La regione più piccola dell'aereo che contiene tutti gli n-omino liberi


27

Su Math Stack Exchange, ho posto una domanda sulla regione più piccola che può contenere tutti i n-ominos gratuiti .

Vorrei aggiungere questa sequenza all'Enciclopedia on-line di sequenze intere una volta che avrò più termini.

Esempio

Una regione a nove celle è il sottoinsieme più piccolo del piano che può contenere tutti i dodici omino 5 liberi , come illustrato di seguito. (Un poliomino libero è uno che può essere ruotato e capovolto.)

Regione a nove celle del piano.

(Una regione di dodici celle è il sottoinsieme più piccolo del piano che può contenere tutti i 35 omino liberi .)


La sfida

Calcola i limiti superiori sulle regioni più piccole del piano che possono contenere tutti gli n-omino in funzione di n.

Tale tabella inizia:

n | size
--+-------
1 | 1*
2 | 2*
3 | 4*
4 | 6*
5 | 9*
6 | 12*
7 | 37
8 | 50
9 | 65

*These values are the smallest possible.

Esempio di presentazione

1-omino: 1
#

2-omino: 2
##

3-omino: 4
###
 #

4-omino: 6
####
 ##

5-omino: 9
 #
#####
 ###

6-omino: 12
####
######
  ##

7-omino: <= 37
#######
######
######
######
######
######

punteggio

Esegui il programma per tutto il tempo che desideri e pubblica l'elenco dei limiti superiori insieme alle forme che raggiungono ciascuno.

Il vincitore sarà il partecipante il cui tavolo è lessicograficamente il primo (con "infinito" aggiunto alle proposte più brevi). Se due partecipanti presentano gli stessi risultati, vince la presentazione precedente.

Ad esempio, se le comunicazioni sono

Aliyah: [1,2,4,6,12,30,50]
Brian:  [1,2,4,6,12,37,49,65]
Clare:  [1,2,4,6,12,30]

poi vince Aliyah. Batte Brian perché 30 <37 e batte Clare perché 50 <infinito.


2
Sfida interessante, ci lavorerò più avanti oggi!
orlp,

1
Polyominos con o senza buchi?
Peter Taylor,

I programmi devono essere eseguiti in modo deterministico?
Nathaniel,

1
@PeterTaylor, poliomino con buchi, ma la mia ingenua aspettativa è che non cambierebbe il risultato.
Peter Kagey,

1
Tecnicamente, penso che tu sia interessato solo ai posizionamenti allineati a reticolo dei poliomino, giusto? Non puoi ruotare il pezzo di 23.42132 gradi, giusto?
isaacg,

Risposte:


7

C # e SAT: 1, 2, 4, 6, 9, 12, 17, 20, 26, 31, 37, 43

Se limitiamo il riquadro di delimitazione, esiste un'espressione abbastanza ovvia del problema in termini di SAT : ogni traduzione di ciascun orientamento di ciascun poliomino libero è una grande congiunzione; per ogni poliomino formiamo una disgiunzione sulle sue congiunzioni; e quindi richiediamo che ogni disgiunzione sia vera e che il numero totale di celle fosse limitato.

Per limitare il numero di celle la mia versione iniziale ha creato un sommatore completo; poi ho usato l'ordinamento bitonico per il conteggio unario (simile a questa risposta precedente ma generalizzato); alla fine ho optato per l'approccio descritto da Bailleux e Boufkhad nella codifica CNF efficiente dei vincoli di cardinalità booleana .

Volevo rendere il post autonomo, quindi ho estratto un'implementazione C # di un solutore SAT con una licenza BSD che era all'avanguardia circa 15 anni fa, sostituendo l'implementazione dell'elenco NIH con System.Collections.Generic.List<T>(guadagnando un fattore di 2 in velocità), lo ha portato da 50 KB a 31 KB per rientrare nel limite di 64 KB, quindi ha fatto un lavoro aggressivo sulla riduzione dell'utilizzo della memoria. Questo codice può ovviamente essere adattato per generare un file DIMACS che può quindi essere passato a solutori più moderni.

Soluzioni trovate

#

##

###
..#

####
.##.

..#..
#####
..###

.####.
######
.##...

....#..
#######
#####..
.####..

########
..######
.....###
.....###

#########
#######..
..#####..
....##...
....###..

##########
########..
..######..
....####..
.....###..

..#######..
..#########
###########
..####.....
..####.....
..##.......

...#######..
...#########
############
..#####....#
..#####.....
...####.....

Per trovare 43 per n = 12 ci sono voluti un po 'più di 7,5 ore.

Codice Polyomino

using MiniSAT;
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;

namespace PPCG167484
{
    internal class SatGenerator
    {
        public static void Main()
        {
            for (int n = 1; n < 13; n++)
            {
                int width = n;
                int height = (n + 1) >> 1;
                var polys = FreePolyomino.All(n);
                (var solver, var unaryWeights) = Generate(polys, width, height);

                int previous = width * height + 1;
                while (true)
                {
                    Stopwatch sw = new Stopwatch(); sw.Start();
                    if (solver.Solve())
                    {
                        // The weight of the solution might be smaller than the target
                        int weight = Enumerable.Range(0, width * height).Count(x => solver.Model[x] == Solver.l_True);

                        Console.Write($"{n}\t<={weight}\t{sw.Elapsed.TotalSeconds:F3}s\t");
                        int cell = 0;
                        for (int y = 0; y < height; y++)
                        {
                            if (y > 0) Console.Write('_');
                            for (int x = 0; x < width; x++) Console.Write(solver.Model[cell++] == Solver.l_True ? '#' : '.');
                        }
                        Console.WriteLine();

                        // Now knock out that weight
                        for (int i = weight - 1; i < previous - 1; i++) solver.AddClause(~unaryWeights[i]);
                        previous = weight;
                    }
                    else
                    {
                        Console.WriteLine("--------");
                        break;
                    }
                }
            }
        }

        public static Tuple<Solver, Solver.Lit[]> Generate(IEnumerable<FreePolyomino> polys, int width, int height)
        {
            var solver = new Solver();

            if (width == 12) solver.Prealloc(6037071 + 448, 72507588 + 6008); // HACK!

