CF395D1C Timofey and remoduling
苦手なタイプだが面白い
問題概要(原文)
素数 $M$ と，長さ $N$ の数列 $A$ が与えられる．$A$ は互いに異なり，どれも $M$ 未満である．
$A$ を並び替えたときに，$\text{MOD}\; M$ で等差数列となるようにすることは可能か？可能なら一例を示せ．
考察
重複がないので，周期的に何度も訪れる，ということはない．
等差数列の特徴に着目しよう． 初項を $a,$ 公差を $d$ とする．
$N$ 項までの総和 $x$ は $\sum_^{N-1} a + id = Na+ \frac{N(N-1)d}{2}$ となる． $N$ 項までの二乗和 $y$ は $\sum_{N-1}(a+id){2} = N a^{2} + ad N(N-1) + \frac{d^{2}N(N-1)(2N-1)}{6}$ となる．
$x,y$ のいずれも $A$ から求めることができる． $1 \ldots N$ のどれが初項になるかを全探索し，$x,y$ の両方の条件を満たすか確認しよう．前者は複数ありうるが，後者の条件も考えると定数個しか条件を満たすものはない(ただし，$d=0,1$ の場合に注意すること)．
計算量は $O(N \log N)$ 程度か．
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 m = sc.Long();
var n = sc.Integer();
if (m == n)
{
IO.Printer.Out.WriteLine("0 1");
return;
}
var a = new ModInteger[n];
for (int i = 0; i < n; i++)
a[i].num = sc.Long();
if (n == 1)
{
IO.Printer.Out.WriteLine("{0} {1}", a[0], 0);
return;
}
solve(m, n, a);
}
void solve(long m, int n, ModInteger[] a)
{
ModInteger.Mod = m;
ModInteger sum = 0;
ModInteger sum2 = 0;
for (int i = 0; i < n; i++)
{
sum += 2 * a[i];
sum2 += a[i] * a[i];
}
sum *= ModInteger.Pow(n, ModInteger.Mod - 2);
for (int i = 0; i < n; i++)
{
var u = sum - 2 * a[i];
u *= ModInteger.Pow(n - 1, ModInteger.Mod - 2);
if (u.num == 0) continue;
var x = a[i];
var y = u;
ModInteger s = x * x * n
+ x * y * n * (n - 1)
+ y * y * (n - 1L) * n * (2 * n - 1L) * ModInteger.Pow(6, ModInteger.Mod - 2);
if (sum2.num != s.num) continue;
var ok = true;
foreach (var z in a)
{
if (((z - x) * ModInteger.Pow(y, ModInteger.Mod - 2)).num >= n) { ok = false; break; }
}
if (ok)
{
IO.Printer.Out.WriteLine("{0} {1}", x, y);
return;
}
}
IO.Printer.Out.WriteLine(-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 ModNumber
public partial struct ModInteger
{
public static long Mod = (long)1e9 + 7;
public long num;
public ModInteger(long n) : this() { num = n % Mod; if (num < 0) num += Mod; }
public override string ToString() { return num.ToString(); }
public static ModInteger operator +(ModInteger l, ModInteger r) { var n = l.num + r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
public static ModInteger operator -(ModInteger l, ModInteger r) { var n = l.num + Mod - r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
public static ModInteger operator *(ModInteger l, ModInteger r) { return new ModInteger(l.num * r.num); }
public static implicit operator ModInteger(long n) { return new ModInteger(n); }
public static ModInteger Pow(ModInteger v, long k)
{
ModInteger ret = 1;
var n = k;
for (; n > 0; n >>= 1, v *= v)
{
if ((n & 1) == 1)
ret = ret * v;
}
return ret;
}
}
#endregion
コメント
- 数論は苦手
