ARC013D 切り分けできるかな？
今となってはABC D問題でもよさそう
問題概要(原文)
$N$ 個の立方体がある．ある立方体を $1$ 回だけ切断して $2$ 個に分割する(このとき，$1$ 辺の長さが整数となるようにしか選べず，斜めに切断してもいけない)． このようにして作れる体積が $K$ 種類だとして， それぞれを $2$ 個ずつ作るには合計何個の立方体が必要か？
考察
$2$ 部グラフの最大マッチングに帰着する．これは最大流で解ける．完
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;
using C = System.Int32;
namespace Program
{
public class Solver
{
public void Solve()
{
var n = sc.Integer();
var a = Enumerate(n, x => sc.Integer(3));
var xs = new List<int>();
foreach (var x in a)
{
var v = x[0] * x[1] * x[2];
for (int k = 0; k < 3; k++)
{
var u = v / x[k];
for (int i = 1; i < x[k]; i++)
xs.Add(u * i);
}
}
xs = xs.Distinct().ToList(); xs.Sort();
var m = xs.Count;
var G = Enumerate(2 * m + 2, x => new List<Edge>());
var src = 2 * m; var sink = 2 * m + 1;
for (int i = 0; i < m; i++)
{
G.AddDirectedEdge(src, i, 1);
G.AddDirectedEdge(m + i, sink, 1);
}
foreach (var x in a)
{
var v = x[0] * x[1] * x[2];
for (int k = 0; k < 3; k++)
{
var u = v / x[k];
for (int i = 1; i < x[k]; i++)
{
var p = u * i;
var q = v - p;
p = xs.BinarySearch(p);
q = xs.BinarySearch(q);
G.AddDirectedEdge(p, m + q, 1);
}
}
}
var f = Flow.GetMaxFlow(G, src, sink);
IO.Printer.Out.WriteLine(2 * m - f);
}
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
#region Edge
public class Edge
{
public int to, rev;
public C cap;
public Edge(int t, int r, C _cap) { to = t; rev = r; cap = _cap; }
public override string ToString() { return string.Format("{0}: Capacity {1}", to, cap); }
}
#endregion
#region AddEdge
static public partial class Flow
{
static public void AddDirectedEdge(this List<Edge>[] G, int from, int to, C cap)
{
G[from].Add(new Edge(to, G[to].Count, cap));
G[to].Add(new Edge(from, G[from].Count - 1, 0));
}
static public void AddUndirectedEdge(this List<Edge>[] G, int from, int to, int cap)
{
G[from].Add(new Edge(to, G[to].Count, cap));
G[to].Add(new Edge(from, G[from].Count - 1, cap));
}
}
#endregion
//MaxFlow
#region Dinic
static public partial class Flow
{
static public C GetMaxFlow(List<Edge>[] G, int s, int t, C INF = -1)
{
var n = G.Length;
var level = new int[n];
var iter = new int[n];
Action<int> bfs = p =>
{
Array.Clear(level, 0, n);
var q = new Queue<int>();
level[s] = 1;
q.Enqueue(s);
while (q.Count > 0)
{
var v = q.Dequeue();
foreach (var e in G[v])
if (e.cap > 0 && level[e.to] == 0)
{
level[e.to] = level[v] + 1;
q.Enqueue(e.to);
}
}
};
Func<int, int, C, C> dfs = null;
dfs = (v, u, f) =>
{
if (v == t) return f;
var ret = 0;
for (; iter[v] < G[v].Count; iter[v]++)
{
var e = G[v][iter[v]];
if (e.cap <= 0 || level[v] >= level[e.to]) continue;
var d = dfs(e.to, u, Math.Min(f, e.cap));
if (d <= 0) continue;
e.cap -= d;
G[e.to][e.rev].cap += d;
ret += d; f -= d;
if (f == 0) break;
}
return ret;
};
C flow = 0;
if (INF == -1) INF = C.MaxValue;
while (INF > 0)
{
bfs(s);
if (level[t] == 0) return flow;
Array.Clear(iter, 0, iter.Length);
C f;
while ((f = dfs(s, t, INF)) > 0) { flow += f; }
}
return flow;
}
}
#endregion
コメント
- はい