            // Variables: 0 to width * height - 1 are the cells available to fill.
            for (int i = 0; i < width * height; i++) solver.NewVar();

            foreach (var poly in polys)
            {
                // We naturally get a DNF: each position of each orientation is a conjunction of poly.Weight variables,
                // and we require any one. Therefore we add an auxiliary variable per.

                var polyAuxs = new List<Solver.Lit>();
                foreach (var orientation in poly.OrientedPolyominos)
                {
                    int maxh = height;
                    // Optimisation: break symmetry
                    if (orientation.BBHeight == 1) maxh = ((height + 1) >> 1);

                    for (int dy = 0; dy + orientation.BBHeight <= maxh; dy++)
                    {
                        for (int dx = 0; dx + orientation.BBWidth <= width; dx++)
                        {
                            var currentAux = solver.NewVar();
                            for (int y = 0; y < orientation.BBHeight; y++)
                            {
                                uint tmp = orientation.Rows[y];
                                for (int x = 0; tmp > 0; x++, tmp >>= 1)
                                {
                                    if ((tmp & 1) == 1) solver.AddClause(~currentAux, new Solver.Lit((y + dy) * width + x + dx));
                                }
                            }

                            polyAuxs.Add(currentAux);
                        }
                    }
                }
                solver.AddClause(polyAuxs.ToArray());
            }

            // Efficient CNF encoding of Boolean cardinality constraints, Bailleux and Boufkhad, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.458.7676&rep=rep1&type=pdf
            var unaryWeights = _BBSum(0, width * height, solver);
            return Tuple.Create(solver, unaryWeights);
        }

        private static Solver.Lit[] _BBSum(int from, int num, Solver solver)
        {
            var sum = new Solver.Lit[num];
            if (num == 1) sum[0] = new Solver.Lit(from);
            else
            {
                var left = _BBSum(from, num >> 1, solver);
                var right = _BBSum(from + left.Length, num - left.Length, solver);

                for (int i = 0; i < num; i++) sum[i] = solver.NewVar();
                for (int alpha = 0; alpha <= left.Length; alpha++)
                {
                    for (int beta = 0; beta <= right.Length; beta++)
                    {
                        var sigma = alpha + beta;
                        // C_1 = ~left[alpha-1] + ~right[beta-1] + sum[sigma-1]
                        if (alpha > 0 && beta > 0) solver.AddClause(~left[alpha - 1], ~right[beta - 1], sum[sigma - 1]);
                        else if (alpha > 0) solver.AddClause(~left[alpha - 1], sum[sigma - 1]);
                        else if (beta > 0) solver.AddClause(~right[beta - 1], sum[sigma - 1]);

                        // C_2 = left[alpha] + right[beta] + ~sum[sigma]
                        if (alpha < left.Length && beta < right.Length) solver.AddClause(left[alpha], right[beta], ~sum[sigma]);
                        else if (alpha < left.Length) solver.AddClause(left[alpha], ~sum[sigma]);
                        else if (beta < right.Length) solver.AddClause(right[beta], ~sum[sigma]);
                    }
                }
            }

            return sum;
        }
    }

    class FreePolyomino : IEquatable<FreePolyomino>
    {
        internal FreePolyomino(OrientedPolyomino orientation)
        {
            var orientations = new HashSet<OrientedPolyomino>();
            orientations.Add(orientation);
            var tmp = orientation.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.FlipV(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);

            OrientedPolyominos = orientations.OrderBy(x => x).ToArray();
        }

        public IReadOnlyList<OrientedPolyomino> OrientedPolyominos { get; private set; }

        public OrientedPolyomino CanonicalOrientation => OrientedPolyominos[0];

        public static IEnumerable<FreePolyomino> All(int numCells)
        {
            if (numCells < 1) throw new ArgumentOutOfRangeException(nameof(numCells));
            if (numCells == 1) return new FreePolyomino[] { new FreePolyomino(OrientedPolyomino.Unit) };

            // We do this in two phases because identifying two equal oriented polyominos is faster than first building
            // free polyominos and then identifying that they're equal.
            var oriented = new HashSet<OrientedPolyomino>();
            foreach (var smaller in All(numCells - 1))
            {
                // We can add a cell to a side. The easiest way to do this is to add to the bottom of one of the rotations.
                // TODO Optimise by distinguishing the symmetries.
                foreach (var orientation in smaller.OrientedPolyominos)
                {
                    int h = orientation.BBHeight;
                    var bottomRow = orientation.Rows[h - 1];
                    for (int deltax = 0; deltax < orientation.BBWidth; deltax++)
                    {
                        if (((bottomRow >> deltax) & 1) == 1)
                        {
                            var rows = orientation.Rows.Concat(Enumerable.Repeat(1U << deltax, 1)).ToArray();
                            oriented.Add(new OrientedPolyomino(rows));
                        }
                    }
                }

