AGC001C Shorten Diameter

本番は根を乱択+木DPで通してしまった.

問題概要(原文)

$N$ 頂点のツリーグラフからいくつか頂点を取り除いて,直径が $K$ 以下の連結グラフを作りたい.取り除く頂点の数の最小値を求めよ.

考察

最小値ではなく,残せる頂点の最大値を求めればよいことはすぐにわかる.

一般の直径 $D$ のツリーグラフにおいて,以下のような点が存在する.

  • $D$ が偶数: その他の頂点への距離が全て $D/2$ 以下
  • $D$ が奇数: ある辺の両端から,その他の頂点への距離のうち,小さいほうを取ったときに全て $(D-1)/2$ 以下

このような位置を全探索すればよい.計算量は $O(N^2)$

本番では根を $0$ とした木DP でほとんどのケースに通ったので,根を乱択して木DPを $50$ 回ぐらい試す解法で通った.

ソースコード(乱択解)

using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var k = sc.Integer();
            var G = Enumerate(n, x => new List<int>());
            for (int i = 0; i < n - 1; i++)
            {
                var a = sc.Integer() - 1;
                var b = sc.Integer() - 1;
                G[a].Add(b);
                G[b].Add(a);
            }
            Func<int, int, int[]> dfs = null;
            dfs = (prev, cur) =>
            {
                // Debug.WriteLine("{0} {1}", prev, cur);
                var S = 1;
                var ret = new int[S];
                ret[0] = 1;
                foreach (var to in G[cur])
                {
                    if (to == prev) continue;
                    var res = dfs(cur, to);
                    var T = res.Length;
                    var next = new int[S + T];

                    for (int i = 0; i < S; i++)
                    {
                        next[i] = Math.Max(next[i], ret[i]);
                        for (int j = 0; j < T; j++)
                        {
                            if (i + j + 1 > k) continue;
                            var go = Math.Max(i, j + 1);
                            next[go] = Math.Max(next[go], ret[i] + res[j]);
                        }
                    }

                    S += T;
                    ret = next;
                }
                return ret;
            };
            var rand = new Random(0);
            var val = int.MinValue;
            for (int _ = 0; _ < 50; _++)
            {
                var x = rand.Next(0, n);
                var ans = dfs(-1, x);
                for (int i = 0; i < ans.Length; i++)
                {
                    if (i <= k) val = Math.Max(val, ans[i]);
                }
            }
            IO.Printer.Out.WriteLine(n - val);
        }

        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }


    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        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 char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion

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