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
コメント
- graph-theory系の問題が苦手
