AGC007D Shik and Game
はい
問題概要(原文)
以下のようなゲームをクリアする最短時間を求めよ
- $N$ 個の鍵のついた箱があり，左から $i$ 番目の箱は座標 $x_i$ にある
- 箱は触ってから $t$ 秒後に開き，中に $1$ 枚のコインがある
- 座標 $0$ から出発し， $N$ 枚のコインを持った状態で座標 $E$ に着くとクリア
- $0 < x_{1} < ,\ldots, < x_{N} < E$ を満たす
考察
とりあえず座圧したあとの世界で考える．また，$E$ にも箱があることにする．
戦略として，ある程度右に進んだ後まだ取ってないコインのうち一番左端にあるコインを取る，というのが最適であるのがすぐに分かる．
$dp[i]$ を $i-1$ 番目以前の箱のコインを全て取って，$i$ 番の箱に触ったときの最短時間，とする． 遷移としては $i \leq j$ を満たす $j$ 番の箱まで触って，$i, i+1, \ldots, j$ 番の箱のコインを回収してから $j+1$ 番の箱に触る，ということになる．
このとき，どれだけの時間がかかるかを考えると $\max(3(x_{j}-x_{i}),t+x_{j}-x_{i}) + x_{j+1}-x_{j}$ だけ時間がかかる． これはどういうことかと言うと，「行って，戻って，行って」か「行って，戻って，$i$ 番の箱の位置で待って，行って」という形の $2$ 通りしかない，ということである．$2$ 点間の距離によってのみ遷移の種類が変化することが見て取れる．
さて，今の形だと遷移が $O(N)$ なのでこれを減らしたい．もらう形に変形して考えていくとデータ構造で高速化できることがあって扱いやすい． 結局以下のような漸化式で書ける
\[ dp[i] = x_{i} - x_{i-1} + \min \begin{cases} 3(x_{i-1}-x_{j}) +dp[j] &(2(x_{i-1}-x_{j}) \geq t)\\ t + x_{i-1}-x_{j} + dp[j] &(otherwise) \end{cases} \]
あとはセグメント木なり，デックなりを使って解ける．
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 xs = new long[n + 1];
long t;
{
var e = sc.Long();
t = sc.Long();
for (int i = 0; i < n; i++)
xs[i] = sc.Long();
xs[n] = e;
}
var dp = Enumerate(n + 1, x => 1L << 60);
dp[0] = xs[0];
var min = 1L << 60;
var deq = new Deque<KeyValuePair<int, long>>();
deq.PushBack(new KeyValuePair<int, long>(0, dp[0]));
for (int i = 1; i <= n; i++)
{
while (deq.Count > 0)
{
var p = deq[0].Key;
if (2 * (xs[i - 1] - xs[p]) >= t)
min = Math.Min(min, 3 * (xs[i - 1] - xs[p]) + deq.PopFront().Value);
else break;
}
dp[i] = Math.Min(dp[i], min);
if (deq.Count > 0)
dp[i] = Math.Min(dp[i], deq[0].Value + t + xs[i - 1] - xs[deq[0].Key]);
/*
for (int j = i + 1; j <= n; j++)
{
var dx = xs[j] - xs[j - 1];
dp[j] = Math.Min(dp[j], Math.Max(3 * d, t + d) + dp[i] + dx);
d += dx;
}
Debug.WriteLine(dp.Items.AsJoinedString());
*/
var dx = xs[i] - xs[i - 1];
min += 3 * dx;
dp[i] += dx;
//
while (deq.Count > 0)
{
var p = deq[deq.Count - 1];
if (p.Value + xs[i] - xs[p.Key] > dp[i]) deq.PopBack();
else break;
}
deq.PushBack(new KeyValuePair<int, Number>(i, dp[i]));
}
IO.Printer.Out.WriteLine(dp[n]);
}
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 Deque<T>
public class Deque<T>
{
T[] buf;
int offset;
int count;
int capacity;
public int Count { get { return count; } }
public Deque(int cap) { buf = new T[capacity = cap]; }
public Deque() { buf = new T[capacity = 8]; }
public T this[int index]
{
get { return buf[getIndex(index)]; }
set { buf[getIndex(index)] = value; }
}
private int getIndex(int index)
{
if (index >= capacity)
throw new IndexOutOfRangeException("out of range");
var ret = index + offset;
if (ret >= capacity)
ret -= capacity;
return ret;
}
public void PushFront(T item)
{
if (count == capacity) Extend();
if (--offset < 0) offset += buf.Length;
buf[offset] = item;
++count;
}
public T PopFront()
{
if (count == 0)
throw new InvalidOperationException("collection is empty");
--count;
var ret = buf[offset++];
if (offset >= capacity) offset -= capacity;
return ret;
}
public void PushBack(T item)
{
if (count == capacity) Extend();
var id = count++ + offset;
if (id >= capacity) id -= capacity;
buf[id] = item;
}
public T PopBack()
{
if (count == 0)
throw new InvalidOperationException("collection is empty");
return buf[getIndex(--count)];
}
public void Insert(T item, int index)
{
if (index > count) throw new IndexOutOfRangeException();
this.PushFront(item);
for (int i = 0; i < index; i++)
this[i] = this[i + 1];
this[index] = item;
}
public T RemoveAt(int index)
{
if (index < 0 || index >= count) throw new IndexOutOfRangeException();
var ret = this[index];
for (int i = index; i > 0; i--)
this[i] = this[i - 1];
this.PopFront();
return ret;
}
private void Extend()
{
T[] newBuffer = new T[capacity << 1];
if (offset > capacity - count)
{
var len = buf.Length - offset;
Array.Copy(buf, offset, newBuffer, 0, len);
Array.Copy(buf, 0, newBuffer, len, count - len);
}
else Array.Copy(buf, offset, newBuffer, 0, count);
buf = newBuffer;
offset = 0;
capacity <<= 1;
}
public T[] Items
{
get
{
var a = new T[count];
for (int i = 0; i < count; i++)
a[i] = this[i];
return a;
}
}
}
#endregion
コメント
- スライド最小値っぽくやる必要はなかったかも
