AGC016D Xor Replace

なんとか解けた

問題概要(原文)

長さ $N$ の数列 $A,B$ が与えられる. $f(A) := A_{1} \oplus A_{2} \oplus, \ldots , \oplus A_{N} $ とする.

$A_{i} = f(A)$ という操作を何回でもできる.最小何回やれば $B$ にできるか? 不可能ならば $-1$ を返せ.

考察

$A_{0} = f(A)$ とする. すると $A_{i} = f(A)$ という操作は実は $\mathrm{swap}(A_{0},A_{i})$ となっている.

つまり,$B_{0} = f(B)$ として,$A$ の要素を操作によってうまくスワップして一致させられるか?という問題になった.

並び替えなので,並び替えて一致しないならば不可能なことは明らか.そうでない場合を考える.

$A_{i} = B_{i} (i \neq 0)$ であるような箇所はあまり関係ない. また,$A_{i} \neq B_{i} (i \neq 0)$ であるような箇所は必ず操作を行わなくてはならない. 値を座圧して潰すことにする.このとき $A_{i}$ から $B_{i}$ へと辺を張った無向グラフにおいて,$A_{0}$ を含まないような連結成分では必ず無駄な操作をすることになるので,コストが $1$ かかる.

よって答えは

  • $A_{i} \neq B_{i} (i \neq 0)$ であるような点の個数
  • $A_{0}$ を含まないような連結成分の個数

の和である.

ソースコード

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
//using System.Numerics;
using Number = System.Int64;
using static System.Math;
//using static MathEx;
//using P = System.Collections.Generic.KeyValuePair<int, int>;

namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            //初めスタックに x が積まれている,xとa_i を入れ替えるという操作が何回でもできる
            //bに入れ替えるには何回操作が必要か?
            //+1手かかるかどうかがよく分からない…
            var n = ri;
            var a = Enumerate(n, x => rl).ToList();
            a.Add(f(a.ToArray()));
            var b = Enumerate(n, x => rl).ToList();
            b.Add(f(b.ToArray()));

            Debug.WriteLine(a.AsJoinedString());
            Debug.WriteLine(b.AsJoinedString());
            var map = new HashMap<long, int>();
            foreach (var x in a)
                map[x]++;
            foreach (var x in b)
                map[x]--;
            if (map.All(x => x.Value == 0))
            {
                var xs = new List<long>();
                for (int i = 0; i <= n; i++)
                    xs.Add(a[i]);
                xs = xs.Distinct().ToList(); xs.Sort();
                for (int i = 0; i <= n; i++)
                {
                    a[i] = xs.BinarySearch(a[i]);
                    b[i] = xs.BinarySearch(b[i]);
                }
                var m = xs.Count;
                var s = new DisjointSet(m);
                Debug.WriteLine(a.AsJoinedString());
                Debug.WriteLine(b.AsJoinedString());
                for (int i = 0; i <= n; i++)
                {
                    if (a[i] != b[i]) s.Unite((int)a[i], (int)b[i]);
                }


                var cnt = 0;
                for (int i = 0; i < n; i++)
                    if (a[i] != b[i]) cnt++;

                if (cnt == 0) IO.Printer.Out.WriteLine(0);
                else
                {

                    for (int i = 0; i < m; i++)
                    {
                        if (s[i] != i) continue;
                        if (s.Size(i) == 1) continue;
                        if (!s.IsUnited(i, (int)a[n])) cnt++;
                    }

                    IO.Printer.Out.WriteLine(cnt);
                }
            }
            else IO.Printer.Out.WriteLine(-1);

        }
        long f(long[] a)
        {
            var ret = 0L;
            foreach (var x in a) ret ^= x;
            return ret;
        }
        //*
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        char rc => sc.Char();

        [System.Diagnostics.Conditional("DEBUG")]
        void put(params object[] a) => Debug.WriteLine(string.Join(" ", a));

        //*/
        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 void Swap<T>(ref T a, ref T b)
        {
            var tmp = a;
            a = b;
            b = tmp;
        }
    }
}

#region main

static class Ex
{
    public static string AsString(this IEnumerable<char> ie)
    {
        return new string(ie.ToArray());
    }

    public static string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ")
    {
        return string.Join(st, ie);
    }

    public static 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(Stream stream) : base(stream, new UTF8Encoding(false, true))
        {
        }

        public Printer(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;

        public bool IsEndOfStream
        {
            get { return isEof; }
        }

        private byte read()
        {
            if (isEof) return 0;
            if (ptr < len) return buf[ptr++];
            ptr = 0;
            if ((len = str.Read(buf, 0, 1024)) > 0) return buf[ptr++];
            isEof = true;
            return 0;
        }

        public char Char()
        {
            byte b;
            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;
            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 (; ; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                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;
        }

        static 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 HashMap<K,V>
public class HashMap<K, V>: Dictionary<K, V>
where V : new()
{
    public HashMap() : base() { }
    public HashMap(int cap) : base(cap) { }
    new public V this[K i]
    {
        get
        {
            V v;
            return TryGetValue(i, out v) ? v :
  //                base[i] = default(V);
  base[i] = new V();
        }
        set { base[i] = value; }
    }
}
#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

  • 後半パートが難しいが,かなり面白い