Problem:
Score : 600 points

Problem Statement

On the xy-plane, Snuke is going to travel from the point (x_s, y_s) to the point (x_t, y_t).
He can move in arbitrary directions with speed 1.
Here, we will consider him as a point without size.

There are N circular barriers deployed on the plane.
The center and the radius of the i-th barrier are (x_i, y_i) and r_i, respectively.
The barriers may overlap or contain each other.

A point on the plane is exposed to cosmic rays if the point is not within any of the barriers.

Snuke wants to avoid exposure to cosmic rays as much as possible during the travel.
Find the minimum possible duration of time he is exposed to cosmic rays during the travel.

Constraints

All input values are integers.

-10^9 ≤ x_s, y_s, x_t, y_t ≤ 10^9

(x_s, y_s) ≠ (x_t, y_t)

1≤N≤1,000

-10^9 ≤ x_i, y_i ≤ 10^9

1 ≤ r_i ≤ 10^9

Input

The input is given from Standard Input in the following format:

x_s y_s x_t y_t
N
x_1 y_1 r_1
x_2 y_2 r_2
:
x_N y_N r_N

Output

Print the minimum possible duration of time Snuke is exposed to cosmic rays during the travel.
The output is considered correct if the absolute or relative error is at most 10^{-9}.

Sample Input 1

-2 -2 2 2
1
0 0 1

Sample Output 1

3.6568542495

An optimal route is as follows:

Sample Input 2

-2 0 2 0
2
-1 0 2
1 0 2

Sample Output 2

0.0000000000

An optimal route is as follows:

Sample Input 3

4 -2 -2 4
3
0 0 2
4 0 1
0 4 1

Sample Output 3

4.0000000000

An optimal route is as follows:

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;
        /// <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()
        {
            double xs = sc.nextDouble();
            double ys = sc.nextDouble();
            double xt = sc.nextDouble();
            double yt = sc.nextDouble();
            int n = sc.nextInt();
            double[] x = new double[n + 2]; 
            double[] y = new double[n + 2];
            double[] r = new double[n + 2];
            x[0] = xs;
            x[1] = xt;
            y[0] = ys;
            y[1] = yt;
            for (int i = 2; i < n + 2; i++)
            {
                x[i] = sc.nextDouble();
                y[i] = sc.nextDouble();
                r[i] = sc.nextDouble();
            }
            UndirectedGraph<double, double> g = new UndirectedGraph<double, double>(n + 2);
            for (int i = 0; i < n + 2; i++)
            {
                for (int j = 0; j < i; j++)
                {
                    g.AddEdge(i, j, Math.Max(0, (Math.Sqrt(Math.Pow(x[i] - x[j], 2) + Math.Pow(y[i] - y[j], 2)) - r[i] - r[j])));
                }
            }
            //var len = g.ShortestPathLengthFrom(0, _ => _);
            var len = g.DijkstraFrom(0, _ => _);
            Console.WriteLine(len[1]);
        }
    }
}
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>

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);
    }
}


/// <summary>
/// 幾何ライブラリ
/// </summary>
namespace Geometry
{
    #region 3次元幾何


    /// <summary>
    /// 3次元幾何
    /// </summary>
    static class Geometry3D
    {
        public static double Eps = 1e-8;
        public static double Inf = 1e12;
        /// <summary>
        /// 点/ベクトル(3次元)
        /// </summary>
        public struct Point3D
        {
            public double X;
            public double Y;
            public double Z;
            public Point3D(double x, double y, double z) { X = x; Y = y; Z = z; }
            public Point3D(double[] ls) { X = ls[0]; Y = ls[1]; Z = ls[2]; }

            public static Point3D operator +(Point3D p) { return new Point3D(p.X, p.Y, p.Z); }
            public static Point3D operator -(Point3D p) { return new Point3D(-p.X, -p.Y, -p.Z); }
            public static Point3D operator /(Point3D p, double r) { return new Point3D(p.X / r, p.Y / r, p.Z / r); }
            public static Point3D operator *(double r, Point3D p) { return new Point3D(p.X * r, p.Y * r, p.Z * r); }
            public static Point3D operator *(Point3D p, double r) { return new Point3D(p.X * r, p.Y * r, p.Z * r); }
            public static Point3D operator +(Point3D p, Point3D q) { return new Point3D(p.X + q.X, p.Y + q.Y, p.Z + q.Z); }
            public static Point3D operator -(Point3D p, Point3D q) { return new Point3D(p.X - q.X, p.Y - q.Y, p.Z - q.Z); }
        }

        /// <summary>
        /// 線分/直線(3次元)
        /// </summary>
        public struct Line3D
        {
            public Point3D a, b;
            public Line3D(Point3D a, Point3D b) { this.a = a; this.b = b; }
        }

        /// <summary>
        /// 内積
        /// </summary>
        /// <param name="a"></param>
        /// <param name="b"></param>
        /// <returns></returns>
        public static double dot(Point3D a, Point3D b)
        {
            return a.X * b.X + a.Y * b.Y + a.Z * b.Z;
        }