                // We can add a cell in the middle, provided it connects up.
                var canon = smaller.CanonicalOrientation;
                uint prev = 0, curr = 0, next = canon.Rows[0];
                for (int y = 0; y < canon.BBHeight; y++)
                {
                    (prev, curr, next ) = (curr, next, y + 1 < canon.BBHeight ? canon.Rows[y + 1] : 0);
                    uint valid = (prev | next | (curr << 1) | (curr >> 1)) & ~curr;
                    for (int x = 0; x < canon.BBWidth; x++)
                    {
                        if (((valid >> x) & 1) == 1)
                        {
                            var rows = canon.Rows.ToArray(); // Copy
                            rows[y] |= 1U << x;
                            oriented.Add(new OrientedPolyomino(rows));
                        }
                    }
                }
            }

            // Now cluster the oriented polyominos into equivalence classes under dihedral symmetry.
            return new HashSet<FreePolyomino>(oriented.Select(orientation => new FreePolyomino(orientation)));
        }

        public bool Equals(FreePolyomino other) => other != null && CanonicalOrientation.Equals(other.CanonicalOrientation);
        public override bool Equals(object obj) => Equals(obj as FreePolyomino);
        public override int GetHashCode() => CanonicalOrientation.GetHashCode();
    }

    [DebuggerDisplay("{ToString()}")]
    struct OrientedPolyomino : IComparable<OrientedPolyomino>, IEquatable<OrientedPolyomino>
    {
        public static readonly OrientedPolyomino Unit = new OrientedPolyomino(1);

        public OrientedPolyomino(params uint[] rows)
        {
            if (rows.Length == 0) throw new ArgumentException("We don't support the empty polyomino", nameof(rows));
            if (rows.Any(row => row == 0) || rows.All(row => (row & 1) == 0)) throw new ArgumentException("Polyomino is not packed into the corner", nameof(rows));
            var colsUsed = rows.Aggregate(0U, (accum, row) => accum | row);
            BBWidth = Helper.Width(colsUsed);
            if (colsUsed != ((1U << BBWidth) - 1)) throw new ArgumentException("Polyomino has empty columns", nameof(rows));
            Rows = rows;
        }

        public IReadOnlyList<uint> Rows { get; private set; }
        public int BBWidth { get; private set; }
        public int BBHeight => Rows.Count;

        #region Dihedral symmetries

        public OrientedPolyomino FlipV() => new OrientedPolyomino(Rows.Reverse().ToArray());

        public OrientedPolyomino Rot90()
        {
            uint[] rot = new uint[BBWidth];
            for (int y = 0; y < BBHeight; y++)
            {
                for (int x = 0; x < BBWidth; x++)
                {
                    rot[x] |= ((Rows[y] >> x) & 1) << (BBHeight - 1 - y);
                }
            }
            return new OrientedPolyomino(rot);
        }

        #endregion

        #region Identity

        public int CompareTo(OrientedPolyomino other)
        {
            // Favour wide-and-short orientations for the canonical one.
            if (BBHeight != other.BBHeight) return BBHeight.CompareTo(other.BBHeight);

            for (int i = 0; i < BBHeight; i++)
            {
                if (Rows[i] != other.Rows[i]) return Rows[i].CompareTo(other.Rows[i]);
            }

            return 0;
        }
        public bool Equals(OrientedPolyomino other) => CompareTo(other) == 0;
        public override int GetHashCode() => Rows.Aggregate(0, (h, row) => h * 37 + (int)row);
        public override bool Equals(object obj) => (obj is OrientedPolyomino other) && Equals(other);
        public override string ToString()
        {
            var width = BBWidth;
            return string.Join("_", Rows.Select(row => Helper.ToString(row, width)));
        }

        #endregion
    }

    static class Helper
    {
        public static int Width(uint x)
        {
            int w = 0;
            if ((x >> 16) != 0) { w += 16; x >>= 16; }
            if ((x >> 8) != 0) { w += 8; x >>= 8; }
            if ((x >> 4) != 0) { w += 4; x >>= 4; }
            if ((x >> 2) != 0) { w += 2; x >>= 2; }
            switch (x)
            {
                case 0: break;
                case 1: w++; break;
                case 2:
                case 3: w += 2; break;
                default: throw new Exception("Unreachable code");
            }

            return w;
        }

        internal static string ToString(uint x, int width)
        {
            char[] chs = new char[width];
            for (int i = 0; i < width; i++)
            {
                chs[i] = (char)('0' + (x & 1));
                x >>= 1;
            }
            return new string(chs);
        }

        internal static uint Weight(uint v)
        {
            // https://graphics.stanford.edu/~seander/bithacks.html
            v = v - ((v >> 1) & 0x55555555);
            v = (v & 0x33333333) + ((v >> 2) & 0x33333333);
            return ((v + (v >> 4) & 0xF0F0F0F) * 0x1010101) >> 24;
        }
    }
}

Codice risolutore SAT

/******************************************************************************************
MiniSat -- Copyright (c) 2003-2005, Niklas Een, Niklas Sorensson
MiniSatCS -- Copyright (c) 2006-2007 Michal Moskal
GolfMiniSat -- Copyright (c) 2018 Peter Taylor

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and
associated documentation files (the "Software"), to deal in the Software without restriction,
including without limitation the rights to use, copy, modify, merge, publish, distribute,
sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or
substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT
NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT
OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
**************************************************************************************************/

using System;
using System.Diagnostics;
using System.Collections.Generic;

