Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i.
Vertex i has an integer a_i written on it.
For every integer k from 1 through N, solve the following problem:

We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.

Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i \leq 10^9

1 \leq u_i , v_i \leq N

u_i \neq v_i

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Sample Input 1

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

#region いつもの
using AtCoderProject;
using System;
using System.Collections.Generic;
using System.Linq;
using static AtCoderProject.Global;
using BigInteger = System.Numerics.BigInteger;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using StringBuilder = System.Text.StringBuilder;
using Unsafe = System.Runtime.CompilerServices.Unsafe;

namespace AtCoderProject { using System.IO; using System.Text; using System.Diagnostics; using System.Runtime.CompilerServices; public static class Global { public static T[] NewArray<T>(int len0, T value) => new T[len0].Fill(value); public static T[] NewArray<T>(int len0, Func<T> factory) { var arr = new T[len0]; for (int i = 0; i < arr.Length; i++) arr[i] = factory(); return arr; } public static T[][] NewArray<T>(int len0, int len1, T value) where T : struct { var arr = new T[len0][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, value); return arr; } public static T[][] NewArray<T>(int len0, int len1, Func<T> factory) { var arr = new T[len0][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, factory); return arr; } public static T[][][] NewArray<T>(int len0, int len1, int len2, T value) where T : struct { var arr = new T[len0][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, value); return arr; } public static T[][][] NewArray<T>(int len0, int len1, int len2, Func<T> factory) { var arr = new T[len0][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, factory); return arr; } public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, T value) where T : struct { var arr = new T[len0][][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, value); return arr; } public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, Func<T> factory) { var arr = new T[len0][][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, factory); return arr; } public static int Pow(int x, int y) { int res = 1; for (; y > 0; y >>= 1) { if ((y & 1) == 1) res *= x; x *= x; } return res; } public static BigInteger ParseBigInteger(ReadOnlySpan<char> s) { /* 自前実装の方が速い */ BigInteger res; if (s.Length % 9 == 0) res = 0; else { res = new BigInteger(int.Parse(s.Slice(0, s.Length % 9))); s = s.Slice(s.Length % 9); } while (s.Length > 0) { var sp = s.Slice(0, 9); res *= 1000_000_000; res += int.Parse(sp); s = s.Slice(9); } return res; } public static int BitCount(int x) { x -= (x >> 1) & 0x55555555; x = (x & 0x33333333) + ((x >> 2) & 0x33333333); x = (x + (x >> 4)) & 0x0f0f0f0f; x += x >> 8; x += x >> 16; return x & 0x3f; } public static int BitCount(long x) { x -= (x >> 1) & 0x5555555555555555; x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333); x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f; x += x >> 8; x += x >> 16; x += x >> 32; return (int)(x & 0x0000007f); } public static int MSB(int x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; return BitCount(x) - 1; } public static int MSB(long x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; x |= x >> 32; return BitCount(x) - 1; } public static int LSB(int x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; return 32 - BitCount(x); } public static int LSB(long x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; x |= x << 32; return 64 - BitCount(x); } } public static class Ext { public static bool UpdateMax(this ref int r, int val) { if (r < val) { r = val; return true; } return false; } public static bool UpdateMax(this ref long r, long val) { if (r < val) { r = val; return true; } return false; } public static bool UpdateMin(this ref int r, int val) { if (r > val) { r = val; return true; } return false; } public static bool UpdateMin(this ref long r, long val) { if (r > val) { r = val; return true; } return false; } public static long ToLong(this int i) => i; public static T[] Fill<T>(this T[] arr, T value) { Array.Fill(arr, value); return arr; } public static T[] Sort<T>(this T[] arr) { Array.Sort(arr); return arr; } public static string[] Sort(this string[] arr) => Sort(arr, StringComparer.OrdinalIgnoreCase); public static T[] Sort<T, U>(this T[] arr, Func<T, U> selector) where U : IComparable<U> => Sort(arr, (a, b) => selector(a).CompareTo(selector(b))); public static T[] Sort<T>(this T[] arr, Comparison<T> comparison) { Array.Sort(arr, comparison); return arr; } public static T[] Sort<T>(this T[] arr, IComparer<T> comparer) { Array.Sort(arr, comparer); return arr; } public static T[] Reverse<T>(this T[] arr) { Array.Reverse(arr); return arr; } public static (int index, T max) MaxBy<T>(this T[] arr) where T : IComparable<T> { T max = arr[0]; int maxIndex = 0; for (int i = 0; i < arr.Length; i++) { if (max.CompareTo(arr[i]) < 0) { max = arr[i]; maxIndex = i; } } return (maxIndex, max); } public static (TSource item, TMax max) MaxBy<TSource, TMax>(this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector) where TMax : IComparable<TMax> { TMax max; TSource maxByItem; var e = source.GetEnumerator(); e.MoveNext(); maxByItem = e.Current; max = maxBySelector(maxByItem); while (e.MoveNext()) { var item = e.Current; var next = maxBySelector(item); if (max.CompareTo(next) < 0) { max = next; maxByItem = item; } } return (maxByItem, max); } public static (int index, T min) MinBy<T>(this T[] arr) where T : IComparable<T> { T min = arr[0]; int minIndex = 0; for (int i = 0; i < arr.Length; i++) { if (min.CompareTo(arr[i]) > 0) { min = arr[i]; minIndex = i; } } return (minIndex, min); } public static (TSource item, TMin min) MinBy<TSource, TMin>(this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector) where TMin : IComparable<TMin> { TMin min; TSource minByItem; var e = source.GetEnumerator(); e.MoveNext(); minByItem = e.Current; min = minBySelector(minByItem); while (e.MoveNext()) { var item = e.Current; var next = minBySelector(item); if (min.CompareTo(next) > 0) { min = next; minByItem = item; } } return (minByItem, min); } public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x)); public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count()); public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i); public static TValue Get<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key) { dic.TryGetValue(key, out var v); return v; } public static TValue GetOrInit<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key, TValue value) { if (dic.TryGetValue(key, out var v)) return v; return dic[key] = value; } } public class ReverseComparer<T> : Comparer<T> where T : IComparable<T> { private static ReverseComparer<T> defaultComparer; public static new IComparer<T> Default => defaultComparer ??= new ReverseComparer<T>(); public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); } public class ΔDebugView<T> { private IEnumerable<T> collection; public ΔDebugView(IEnumerable<T> collection) { this.collection = collection ?? throw new ArgumentNullException(nameof(collection)); }[DebuggerBrowsable(DebuggerBrowsableState.RootHidden)] public T[] Items => collection.ToArray(); }[DebuggerStepThrough] public class ConsoleReader { const int BufSize = 1 << 12; private readonly byte[] buffer = new byte[BufSize]; private readonly Stream input; private readonly Encoding encoding; private int pos = 0; private int len = 0; public ConsoleReader(Stream input, Encoding encoding) { this.input = input; this.encoding = encoding; } public ConsoleReader(Stream input) : this(input, Console.InputEncoding) { } private void MoveNext() { if (++pos >= len) { len = input.Read(buffer, 0, buffer.Length); if (len == 0) { buffer[0] = 10; } pos = 0; } } public int Int { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { int res = 0; bool neg = false; while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); } do { res = checked(res * 10 + (buffer[pos] ^ 48)); MoveNext(); } while (48 <= buffer[pos]); return neg ? -res : res; } } public int Int0 => this.Int - 1; public long Long { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { long res = 0; bool neg = false; while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); } do { res = res * 10 + (buffer[pos] ^ 48); MoveNext(); } while (48 <= buffer[pos]); return neg ? -res : res; } } public long Long0 => this.Long - 1; public string String { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new List<byte>(); while (buffer[pos] <= 32) MoveNext(); do { sb.Add(buffer[pos]); MoveNext(); } while (32 < buffer[pos]); return this.encoding.GetString(sb.ToArray()); } } public string Ascii { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new StringBuilder(); while (buffer[pos] <= 32) MoveNext(); do { sb.Append((char)buffer[pos]); MoveNext(); } while (32 < buffer[pos]); return sb.ToString(); } } public string Line { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new List<byte>(); while (buffer[pos] < 32) MoveNext(); do { sb.Add(buffer[pos]); MoveNext(); } while (buffer[pos] != 10 && buffer[pos] != 13); return this.encoding.GetString(sb.ToArray()); } } public char Char { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { while (buffer[pos] < 32) MoveNext(); char res = (char)buffer[pos]; MoveNext(); return res; } } public double Double => double.Parse(this.Ascii);[DebuggerStepThrough] public struct RepeatReader { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public T[] Select<T>(Func<ConsoleReader, T> factory) { var arr = new T[count]; for (var i = 0; i < count; i++) arr[i] = factory(cr); return arr; } public T[] Select<T>(Func<ConsoleReader, int, T> factory) { var arr = new T[count]; for (var i = 0; i < count; i++) arr[i] = factory(cr, i); return arr; } public string[] Line { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Line; return arr; } } public string[] String { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.String; return arr; } } public string[] Ascii { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Ascii; return arr; } } public int[] Int { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int; return arr; } } public int[] Int0 { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int0; return arr; } } public long[] Long { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long; return arr; } } public long[] Long0 { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long0; return arr; } } public double[] Double { get { var arr = new double[count]; for (var i = 0; i < count; i++) arr[i] = cr.Double; return arr; } } public static implicit operator int[](RepeatReader rr) => rr.Int; public static implicit operator long[](RepeatReader rr) => rr.Long; public static implicit operator double[](RepeatReader rr) => rr.Double; public static implicit operator string[](RepeatReader rr) => rr.Ascii; } public RepeatReader Repeat(int count) => new RepeatReader(this, count);[DebuggerStepThrough] public struct SplitReader { ConsoleReader cr; public SplitReader(ConsoleReader cr) { this.cr = cr; } public string[] String { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.String); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public string[] Ascii { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Ascii); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public int[] Int { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public int[] Int0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public long[] Long { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public long[] Long0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public double[] Double { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<double>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Double); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public static implicit operator int[](SplitReader sr) => sr.Int; public static implicit operator long[](SplitReader sr) => sr.Long; public static implicit operator double[](SplitReader sr) => sr.Double; public static implicit operator string[](SplitReader sr) => sr.Ascii; } public SplitReader Split => new SplitReader(this); public static implicit operator int(ConsoleReader cr) => cr.Int; public static implicit operator long(ConsoleReader cr) => cr.Long; public static implicit operator double(ConsoleReader cr) => cr.Double; public static implicit operator string(ConsoleReader cr) => cr.Ascii; public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2) => (o1, o2) = (this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3) => (o1, o2, o3) = (this, this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3, out ConsoleReader o4) => (o1, o2, o3, o4) = (this, this, this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3, out ConsoleReader o4, out ConsoleReader o5) => (o1, o2, o3, o4, o5) = (this, this, this, this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3, out ConsoleReader o4, out ConsoleReader o5, out ConsoleReader o6) => (o1, o2, o3, o4, o5, o6) = (this, this, this, this, this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3, out ConsoleReader o4, out ConsoleReader o5, out ConsoleReader o6, out ConsoleReader o7) => (o1, o2, o3, o4, o5, o6, o7) = (this, this, this, this, this, this, this); public void Deconstruct(out ConsoleReader o1, out ConsoleReader o2, out ConsoleReader o3, out ConsoleReader o4, out ConsoleReader o5, out ConsoleReader o6, out ConsoleReader o7, out ConsoleReader o8) => (o1, o2, o3, o4, o5, o6, o7, o8) = (this, this, this, this, this, this, this, this); }[DebuggerStepThrough] public class ConsoleWriter { private readonly StreamWriter sw; public ConsoleWriter(Stream output) : this(output, Console.OutputEncoding) { } public ConsoleWriter(Stream output, Encoding encoding) { sw = new StreamWriter(output, encoding); } public void Flush() => sw.Flush(); public ConsoleWriter WriteLine<T>(T obj) { sw.WriteLine(obj.ToString()); return this; } public ConsoleWriter WriteLineJoin<T>(IEnumerable<T> col) => WriteMany(' ', col); public ConsoleWriter WriteLines<T>(IEnumerable<T> col) => WriteMany('\n', col); public ConsoleWriter WriteLineGrid<T>(IEnumerable<IEnumerable<T>> cols) { var en = cols.GetEnumerator(); while (en.MoveNext()) WriteLineJoin(en.Current); return this; } private ConsoleWriter WriteMany<T>(char sep, IEnumerable<T> col) { var en = col.GetEnumerator(); if (!en.MoveNext()) return this; sw.Write(en.Current.ToString()); while (en.MoveNext()) { sw.Write(sep); sw.Write(en.Current.ToString()); } sw.WriteLine(); return this; } } }
public partial class Program {[System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] public ConsoleReader cr;[System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] public ConsoleWriter cw; public Program(ConsoleReader reader, ConsoleWriter writer) { this.cr = reader; this.cw = writer; System.Globalization.CultureInfo.CurrentCulture = System.Globalization.CultureInfo.InvariantCulture; } static void Main() => new Program(new ConsoleReader(Console.OpenStandardInput()), new ConsoleWriter(Console.OpenStandardOutput())).Run(); public void Run() { var res = Calc(); if (res is double) cw.WriteLine(Result((double)res)); else if (res is bool) cw.WriteLine(Result((bool)res)); else if (res != null) cw.WriteLine(res.ToString()); cw.Flush(); } }
public partial class Program
{
    public static string Result(double d) => d.ToString("0.####################", System.Globalization.CultureInfo.InvariantCulture);
    #endregion
    public static string Result(bool b) => b ? "Yes" : "No";
    private object Calc()
    {
        N = cr;
        a = cr.Repeat(N).Int;//Compressed(cr.Repeat(N).Int);
        graph = new GraphBuilder(N, cr, N - 1, false).ToTree(0);
        for (int i = 0; i < N; i++)
            graph[i].a = a[i];
        res = new int[N];
        ld = NewArray(N, int.MaxValue >> 1);
        Dfs(0);
        cw.WriteLines(res);
        return null;
    }

