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

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