JAGSummer2014Day2D Dense Amidakuji
得意
問題概要(原文)
$w(w \equiv 0 (\mathrm{mod}\; 2)))$ 本の縦棒， $h$ 段からなるのあみだくじがある．以下のように横線を引いた．
- 上から $i$ 段目，左から $j$ 番目の縦棒と $j+1$ 番目の縦棒の間に横棒を入れる．
- ただし $i \equiv j (\mathrm{mod}\; 2)$ を満たす
ここから $n$ 本だけ横棒を取り除いた．$1 \leq i \leq w$ を満たす $i$ について，左から $i$ 番目を選んだときどこに行くかを求めよ．
考察
奇数番目と偶数番目がスワップされる．
上から奇数段目では以下のようにスワップがなされる．
1 2 3 4 5 6
2 1 4 3 6 5
また，上から偶数段目では以下のようにスワップがなされる．
1 2 3 4 5 6
1 3 2 5 4 6
ここで奇数番目と偶数番目を分けると以下のように表せる
1 3 5
2 4 6
2 4 6
1 3 5
1 3 5
2 4 6
1 2 4
3 5 6
前者は $2$ つの列のスワップ，後者は先頭と末尾のみを固定して，その他はスワップ，という感じで動く．
これはdequeを用いて $O(1)$ で可能である．
横棒が取り除かれた点は単にもう $1$ 度スワップすればよい．
計算量は $O(h+w+n)$
ソースコード
using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using Number = System.Int64;
namespace Program
{
public class Solver
{
public void Solve()
{
var h = sc.Integer();
var w = sc.Integer();
var m = sc.Integer();
var a = new Deque<int>();
var b = new Deque<int>();
for (int i = 0; i < w; i += 2)
{
a.PushBack(i);
b.PushBack(i + 1);
}
var q = Enumerate(h, x => new List<int>());
for (int i = 0; i < m; i++)
{
var u = sc.Integer() - 1;
var v = sc.Integer() - 1;
q[u].Add(v);
}
for (int i = 0; i < h; i++)
{
if (i % 2 == 0)
{
Swap(ref a, ref b);
}
else
{
var l = a.PopFront();
var r = b.PopBack();
Swap(ref a, ref b);
a.PushFront(l);
b.PushBack(r);
}
foreach (var x in q[i])
{
if (x % 2 == 0)
{
var p = x / 2;
var u = a[p];
var v = b[p];
a[p] = v;
b[p] = u;
}
else
{
var p = x / 2;
var pp = x / 2 + 1;
var u = b[p];
var v = a[pp];
a[pp] = u;
b[p] = v;
}
}
}
var ret = new int[w];
for (int i = 0; i < w / 2; i++)
{
var u = a[i];
var v = b[i];
ret[u] = i * 2;
ret[v] = i * 2 + 1;
}
foreach (var x in ret)
IO.Printer.Out.WriteLine(x + 1);
}
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
- 偶奇でスワップといえばdequeみたいなところがある
- その他の解法として，階段状に下っていって折り返して，みたいな周期性を利用する解法もある
