8VC2017 Elim F PolandBall and Gifts
典型ではある
問題概要(原文)
長さ $N$ の完全順列 $P$ が与えられる．$1,2,3, \ldots, N$ たちは白く塗られていることにする． $K$ 回以下の操作をする．
- 過去に選んでいないような整数 $i$ を選ぶ．
- $i$ と $P_j = i$ なる $j$ を黒く塗る．
黒く塗られた数の最小値と最大値を求めよ．
考察
完全順列と言っているが，要するに $2$ 個以上の頂点からなるサイクルがたくさんある，と考えてよい．
最大値から考える．基本的には $2$ つ取り除ける．余った回数 $1$ ずつ取る．
最小値から考える．基本的にはサイクルを全部塗る，というのを繰り返す． これで $K$ 個取れたら答えは $K$ ，そうでなければ $K+1$． $K$ 個選んだうち，少なくとも $1$ つは無駄な操作をしないといけないと考えればよい．
$N \leq 10^{6}$ ととても大きい．しかし，サイクルの長さの種類数は $O(\sqrt{N})$ 程度しかない．個数制限ナップサックを bitset で加速すれば $\frac{N \sqrt{N}}{64}$ ステップぐらいになって間に合う．
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 n = sc.Integer();
var k = sc.Integer();
var s = new DisjointSet(n);
for (int i = 0; i < n; i++)
s.Unite(i, sc.Integer() - 1);
var a = new int[n + 1];
for (int i = 0; i < n; i++)
if (s[i] == i) a[s.Size(i)]++;
var max = solve(n, k, a);
Debug.WriteLine(max);
var min = solve2(n, k, a);
Debug.WriteLine(min);
IO.Printer.Out.WriteLine("{0} {1}", min, max);
}
int solve2(int n, int k, int[] a)
{
var bitset = new BitSet(n);
bitset[0] = true;
for (int i = 0; i <= n; i++)
{
if (a[i] == 0) continue;
var x = a[i];
a[i] = 0;
for (int j = 1; j <= x; j *= 2)
{
a[j * i]++;
x -= j;
}
a[x * i]++;
bitset.OrTo(bitset, 0, i, n - i + 1);
if (a[i] == 2) { bitset.OrTo(bitset, 0, i, n - i); }
}
Debug.WriteLine(bitset.Items.AsJoinedString());
if (bitset[k]) return k;
return k + 1;
}
int solve(int n, int k, int[] a)
{
var sum = 0;
for (int i = 2; i <= n; i += 2)
{
var t = a[i] * i;
while (k > 0 && t >= 2)
{
t -= 2;
sum += 2;
k--;
}
}
var rem = 0;
for (int i = n; i >= 0; i--)
{
if (i % 2 == 0) continue;
var t = a[i] * (i - 1);
rem += a[i];
while (k > 0 && t >= 2)
{
t -= 2;
sum += 2;
k--;
}
}
sum += Math.Min(k, rem);
return sum;
}
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 BitArray
public struct BitSet
{
const int SIZE = 1000000;
const int size = SIZE / 64 + 50;
static public readonly ulong[] mask = new ulong[65];
static BitSet()
{
mask[0] = 0;
for (int i = 0; i < 64; i++)
mask[i + 1] = (mask[i] << 1) | 1ul;
}
ulong[] bits;
int n;
public BitSet(int N) : this()
{
n = N;
bits = new ulong[size];
}
public int[] Items
{
get
{
var ret = new int[n];
for (int i = 0; i < n; i++)
if (this[i]) ret[i] = 1;
return ret;
}
}
public bool this[int index]
{
get { return (bits[index / 64] >> (index % 64) & 1) == 1; }
set
{
if (value)
bits[index / 64] |= 1ul << (index % 64);
else bits[index / 64] &= ~(1ul << (index % 64));
}
}
void align(ref int p, ref int q, ref int len)
{
var xr = Math.Min(SIZE, p + len);
len = xr - p;
if (q < 0)
{
p += Math.Abs(q);
len -= Math.Abs(q);
q = 0;
}
var yr = Math.Min(SIZE, q + len);
len = yr - q;
}
/// <summary>
/// a->b. a[p,p+len)->b[q,q+len)
/// </summary>
public void XorTo(BitSet b, int p, int q, int len)//xor b[l,l+len)^=a[r,r+len)
{
align(ref p, ref q, ref len);
var lx = p / 64;
var ly = p % 64;
var rx = (p + len) / 64;
var ry = (p + len) % 64;
var d = p - q;
var dx = d / 64;
var dy = d % 64;
if (dy < 0) { dy += 64; dx--; }
if (p > q)
{
if (lx == rx)
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] ^= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
b.bits[lx - dx] ^= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
}
else
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] ^= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
b.bits[lx - dx] ^= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
for (var i = lx + 1; i < rx; i++)
{
if (dy > 0) b.bits[i - dx - 1] ^= bits[i] << (64 - dy);
b.bits[i - dx] ^= bits[i] >> dy;
}
if (dy > 0) b.bits[rx - dx - 1] ^= (bits[rx] & (mask[ry])) << (64 - dy);
b.bits[rx - dx] ^= (bits[rx] & (mask[ry])) >> dy;
}
}
else
{
if (lx == rx)
{
b.bits[lx - dx] ^= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
if (dy > 0) b.bits[lx - dx - 1] ^= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
}
else
{
b.bits[rx - dx] ^= (bits[rx] & (mask[ry])) >> dy;
if (dy > 0) b.bits[rx - dx - 1] ^= (bits[rx] & (mask[ry])) << (64 - dy);
for (var i = rx - 1; i > lx; i--)
{
b.bits[i - dx] ^= bits[i] >> dy;
if (dy > 0) b.bits[i - dx - 1] ^= bits[i] << (64 - dy);