// NOTE! Variables are just integers. No abstraction here. They should be chosen from 0..N, so that they can be used as array indices.
using Var = System.Int32;
using System.Linq;

namespace MiniSAT
{
    public static class Ext
    {
        private static int TargetCapacity(int size) =>
            size < 65536 ? (size << 1) :
            size < 1048576 ? (size + (size >> 1)) :
                           size + (size >> 2);

        public static void Push<T>(this List<T> list, T elem)
        {
            // Similar to List<T>.Add but with a slower growth rate for large lists
            if (list.Count == list.Capacity) list.Capacity = TargetCapacity(list.Count + 1);
            list.Add(elem);
        }
        public static void Pop<T>(this List<T> list) => list.RemoveAt(list.Count - 1);
        public static T Peek<T>(this List<T> list) => list[list.Count - 1];
        public static void GrowTo<T>(this List<T> list, int size, T pad)
        {
            if (size > list.Count)
            {
                // Minimise resizing
                if (size > list.Capacity) list.Capacity = size;
                while (list.Count < size) list.Add(pad);
            }
        }
        public static void ShrinkTo<T>(this List<T> list, int size)
        {
            list.RemoveRange(size, list.Count - size);

            int targetCap = TargetCapacity(size);
            if (list.Capacity > targetCap) list.Capacity = targetCap;
        }
    }

    public delegate bool IntLess(int i1, int i2);

    public class Heap
    {
        IntLess Cmp;
        List<int> Heap_ = new List<int>(); // heap of ints
        List<int> Indices = new List<int>(); // index in Heap_

        static int Left(int i) => i << 1;
        static int Right(int i) => (i << 1) | 1;
        static int Parent(int i) => i >> 1;

        void UpHead(int i)
        {
            int x = Heap_[i];
            while (Parent(i) != 0 && Cmp(x, Heap_[Parent(i)]))
            {
                Heap_[i] = Heap_[Parent(i)];
                Indices[Heap_[i]] = i;
                i = Parent(i);
            }
            Heap_[i] = x;
            Indices[x] = i;
        }

        void DownHeap(int i)
        {
            int x = Heap_[i];
            while (Left(i) < Heap_.Count)
            {
                int child = Right(i) < Heap_.Count && Cmp(Heap_[Right(i)], Heap_[Left(i)]) ? Right(i) : Left(i);
                if (!Cmp(Heap_[child], x)) break;
                Heap_[i] = Heap_[child];
                Indices[Heap_[i]] = i;
                i = child;
            }
            Heap_[i] = x;
            Indices[x] = i;
        }

        bool Ok(int n) => n >= 0 && n < Indices.Count;

        public Heap(IntLess c) { Cmp = c; Heap_.Add(-1); }

        public void SetBounds(int size) { Solver.Assert(size >= 0); Indices.GrowTo(size, 0); if (size > Heap_.Capacity) Heap_.Capacity = size; }
        public bool InHeap(int n) { Solver.Assert(Ok(n)); return Indices[n] != 0; }
        public void Increase(int n) { Solver.Assert(Ok(n)); Solver.Assert(InHeap(n)); UpHead(Indices[n]); }
        public bool IsEmpty => Heap_.Count == 1;

        public void Push(int n)
        {
            Solver.Assert(Ok(n));
            Indices[n] = Heap_.Count;
            Heap_.Add(n);
            UpHead(Indices[n]);
        }

        public int Pop()
        {
            int r = Heap_[1];
            Heap_[1] = Heap_.Peek();
            Indices[Heap_[1]] = 1;
            Indices[r] = 0;
            Heap_.Pop();
            if (Heap_.Count > 1) DownHeap(1);
            return r;
        }
    }

    public class Solver
    {
        #region lbool ~= Nullable<bool>

        public struct LBool
        {
            public static readonly LBool True = new LBool { Content = 1 };
            public static readonly LBool False = new LBool { Content = -1 };
            public static readonly LBool Undef = new LBool { Content = 0 };

            private sbyte Content;

            public static bool operator ==(LBool a, LBool b) => a.Content == b.Content;
            public static bool operator !=(LBool a, LBool b) => a.Content != b.Content;
            public static LBool operator ~(LBool a) => new LBool { Content = (sbyte)-a.Content };

            public static implicit operator LBool(bool b) => b ? True : False;
        }

        public static readonly LBool l_True = LBool.True;
        public static readonly LBool l_False = LBool.False;
        public static readonly LBool l_Undef = LBool.Undef;

        #endregion

        #region Literals

        const int var_Undef = -1;

        public struct Lit
        {
            public Lit(Var var) { Index = var << 1; }

            public bool Sign => (Index & 1) != 0;
            public int Index { get; private set; }
            public int Var => Index >> 1;

            public bool SatisfiedBy(List<LBool> assignment) => assignment[Var] == (Sign ? l_False : l_True);

            public static Lit operator ~(Lit p) => new Lit { Index = p.Index ^ 1 };
            public static bool operator ==(Lit p, Lit q) => p.Index == q.Index;
            public static bool operator !=(Lit p, Lit q) => !(p == q);

            public override int GetHashCode() => Index;
            public override bool Equals(object other) => other is Lit that && this == that;
            public override string ToString() => (Sign ? "-" : "") + "x" + Var;
        }

        static public readonly Lit lit_Undef = ~new Lit(var_Undef);