    void Dfs(int v)
    {
        var oi = ld.LowerBound(graph[v].a);
        var ov = ld[oi];
        ld[oi] = graph[v].a;
        res[v] = ld.LowerBound(int.MaxValue >> 1);
        foreach (var child in graph[v].children)
            Dfs(child);
        ld[oi] = ov;
    }

    int[] ld;
    TreeNode[] graph;
    int N;
    int[] res;
    int[] a;
    static Dictionary<T, int> Compress<T>(IEnumerable<T> orig) { var ox = new HashSet<T>(orig).ToArray(); if (typeof(IComparable<T>).IsAssignableFrom(typeof(T))) ox.Sort(); var zip = new Dictionary<T, int>(); for (int i = 0; i < ox.Length; i++) zip[ox[i]] = i; return zip; }
    static int[] Compressed<T>(T[] orig) => Compressed(orig, Compress(orig)); static int[] Compressed<T>(T[] orig, Dictionary<T, int> zip) { var res = new int[orig.Length]; for (int i = 0; i < res.Length; i++) res[i] = zip[orig[i]]; return res; }

}

class GraphBuilder { private List<int>[] roots; private List<int>[] children; public GraphBuilder(int count, bool isOriented) { this.roots = new List<int>[count]; this.children = new List<int>[count]; for (var i = 0; i < count; i++) { if (isOriented) { this.roots[i] = new List<int>(); this.children[i] = new List<int>(); } else { this.roots[i] = this.children[i] = new List<int>(); } } } public GraphBuilder(int count, ConsoleReader cr, int edgeCount, bool isOriented) : this(count, isOriented) { for (var i = 0; i < edgeCount; i++) this.Add(cr.Int0, cr.Int0); } public void Add(int from, int to) { children[from].Add(to); roots[to].Add(from); } public TreeNode[] ToTree(int root) { if (this.roots[0] != this.children[0]) throw new Exception("木には無向グラフをしたほうが良い"); var res = new TreeNode[this.children.Length]; res[root] = new TreeNode(root, -1, 0, this.children[root].ToArray()); var queue = new Queue<int>(); foreach (var child in res[root].children) { res[child] = new TreeNode(child, root, 1, Array.Empty<int>()); queue.Enqueue(child); } while (queue.Count > 0) { var from = queue.Dequeue(); if (res[from].root == -1) res[from].children = this.children[from].ToArray(); else { var children = new List<int>(this.children[from].Count); foreach (var c in this.children[from]) if (c != res[from].root) children.Add(c); res[from].children = children.ToArray(); } foreach (var child in res[from].children) { res[child] = new TreeNode(child, from, res[from].depth + 1, Array.Empty<int>()); queue.Enqueue(child); } } return res; } public Node[] ToArray() => Enumerable.Zip(roots, children, (root, child) => (root, child)).Select((t, i) => new Node(i, t.root.ToArray(), t.child.ToArray())).ToArray(); }
class TreeNode { public int a; public TreeNode(int i, int root, int depth, int[] children) { this.index = i; this.root = root; this.children = children; this.depth = depth; } public readonly int index; public readonly int root; public readonly int depth; public int[] children; public override string ToString() => $"children: {string.Join(",", children)}"; public override bool Equals(object obj) { if (obj is TreeNode) return this.Equals((TreeNode)obj); else return false; } public bool Equals(TreeNode other) => this.index == other.index; public override int GetHashCode() => this.index; }
class Node { public Node(int i, int[] roots, int[] children) { this.index = i; this.roots = roots; this.children = children; } public readonly int index; public readonly int[] roots; public readonly int[] children; public override string ToString() => $"children: {string.Join(",", children)}"; public override bool Equals(object obj) { if (obj is Node) return this.Equals((Node)obj); else return false; } public bool Equals(Node other) => this.index == other.index; public override int GetHashCode() => this.index; }