}
b.bits[lx - dx] ^= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] ^= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
}
}
}
/// <summary>
/// a->b. a[p,p+len)->b[q,q+len)
/// </summary>
public void AndTo(BitSet b, int p, int q, int len)//and b[l,l+len)&=a[r,r+len)
{
align(ref p, ref q, ref len);
var lx = p / 64;
var ly = p % 64;
var rx = (p + len) / 64;
var ry = (p + len) % 64;
var d = p - q;
var dx = d / 64;
var dy = d % 64;
if (dy < 0) { dy += 64; dx--; }
if (p > q)
{
if (lx == rx)
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] &= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
b.bits[lx - dx] &= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
}
else
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] &= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
b.bits[lx - dx] &= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
for (var i = lx + 1; i < rx; i++)
{
if (dy > 0) b.bits[i - dx - 1] &= bits[i] << (64 - dy);
b.bits[i - dx] &= bits[i] >> dy;
}
if (dy > 0) b.bits[rx - dx - 1] &= (bits[rx] & (mask[ry])) << (64 - dy);
b.bits[rx - dx] &= (bits[rx] & (mask[ry])) >> dy;
}
}
else
{
if (lx == rx)
{
b.bits[lx - dx] &= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
if (dy > 0) b.bits[lx - dx - 1] &= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
}
else
{
b.bits[rx - dx] &= (bits[rx] & (mask[ry])) >> dy;
if (dy > 0) b.bits[rx - dx - 1] &= (bits[rx] & (mask[ry])) << (64 - dy);
for (var i = rx - 1; i > lx; i--)
{
b.bits[i - dx] &= bits[i] >> dy;
if (dy > 0) b.bits[i - dx - 1] ^= bits[i] << (64 - dy);
}
b.bits[lx - dx] &= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] &= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
}
}
}
/// <summary>
/// a->b. a[p,p+len)->b[q,q+len)
/// </summary>
public void OrTo(BitSet b, int p, int q, int len)//and b[l,l+len)&=a[r,r+len)
{
align(ref p, ref q, ref len);
var lx = p / 64;
var ly = p % 64;
var rx = (p + len) / 64;
var ry = (p + len) % 64;
var d = p - q;
var dx = d / 64;
var dy = d % 64;
if (dy < 0) { dy += 64; dx--; }
if (p > q)
{
if (lx == rx)
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] |= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
b.bits[lx - dx] |= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
}
else
{
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] |= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
b.bits[lx - dx] |= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
for (var i = lx + 1; i < rx; i++)
{
if (dy > 0) b.bits[i - dx - 1] |= bits[i] << (64 - dy);
b.bits[i - dx] |= bits[i] >> dy;
}
if (dy > 0) b.bits[rx - dx - 1] |= (bits[rx] & (mask[ry])) << (64 - dy);
b.bits[rx - dx] |= (bits[rx] & (mask[ry])) >> dy;
}
}
else
{
if (lx == rx)
{
b.bits[lx - dx] |= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) >> dy;
if (dy > 0) b.bits[lx - dx - 1] |= (bits[lx] & (mask[ry] & (mask[64] ^ mask[ly]))) << (64 - dy);
}
else
{
b.bits[rx - dx] |= (bits[rx] & (mask[ry])) >> dy;
if (dy > 0) b.bits[rx - dx - 1] |= (bits[rx] & (mask[ry])) << (64 - dy);
for (var i = rx - 1; i > lx; i--)
{
b.bits[i - dx] |= bits[i] >> dy;
if (dy > 0) b.bits[i - dx - 1] |= bits[i] << (64 - dy);
}
b.bits[lx - dx] |= (bits[lx] & (mask[64] ^ mask[ly])) >> dy;
if (dy > 0 && lx - dx - 1 >= 0) b.bits[lx - dx - 1] |= (bits[lx] & (mask[64] ^ mask[ly])) << (64 - dy);
}
}
}
}
#endregion
#region DisjointSet
public class DisjointSet
{
int[] par;
byte[] rank;
public DisjointSet(int n)
{
par = new int[n];
for (int i = 0; i < n; i++)
par[i] = -1;
rank = new byte[n];
}
public int this[int id]
{
get
{
if ((par[id] < 0)) return id;
return par[id] = this[par[id]];
}
}
public bool Unite(int x, int y)
{
x = this[x]; y = this[y];
if (x == y) return false;
if (rank[x] < rank[y])
{
par[y] += par[x];
par[x] = y;
}
else
{
par[x] += par[y];
par[y] = x;
if (rank[x] == rank[y])
rank[x]++;
}
return true;
}
public int Size(int x) { return -par[this[x]]; }
public bool IsUnited(int x, int y) { return this[x] == this[y]; }
}
#endregion
コメント
- ちょっとおもしろいが最小値だけで良い気もする