        #endregion

        #region Clauses

        public abstract class Clause
        {
            protected Clause(bool learnt)
            {
                IsLearnt = learnt;
            }

            public bool IsLearnt { get; private set; }
            public float Activity;

            public abstract int Size { get; }
            public abstract Lit this[int i] { get;set; }

            public abstract bool SatisfiedBy(List<LBool> assigns);

            public static Clause Create(bool learnt, List<Lit> ps)
            {
                if (ps.Count < 2) throw new ArgumentOutOfRangeException(nameof(ps));
                if (ps.Count == 2) return new BinaryClause(learnt, ps[0], ps[1]);
                return new LargeClause(learnt, ps);
            }
        }

        public class BinaryClause : Clause
        {
            public BinaryClause(bool learnt, Lit p0, Lit p1) : base(learnt)
            {
                l0 = p0; l1 = p1;
            }

            private Lit l0;
            private Lit l1;

            public override Lit this[int i]
            {
                get { return i == 0 ? l0 : l1; }
                set { if (i == 0) l0 = value; else l1 = value; }
            }

            public override int Size => 2;

            public override bool SatisfiedBy(List<LBool> assigns) => l0.SatisfiedBy(assigns) || l1.SatisfiedBy(assigns);
        }

        public class LargeClause : Clause
        {
            public static int[] SizeDistrib = new int[10];

            internal LargeClause(bool learnt, List<Lit> ps) : base(learnt)
            {
                Data = ps.ToArray();

                SizeDistrib[Size >= SizeDistrib.Length ? SizeDistrib.Length - 1 : Size]++;
            }

            public Lit[] Data { get; private set; }

            public override int Size => Data.Length;

            public override Lit this[int i]
            {
                get { return Data[i]; }
                set { Data[i] = value; }
            }

            public override bool SatisfiedBy(List<LBool> assigns) => Data.Any(lit => lit.SatisfiedBy(assigns));

            public override string ToString() => "[" + string.Join(", ", Data) + "]";
        }

        #endregion

        #region Utilities

        // Returns a random float 0 <= x < 1. Seed must never be 0.
        static double Rnd(ref double seed)
        {
            seed *= 1389796;
            int k = 2147483647;
            int q = (int)(seed / k);
            seed -= (double)q * k;
            return seed / k;
        }

        [Conditional("DEBUG")]
        static public void Assert(bool expr) => Check(expr);

        // Just like 'assert()' but expression will be evaluated in the release version as well.
        static void Check(bool expr) { if (!expr) throw new Exception("assertion violated"); }

        #endregion

        #region VarOrder

        public class VarOrder
        {
            readonly List<LBool> Assigns; // Pointer to external assignment table.
            readonly List<float> Activity; // Pointer to external activity table.
            internal Heap Heap_;
            double RandomSeed;

            public VarOrder(List<LBool> ass, List<float> act)
            {
                Assigns = ass;
                Activity = act;
                Heap_ = new Heap(Lt);
                RandomSeed = 91648253;
            }

            bool Lt(Var x, Var y) => Activity[x] > Activity[y];

            public virtual void NewVar()
            {
                Heap_.SetBounds(Assigns.Count);
                Heap_.Push(Assigns.Count - 1);
            }

            // Called when variable increased in activity.
            public virtual void Update(Var x) { if (Heap_.InHeap(x)) Heap_.Increase(x); }

            // Called when variable is unassigned and may be selected again.
            public virtual void Undo(Var x) { if (!Heap_.InHeap(x)) Heap_.Push(x); }

            // Selects a new, unassigned variable (or 'var_Undef' if none exists).
            public virtual Lit Select(double random_var_freq)
            {
                // Random decision:
                if (Rnd(ref RandomSeed) < random_var_freq && !Heap_.IsEmpty)
                {
                    Var next = (Var)(Rnd(ref RandomSeed) * Assigns.Count);
                    if (Assigns[next] == l_Undef) return ~new Lit(next);
                }

                // Activity based decision:
                while (!Heap_.IsEmpty)
                {
                    Var next = Heap_.Pop();
                    if (Assigns[next] == l_Undef) return ~new Lit(next);
                }

                return lit_Undef;
            }
        }

        #endregion

        #region Solver state

        public bool Ok { get; private set; } // If false, the constraints are already unsatisfiable. No part of the solver state may be used!
        List<Clause> Clauses = new List<Clause>(); // List of problem clauses.
        List<Clause> Learnts = new List<Clause>(); // List of learnt clauses.
        double ClaInc = 1; // Amount to bump next clause with.
        const double ClaDecay = 1 / 0.999; // INVERSE decay factor for clause activity: stores 1/decay.

        public List<float> Activity = new List<float>(); // A heuristic measurement of the activity of a variable.
        float VarInc = 1; // Amount to bump next variable with.
        const float VarDecay = 1 / 0.95f; // INVERSE decay factor for variable activity: stores 1/decay. Use negative value for static variable order.
        VarOrder Order; // Keeps track of the decision variable order.
        const double RandomVarFreq = 0.02;

        List<List<Clause>> Watches = new List<List<Clause>>(); // 'watches[lit]' is a list of constraints watching 'lit' (will go there if literal becomes true).
        public List<LBool> Assigns = new List<LBool>(); // The current assignments.
        public List<Lit> Trail = new List<Lit>(); // Assignment stack; stores all assigments made in the order they were made.
        List<int> TrailLim = new List<int>(); // Separator indices for different decision levels in 'trail'.
        List<Clause> Reason = new List<Clause>(); // 'reason[var]' is the clause that implied the variables current value, or 'null' if none.
        List<int> Level = new List<int>(); // 'level[var]' is the decision level at which assignment was made.
        List<int> TrailPos = new List<int>(); // 'trail_pos[var]' is the variable's position in 'trail[]'. This supersedes 'level[]' in some sense, and 'level[]' will probably be removed in future releases.
        int QHead = 0; // Head of queue (as index into the trail -- no more explicit propagation queue in MiniSat).
        int SimpDBAssigns = 0; // Number of top-level assignments since last execution of 'simplifyDB()'.
        long SimpDBProps = 0; // Remaining number of propagations that must be made before next execution of 'simplifyDB()'.

