AIMTech Round3 D1C Centroids
複雑だが設定は簡潔
問題概要(原文)
$N$ 頂点のツリーグラフが与えられる．辺を一本削除，追加したあともツリーグラフになるようにする．
この操作で各頂点を重心にすることが可能か判定せよ．
重心とは，その頂点を取り除いたとき，各連結成分の大きさが $\frac{N}{2}$ を超えない．
考察
重心は最大でも $2$ 個しかない．どちらかを適当に選んで根付き木にする． すると考えるべき操作は
- 重心から生えている場所のどこかを切って重心にしたいところにつける
- 重心への根までのどこかを切って重心にしたいところにつける
でよい． 切るのはサイズが $\frac{N}{2}$ を超えない最大のものを切ることにするのが最適．
ソースコード
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 G = Enumerate(n, x => new List<int>());
int r = -1;
for (int i = 0; i < n - 1; i++)
{
var f = sc.Integer() - 1;
var t = sc.Integer() - 1;
G[f].Add(t); G[t].Add(f);
}
{
var par = new int[n];
var size = new int[n];
Func<int, int, int> dfs = null;
dfs = (prev, cur) =>
{
par[cur] = prev;
size[cur] = 1;
foreach (var to in G[cur])
if (to != prev) size[cur] += dfs(cur, to);
return size[cur];
};
dfs(-1, 0);
for (int i = 0; i < n; i++)
{
var max = n - size[i];
foreach (var to in G[i])
if (par[i] != to) max = Math.Max(max, size[to]);
if (max * 2 <= n)
r = i;
}
}
{
var par = new int[n];
var size = new int[n];
Func<int, int, int> dfs = null;
dfs = (prev, cur) =>
{
par[cur] = prev;
size[cur] = 1;
foreach (var to in G[cur])
if (to != prev) size[cur] += dfs(cur, to);
return size[cur];
};
dfs(-1, r);
Debug.WriteLine(size.AsJoinedString());
var ans = new int[n];
ans[r] = 1;
Action<int, int, long> efs = null;
efs = (prev, cur, max) =>
{
if (cur == r)
{
var set = new SortedSet<long>();
foreach (var to in G[cur])
{
set.Add(size[to] * 1L * n + to);
}
foreach (var to in G[cur])
{
set.Remove(size[to] * 1L * n + to);
efs(cur, to, set.Max / n);
set.Add(size[to] * 1L * n + to);
}
}
else
{
long up = n - size[cur];
if (up * 2 <= n)
{
ans[cur] = 1;
max = Math.Max(max, up);
}
else
{
up -= max;
if (up * 2 <= n)
ans[cur] = 1;
}
foreach (var to in G[cur])
if (prev != to) efs(cur, to, max);
}
};
efs(-1, r, -1);
IO.Printer.Out.WriteLine(ans.AsJoinedString());
}
}
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
- 重心系ニガテ
