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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