        // Temporaries (to reduce allocation overhead)
        List<LBool> AnalyzeSeen = new List<LBool>();
        List<Lit> AnalyzeStack = new List<Lit>();
        List<Lit> AnalyzeToClear = new List<Lit>();

        #endregion

        #region Main internal methods:

        // Activity
        void VarBumpActivity(Lit p)
        {
            if (VarDecay < 0) return; // (negative decay means static variable order -- don't bump)
            if ((Activity[p.Var] += VarInc) > 1e100) VarRescaleActivity();
            Order.Update(p.Var);
        }
        void VarDecayActivity() { if (VarDecay >= 0) VarInc *= VarDecay; }
        void ClaDecayActivity() { ClaInc *= ClaDecay; }

        // Operations on clauses
        void ClaBumpActivity(Clause c) { if ((c.Activity += (float)ClaInc) > 1e20) ClaRescaleActivity(); }
        // Disposes of clause and removes it from watcher lists. NOTE! Low-level; does NOT change the 'clauses' and 'learnts' vector.
        void Remove(Clause c)
        {
            RemoveWatch(Watches[(~c[0]).Index], c);
            RemoveWatch(Watches[(~c[1]).Index], c);

            if (c.IsLearnt) LearntsLiterals -= c.Size;
            else ClausesLiterals -= c.Size;
        }
        bool IsLocked(Clause c) => c == Reason[c[0].Var];

        int DecisionLevel => TrailLim.Count;

        #endregion

        #region Public interface

        public Solver()
        {
            Ok = true;
            Order = new VarOrder(Assigns, Activity);
        }

        public void Prealloc(int numVars, int numClauses)
        {
            Activity.Capacity = numVars;
            AnalyzeSeen.Capacity = numVars;
            Assigns.Capacity = numVars;
            Level.Capacity = numVars;
            Reason.Capacity = numVars;
            Watches.Capacity = numVars << 1;
            Order.Heap_.SetBounds(numVars + 1);
            Trail.Capacity = numVars;
            TrailPos.Capacity = numVars;

            Clauses.Capacity = numClauses;
        }

        // Helpers (semi-internal)
        public LBool Value(Lit p) => p.Sign ? ~Assigns[p.Var] : Assigns[p.Var];

        public int nAssigns => Trail.Count;
        public int nClauses => Clauses.Count;
        public int nLearnts => Learnts.Count;

        // Statistics
        public long ClausesLiterals, LearntsLiterals;

        // Problem specification
        public int nVars => Assigns.Count;
        public void AddClause(params Lit[] ps) => NewClause(new List<Lit>(ps), false);

        // Solving
        public List<LBool> Model = new List<LBool>(); // If problem is satisfiable, this vector contains the model (if any).

        #endregion

        #region Operations on clauses:

        List<Lit> BasicClauseSimplification(List<Lit> ps)
        {
            List<Lit> qs = new List<Lit>(ps);

            var dict = new Dictionary<Var, Lit>(qs.Count);
            int ptr = 0;

            for (int i = 0; i < qs.Count; i++)
            {
                Lit l = qs[i];
                Var v = l.Var;
                if (dict.TryGetValue(v, out var other))
                {
                    if (other != l) return null; // other = ~l, so always satisfied
                }
                else
                {
                    dict[v] = l;
                    qs[ptr++] = l;
                }
            }
            qs.ShrinkTo(ptr);

            return qs;
        }

        void NewClause(List<Lit> ps, bool learnt)
        {
            if (!Ok) return;
            Assert(ps != null);

            if (!learnt)
            {
                Assert(DecisionLevel == 0);

                ps = BasicClauseSimplification(ps);
                if (ps == null) return;

                int j = 0;
                for (int i = 0; i < ps.Count; i++)
                {
                    var lit = ps[i];
                    if (Level[lit.Var] == 0)
                    {
                        if (Value(lit) == l_True) return; // Clause already sat
                        if (Value(lit) == l_False) continue; // Literal already eliminated
                    }
                    ps[j++] = lit;
                }
                ps.ShrinkTo(j);
            }

            // 'ps' is now the (possibly) reduced vector of literals.
            if (ps.Count == 0) Ok = false;
            else if (ps.Count == 1)
            {
                if (!Enqueue(ps[0], null)) Ok = false;
            }
            else
            {
                var c = Clause.Create(learnt, ps);

                if (!learnt)
                {
                    Clauses.Add(c);
                    ClausesLiterals += c.Size;
                }
                else
                {
                    // Put the second watch on the literal with highest decision level:
                    int max_i = 1;
                    int max = Level[ps[1].Var];
                    for (int i = 2; i < ps.Count; i++)
                        if (Level[ps[i].Var] > max)
                        {
                            max = Level[ps[i].Var];
                            max_i = i;
                        }
                    c[1] = ps[max_i];
                    c[max_i] = ps[1];

                    Check(Enqueue(c[0], c));

                    // Bumping:
                    ClaBumpActivity(c); // (newly learnt clauses should be considered active)
                    Learnts.Push(c);
                    LearntsLiterals += c.Size;
                }

                // Watch clause:
                Watches[(~c[0]).Index].Push(c);
                Watches[(~c[1]).Index].Push(c);
            }
        }

