yukicoder No.409 ダイエット
体重くんこわれる
問題概要(原文)
$N$ 日間のうち，毎日ドーナツを食べるか食べないか決める．
- $i$ 日目に食べた: $D_i$ だけ体重が増える
- $i$ 日目に食べない: $A$ だけ体重が減ったあと， $B \times t$ だけ体重が増える ($t$ は最後にドーナツを食べた日からの日数)．
体重を最小化せよ．
考察
$dp(i) = i$ 日目にドーナツを食べたときの体重の最小値とする．
\[ \begin{aligned} dp(i+1) & = D_i + min \{ dp(j) - (i-j) \times A + (i-j) \times (i-j+1) / 2 \times B \} \\ dp(i+1) & = D_i - iA + min \{ dp(j) + jA + (i^2 + (1-2j)i + j^2 -j) / 2 \times B \} \\ dp(i+1) & = D_i -iA + \frac{(i^2+i)B}{2} + min \left \{ dp(j) + Aj -Bji + \frac{(j^2-j)B}{2} \right \} \end{aligned} \]
このように式変形をすると， $min$ の内側の式は $i$ に関する $1$ 次式 $\alpha i + \beta$ の形になっている． このような形のDPは convex hull trickと呼ばれるテクニックが使えて $O(1)$ とか $O(logN)$ で遷移が実は試せる．
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 A = sc.Integer();
var B = sc.Integer();
var dp = new long[n + 1];
dp[0] = sc.Integer();
var f = new Optimization.SimpleCHT();
Action<long, long> addline = (j, val) =>
{
f.AddLine(-j * B, A * j + (j * j - j) * B / 2 + val);
};
for (long i = 0; i < n; i++)
{
addline(i, dp[i]);
var min = f.Get((int)(i));
var v = sc.Integer();
dp[i + 1] = -A * i + B * (i * i + i) / 2 + v + min;
}
Debug.WriteLine(dp.AsJoinedString());
var ans = long.MaxValue;
for (int i = 0; i <= n; i++)
ans = Math.Min(ans, dp[i] - 1L * A * (n - i) + 1L * B * Math.BigMul(n - i, n - i + 1) / 2);
IO.Printer.Out.WriteLine(ans);
}
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 CHT
namespace Optimization
{
using N = System.Int64;
struct L
{
internal N a, b;
/// <summary>
/// u*x+v
/// </summary>
public L(N u, N v) : this() { a = u; b = v; }
/// <summary>
/// F(x)=a*x+b
/// </summary>
public N F(int x) { return a * x + b; }
}
/// <summary>
/// 追加される直線の傾きが単調減少
/// </summary>
public class SimpleCHT
{
int s, t;
List<L> deq = new List<L>();
/// <summary>
/// ax+b を追加
/// aは単調減少でないとダメ,ならしO(1)
/// </summary>
public void AddLine(N a, N b)
{
var l = new L(a, b);
while (s + 1 < t && check_slow(deq[t - 2], deq[t - 1], l)) { t--; deq.RemoveAt(t); }
deq.Add(l);
t++;
}
/// <summary>
/// xが単調増加のときのみ使ってよい,ならしO(1)
/// </summary>
public N Get(int x)
{
while (s + 1 < t && deq[s].F(x) >= deq[s + 1].F(x)) s++;
return deq[s].F(x);
}
/// <summary>
/// xが単調増加でないときでも使える,O(logN)
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public N Query(int x)
{
var l = s - 1; var h = t - 1;
while (l + 1 < h)
{
var m = (l + h) / 2;
if (isR_fast(deq[m], deq[m + 1], x)) l = m;
else h = m;
}
return deq[h].F(x);
}
/// <summary>
/// ax,bが小さい
/// </summary>
private bool isR_fast(L l1, L l2, int x)
{
return (l1.b - l2.b) >= x * (l2.a - l1.a);
}
/// <summary>
/// ax,bが大きい
/// </summary>
private bool isR_slow(L l1, L l2, int x)
{
BigInteger b1 = l1.b, b2 = l2.b;
BigInteger a1 = l1.a, a2 = l2.a;
return (b1 - b2) >= x * (a2 - a1);
}
/// <summary>
/// a*bが小さい
/// </summary>
private bool check_fast(L l1, L l2, L l3)
{
return (l2.a - l1.a) * (l3.b - l2.b) >= (l2.b - l1.b) * (l3.a - l2.a);
}
/// <summary>
/// a*bが大きい
/// </summary>
private bool check_slow(L l1, L l2, L l3)
{
BigInteger a1 = l1.a, a2 = l2.a, a3 = l3.a;
BigInteger b1 = l1.b, b2 = l2.b, b3 = l3.b;
return (a2 - a1) * (b3 - b2) >= (b2 - b1) * (a3 - a2);
}
}
}
#endregion
コメント
- 式変形がややこしい系は wolfram alphaに投げるとよい
