouble());
        for (var i = 0; i < N; i++) Console.Write(avl.Remove(toBeRemoved[i]) ? "" : "!!!NOT REMOVED!!! => " + toBeRemoved[i] + "\n");
        var insertErrors = toBeLeft.All(x => avl.Contains(x));
        var deleteErrors = avl.Count == N;
        //Console.WriteLine("【AVL木の構造】");
        //Console.WriteLine(avl);
        if (insertErrors && deleteErrors) Console.WriteLine("○\t挿入, 削除操作が正しく行われています.");
        else if (insertErrors) Console.WriteLine("×\t挿入(または削除)操作に問題があります.");
        else Console.WriteLine("×\t削除(または挿入)操作に問題があります.");
        if (avl.IsBalanced()) Console.WriteLine("○\tAVL木は平衡条件を保っています.");
        else Console.WriteLine("×\tAVL木の平衡条件が破れています.");
        if (avl.IsValidBinarySearchTree()) Console.WriteLine("○\tAVL木は二分探索木になっています.");
        else Console.WriteLine("×\tAVL木は二分探索木になっていません.");
        Array.Sort(toBeLeft, comp);
        Console.WriteLine($"最小値 : {avl.Min()} ≡ {toBeLeft.First()}");
        Console.WriteLine($"最大値 : {avl.Max()} ≡ {toBeLeft.Last()}");
        Console.WriteLine($"要素数 : {avl.Count} 個");
    }
}
#endregion グラフ