        // Can assume everything has been propagated! (esp. the first two literals are != l_False, unless
        // the clause is binary and satisfied, in which case the first literal is true)
        bool IsSatisfied(Clause c)
        {
            Assert(DecisionLevel == 0);
            return c.SatisfiedBy(Assigns);
        }

        #endregion

        #region Minor methods

        static bool RemoveWatch(List<Clause> ws, Clause elem) // Pre-condition: 'elem' must exists in 'ws' OR 'ws' must be empty.
        {
            if (ws.Count == 0) return false; // (skip lists that are already cleared)
            int j = 0;
            for (; ws[j] != elem; j++) Assert(j < ws.Count - 1);
            for (; j < ws.Count - 1; j++) ws[j] = ws[j + 1];
            ws.Pop();
            return true;
        }

        public Lit NewVar()
        {
            int index = nVars;
            Watches.Add(new List<Clause>()); // (list for positive literal)
            Watches.Add(new List<Clause>()); // (list for negative literal)
            Reason.Add(null);
            Assigns.Add(l_Undef);
            Level.Add(-1);
            TrailPos.Add(-1);
            Activity.Add(0);
            Order.NewVar();
            AnalyzeSeen.Add(l_Undef);
            return new Lit(index);
        }

        // Returns FALSE if immediate conflict.
        bool Assume(Lit p)
        {
            TrailLim.Add(Trail.Count);
            return Enqueue(p, null);
        }

        // Revert to the state at given level.
        void CancelUntil(int level)
        {
            if (DecisionLevel > level)
            {
                for (int c = Trail.Count - 1; c >= TrailLim[level]; c--)
                {
                    Var x = Trail[c].Var;
                    Assigns[x] = l_Undef;
                    Reason[x] = null;
                    Order.Undo(x);
                }
                Trail.RemoveRange(TrailLim[level], Trail.Count - TrailLim[level]);
                TrailLim.ShrinkTo(level);
                QHead = Trail.Count;
            }
        }

        #endregion

        #region Major methods:

        int Analyze(Clause confl, List<Lit> out_learnt)
        {
            List<LBool> seen = AnalyzeSeen;
            int pathC = 0;
            Lit p = lit_Undef;

            // Generate conflict clause
            out_learnt.Push(lit_Undef); // (placeholder for the asserting literal)
            var out_btlevel = 0;
            int index = Trail.Count - 1;
            do
            {
                Assert(confl != null); // (otherwise should be UIP)

                if (confl.IsLearnt) ClaBumpActivity(confl);

                for (int j = (p == lit_Undef) ? 0 : 1; j < confl.Size; j++)
                {
                    Lit q = confl[j];
                    var v = q.Var;
                    if (seen[v] == l_Undef && Level[v] > 0)
                    {
                        VarBumpActivity(q);
                        seen[v] = l_True;
                        if (Level[v] == DecisionLevel) pathC++;
                        else
                        {
                            out_learnt.Push(q);
                            out_btlevel = Math.Max(out_btlevel, Level[v]);
                        }
                    }
                }

                // Select next clause to look at
                while (seen[Trail[index--].Var] == l_Undef) ;
                p = Trail[index + 1];
                confl = Reason[p.Var];
                seen[p.Var] = l_Undef;
                pathC--;
            } while (pathC > 0);
            out_learnt[0] = ~p;

            // Conflict clause minimization
            {
                uint min_level = 0;
                for (int i = 1; i < out_learnt.Count; i++) min_level |= (uint)(1 << (Level[out_learnt[i].Var] & 31)); // (maintain an abstraction of levels involved in conflict)

                AnalyzeToClear.Clear();
                int j = 1;
                for (int i = 1; i < out_learnt.Count; i++)
                    if (Reason[out_learnt[i].Var] == null || !AnalyzeRemovable(out_learnt[i], min_level)) out_learnt[j++] = out_learnt[i];

                // Clean up
                for (int jj = 0; jj < out_learnt.Count; jj++) seen[out_learnt[jj].Var] = l_Undef;
                for (int jj = 0; jj < AnalyzeToClear.Count; jj++) seen[AnalyzeToClear[jj].Var] = l_Undef; // ('seen[]' is now cleared)

                out_learnt.ShrinkTo(j);
            }

            return out_btlevel;
        }

        // Check if 'p' can be removed. 'min_level' is used to abort early if visiting literals at a level that cannot be removed.
        bool AnalyzeRemovable(Lit p_, uint min_level)
        {
            Assert(Reason[p_.Var] != null);
            AnalyzeStack.Clear(); AnalyzeStack.Add(p_);
            int top = AnalyzeToClear.Count;
            while (AnalyzeStack.Count > 0)
            {
                Clause c = Reason[AnalyzeStack.Peek().Var];
                Assert(c != null);
                AnalyzeStack.Pop();
                for (int i = 1; i < c.Size; i++)
                {
                    Lit p = c[i];
                    if (AnalyzeSeen[p.Var] == l_Undef && Level[p.Var] != 0)
                    {
                        if (Reason[p.Var] != null && ((1 << (Level[p.Var] & 31)) & min_level) != 0)
                        {
                            AnalyzeSeen[p.Var] = l_True;
                            AnalyzeStack.Push(p);
                            AnalyzeToClear.Push(p);
                        }
                        else
                        {
                            for (int j = top; j < AnalyzeToClear.Count; j++) AnalyzeSeen[AnalyzeToClear[j].Var] = l_Undef;
                            AnalyzeToClear.ShrinkTo(top);
                            return false;
                        }
                    }
                }
            }

