K4PC E はじめての動的計画法(Easy Dynamic Programming)
問題文が…
問題概要(原文)
$N(1 \leq N \leq 1{,}000)$ 個のものがあり， $i$ 番のものの重さは $A_i (1 \leq A_i \leq 1{,}000)$ である．$2^{N}$ 通りのものの選び方があるうち， ものの重さの総和が $W(1 \leq W \leq 1{,}000)$ 以下であるという条件を満たす選び方は何通りか？
という問題を考える．答えが $X(1 \leq X \leq 10^{18})$ となる入力をどれか $1$ つ作成せよ．
考察
答えが $1$ となるものは， $N=1, A_{1} = 10, W=1$ などとすればよい．その他の場合を考えよう．
$X$ を $2$ 進表記にしたとき，最大の桁を $K$ とする． $K$ 個 $1$ を用意しよう．すると $K$ 以下で $2^{K}$ 通りを作ることができる．
$Y = X - 2^{K}$ としよう．$Y$ 通り追加するために要素を追加っしていくことを考える． $m + x = W (m \leq K)$ となるような $x$ を $1$ 個追加したとき， $\binom{K}{m}$ 通り答えが増える．これを $m = K$ から降順に見ていって可能な限り追加していけばよい．
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 X = sc.Long();
if (X == 1)
{
IO.Printer.Out.WriteLine("1 1");
IO.Printer.Out.WriteLine("1000");
return;
}
var one = 0;
for (int i = 0; i < 63; i++)
if ((X >> i & 1) == 1) one = i;
X -= 1L << one;
var A = Enumerable.Repeat(1, one).ToList();
var dp = new long[1001];
{
dp[0] = 1;
for (int _ = 0; _ < one; _++)
for (int i = 100; i >= 0; i--)
dp[i + 1] += dp[i];
for (int i = 1; i <= 1000; i++)
dp[i] += dp[i - 1];
}
for (int i = 59; i >= 0; i--)
{
var j = 1000 - i;
var k = X / dp[i];
X %= dp[i];
for (int t = 0; t < k; t++)
A.Add(j);
}
IO.Printer.Out.WriteLine("{0} 1000", A.Count);
foreach (var x in A)
IO.Printer.Out.WriteLine(x);
check(A, 1000);
}
void check(List<int> A, int W)
{
var dp = new long[1001];
dp[0] = 1;
foreach (var x in A)
{
for (int i = 1000; i >= x; i--)
dp[i] += dp[i - x];
}
for (int i = 1; i <= W; i++)
dp[i] += dp[i - 1];
Debug.WriteLine(dp[W]);
}
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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