static class 二分探索
{
    public static int BinarySearch(int ok, int ng, Predicate<int> isOK)
    {
        while (Math.Abs(ok - ng) > 1)
        {
            var m = (ok + ng) / 2;
            if (isOK(m)) ok = m;
            else ng = m;
        }
        return ok;
    }



    /* https://bitbucket.org/camypaper/complib/src/master/lib/Algorithms/BinarySearch.cs */
    /**<summary>与えられた比較関数に従って，<paramref name="a"/> の要素のうち，<paramref name="v"/> 以上の要素であるような最小のインデックスを取得します．</summary><typeparam name="T"><see cref="IList{T}"/> の要素の型を指定します．</typeparam><param name="a">対象となるコレクション</param><param name="v">対象となる要素</param><param name="f"></param><returns><paramref name="v"/> 以上の要素であるような最小の o-indexed でのインデックス．</returns><remarks> <paramref name="a"/> は比較関数に対して昇順であることを仮定しています．この関数は O(log N) で実行されます．</remarks> */
    public static int LowerBound<T>(this IList<T> a, T v, IComparer<T> cmp) => BinarySearch(a, v, cmp, true);

    /**<summary> デフォルトの比較関数に従って，<paramref name="a"/> の要素のうち，<paramref name="v"/> 以上の要素であるような最小のインデックスを取得します．</summary><typeparam name="T"><see cref="IList{T}"/> の要素の型を指定します．</typeparam><param name="a">対象となるコレクション</param><param name="v">対象となる要素</param><param name="f"></param><returns><paramref name="v"/> 以上の要素であるような最小の o-indexed でのインデックス．</returns><remarks> <paramref name="a"/> は比較関数に対して昇順であることを仮定しています．この関数は O(log N) で実行されます．</remarks> */
    public static int LowerBound<T>(this IList<T> a, T v) => BinarySearch(a, v, Comparer<T>.Default, true);

    /**<summary>与えられた比較関数に従って，<paramref name="a"/> の要素のうち，<paramref name="v"/> より真に大きい要素が現れる最小のインデックスを取得します．</summary><typeparam name="T"><see cref="IList{T}"/> の要素の型を指定します．</typeparam><param name="a">対象となるコレクション</param><param name="v">対象となる要素</param><param name="f"></param><returns><paramref name="v"/> 以上の要素であるような最小の o-indexed でのインデックス．</returns><remarks> <paramref name="a"/> は比較関数に対して昇順であることを仮定しています．この関数は O(log N) で実行されます．</remarks> */
    public static int UpperBound<T>(this IList<T> a, T v, IComparer<T> cmp) => BinarySearch(a, v, cmp, false);

    /**<summary>デフォルトの比較関数に従って，<paramref name="a"/> の要素のうち，<paramref name="v"/> より真に大きい要素が現れる最小のインデックスを取得します．</summary><typeparam name="T"><see cref="IList{T}"/> の要素の型を指定します．</typeparam><param name="a">対象となるコレクション</param><param name="v">対象となる要素</param><param name="f"></param><returns><paramref name="v"/> 以上の要素であるような最小の o-indexed でのインデックス．</returns><remarks> <paramref name="a"/> は比較関数に対して昇順であることを仮定しています．この関数は O(log N) で実行されます．</remarks> */
    public static int UpperBound<T>(this IList<T> a, T v) => BinarySearch(a, v, Comparer<T>.Default, false);

    private static int BinarySearch<T>(this IList<T> a, T v, IComparer<T> cmp, bool isLowerBound)
    {
        int ok = a.Count;
        int ng = -1;
        while (ok - ng > 1)
        {
            var m = (ok + ng) / 2;
            var c = cmp.Compare(a[m], v);
            if (c > 0 || (c == 0 && isLowerBound)) ok = m;
            else ng = m;
        }
        return ok;
    }
}