        /// <summary>
        /// 外積
        /// </summary>
        /// <param name="a"></param>
        /// <param name="b"></param>
        /// <returns></returns>
        public static Point3D cross(Point3D a, Point3D b)
        {
            return new Point3D(a.Y * b.Z - a.Z * b.Y, a.Z * b.X - a.X * b.Z, a.X * b.Y - a.Y * b.X);
        }

        public static double norm(Point3D a)
        {
            return dot(a, a);
        }

        public static double dist(Point3D a)
        {
            return Math.Sqrt(norm(a));
        }

        #region line
        /// <summary>
        /// 点が線分上にあるか
        /// </summary>
        /// <param name="l">線分</param>
        /// <param name="p">点</param>
        /// <returns></returns>
        public static bool is_in_segment(Line3D l, Point3D p)
        {
            return Math.Abs(dist(l.a - p) + dist(l.b - p) - dist(l.a - l.b)) < Eps;
        }

        /// <summary>
        /// 点の射影(点から直線に引いた垂線と線分の交点)
        /// </summary>
        /// <param name="l">直線</param>
        /// <param name="p">点</param>
        /// <returns></returns>
        public static Point3D project_lp(Line3D l, Point3D p)
        {
            Point3D point = l.a, vec = l.b - l.a;
            return point + dot(p - point, vec) / norm(vec) * vec;
        }

        /// <summary>
        /// 点と直線の距離
        /// </summary>
        /// <param name="l">直線</param>
        /// <param name="p">点</param>
        /// <returns></returns>
        public static double distance_lp(Line3D l, Point3D p)
        {
            return dist(p - project_lp(l, p));
        }

        /// <summary>
        /// 点と線分の距離
        /// </summary>
        /// <param name="l">線分</param>
        /// <param name="p">点</param>
        /// <returns></returns>
        public static double distance_sp(Line3D l, Point3D p)
        {
            Point3D proj = project_lp(l, p);
            if (is_in_segment(l, proj))
            {
                return dist(p - proj);
            }
            else
            {
                return Math.Min(dist(p - l.a), dist(p - l.b));
            }
        }

        /// <summary>
        /// 直線と直線の距離
        /// </summary>
        /// <param name="l"></param>
        /// <param name="m"></param>
        /// <returns></returns>
        public static double distance_ll(Line3D l, Line3D m)
        {
            Point3D v = cross(l.b - l.a, m.b - m.a), p = m.a - l.a;
            if (dist(v) < Eps)
            {
                return distance_lp(l, m.a);
            }
            else
            {
                return Math.Abs(dot(v, p)) / dist(v);
            }
        }
        #endregion

        //TODO Plane関連以下の実装
    }

    #endregion
}


#region 未整理ライブラリ

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 double[] ShortestPathLengthFrom(int from, Func<E, double> cost)
    {
        var d = Enumerable.Repeat(double.MaxValue/3, 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 double[] DijkstraFrom(int from, Func<E, double> cost)
    {
        var d = Enumerable.Repeat(double.MaxValue/3, numberOfNodes).ToArray();
        var queue = new PriorityQueue<Pair<int, double>>((x, y) => x.Second.CompareTo(y.Second));
        d[from] = 0;
        queue.Enqueue(new Pair<int, double>(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, double>(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");
    }
}

class PointInt : Pair<int, int>
{
    public int X { get { return First; } set { First = value; } }
    public int Y { get { return Second; } set { Second = value; } }
    public PointInt() : base(0, 0) { }
    public PointInt(int x, int y) : base(x, y) { }
    public IEnumerable<PointInt> Neighbors4()
    {
        yield return new PointInt(X - 1, Y);
        yield return new PointInt(X, Y - 1);
        yield return new PointInt(X, Y + 1);
        yield return new PointInt(X + 1, Y);
    }
    public IEnumerable<PointInt> Neighbors8()
    {
        yield return new PointInt(X - 1, Y - 1);
        yield return new PointInt(X - 1, Y);
        yield return new PointInt(X - 1, Y + 1);
        yield return new PointInt(X, Y - 1);
        yield return new PointInt(X, Y + 1);
        yield return new PointInt(X + 1, Y - 1);
        yield return new PointInt(X + 1, Y);
        yield return new PointInt(X + 1, Y + 1);
    }
    public static PointInt operator +(PointInt p) { return new PointInt(p.X, p.Y); }
    public static PointInt operator -(PointInt p) { return new PointInt(-p.X, -p.Y); }
    public static PointInt operator /(PointInt p, int r) { return new PointInt(p.X / r, p.Y / r); }
    public static PointInt operator *(int r, PointInt p) { return new PointInt(p.X * r, p.Y * r); }
    public static PointInt operator *(PointInt p, int r) { return new PointInt(p.X * r, p.Y * r); }
    public static PointInt operator +(PointInt p, PointInt q) { return new PointInt(p.X + q.X, p.Y + q.Y); }
    public static PointInt operator -(PointInt p, PointInt q) { return new PointInt(p.X - q.X, p.Y - q.Y); }
}
#endregion