            AnalyzeToClear.Push(p_);
            return true;
        }

        bool Enqueue(Lit p, Clause from)
        {
            if (Value(p) != l_Undef) return Value(p) == l_True;

            Var x = p.Var;
            Assigns[x] = !p.Sign;
            Level[x] = DecisionLevel;
            TrailPos[x] = Trail.Count;
            Reason[x] = from;
            Trail.Add(p);
            return true;
        }

        Clause Propagate()
        {
            Clause confl = null;
            while (QHead < Trail.Count)
            {
                SimpDBProps--;

                Lit p = Trail[QHead++]; // 'p' is enqueued fact to propagate.
                List<Clause> ws = Watches[p.Index];
                int i, j, end;
                for (i = j = 0, end = ws.Count; i != end;)
                {
                    Clause c = ws[i++];
                    // Make sure the false literal is data[1]
                    Lit false_lit = ~p;
                    if (c[0] == false_lit) { c[0] = c[1]; c[1] = false_lit; }

                    Assert(c[1] == false_lit);

                    // If 0th watch is true, then clause is already satisfied.
                    Lit first = c[0];
                    LBool val = Value(first);
                    if (val == l_True) ws[j++] = c;
                    else
                    {
                        // Look for new watch
                        for (int k = 2; k < c.Size; k++)
                            if (Value(c[k]) != l_False)
                            {
                                c[1] = c[k]; c[k] = false_lit;
                                Watches[(~c[1]).Index].Push(c);
                                goto FoundWatch;
                            }

                        // Did not find watch -- clause is unit under assignment
                        ws[j++] = c;
                        if (!Enqueue(first, c))
                        {
                            if (DecisionLevel == 0) Ok = false;
                            confl = c;
                            QHead = Trail.Count;
                            while (i < end) ws[j++] = ws[i++]; // Copy the remaining watches
                        }
                        FoundWatch:;
                    }
                }
                ws.ShrinkTo(j);
            }

            return confl;
        }

        void ReduceDB()
        {
            double extra_lim = ClaInc / Learnts.Count; // Remove any clause below this activity

            Learnts.Sort((x, y) => x.Size > 2 && (y.Size == 2 || x.Activity < y.Activity) ? -1 : 1);

            int i, j;
            for (i = j = 0; i < Learnts.Count / 2; i++)
            {
                if (Learnts[i].Size > 2 && !IsLocked(Learnts[i])) Remove(Learnts[i]);
                else Learnts[j++] = Learnts[i];
            }
            for (; i < Learnts.Count; i++)
            {
                if (Learnts[i].Size > 2 && !IsLocked(Learnts[i]) && Learnts[i].Activity < extra_lim) Remove(Learnts[i]);
                else Learnts[j++] = Learnts[i];
            }
            Learnts.ShrinkTo(j);
        }

        void SimplifyDB()
        {
            if (!Ok) return;
            Assert(DecisionLevel == 0);

            if (Propagate() != null) { Ok = false; return; }
            if (nAssigns == SimpDBAssigns || SimpDBProps > 0) return; // (nothing has changed or performed a simplification too recently)

            // Clear watcher lists:
            for (int i = SimpDBAssigns; i < nAssigns; i++)
            {
                Lit p = Trail[i];
                Watches[p.Index].Clear();
                Watches[(~p).Index].Clear();
            }

            // Remove satisfied clauses:
            for (int type = 0; type < 2; type++)
            {
                List<Clause> cs = type != 0 ? Learnts : Clauses;
                int j = 0;
                for (int i = 0; i < cs.Count; i++)
                {
                    if (!IsLocked(cs[i]) && IsSatisfied(cs[i])) Remove(cs[i]);
                    else cs[j++] = cs[i];
                }
                cs.ShrinkTo(j);
            }

            SimpDBAssigns = nAssigns;
            SimpDBProps = ClausesLiterals + LearntsLiterals;
        }

        LBool Search(int nof_conflicts, int nof_learnts)
        {
            if (!Ok) return l_False;
            Assert(0 == DecisionLevel);

            int conflictC = 0;
            Model.Clear();

            while (true)
            {
                Clause confl = Propagate();
                if (confl != null)
                {
                    // CONFLICT
                    conflictC++;
                    var learnt_clause = new List<Lit>();
                    if (DecisionLevel == 0) return l_False; // Contradiction found
                    CancelUntil(Analyze(confl, learnt_clause));
                    NewClause(learnt_clause, true);
                    if (learnt_clause.Count == 1) Level[learnt_clause[0].Var] = 0;
                    VarDecayActivity();
                    ClaDecayActivity();
                }
                else
                {
                    // NO CONFLICT
                    if (nof_conflicts >= 0 && conflictC >= nof_conflicts)
                    {
                        // Reached bound on number of conflicts
                        CancelUntil(0);
                        return l_Undef;
                    }

                    // Simplify the set of problem clauses
                    if (DecisionLevel == 0) { SimplifyDB(); if (!Ok) return l_False; }

                    // Reduce the set of learnt clauses
                    if (nof_learnts >= 0 && Learnts.Count - nAssigns >= nof_learnts) ReduceDB();

                    // New variable decision
                    Lit next = Order.Select(RandomVarFreq);

                    if (next == lit_Undef)
                    {
                        // Model found
                        Model.Clear();
                        Model.Capacity = nVars;
                        Model.AddRange(Assigns);
                        CancelUntil(0);
                        return l_True;
                    }

                    Check(Assume(next));
                }
            }
        }

        void VarRescaleActivity()
        {
            for (int i = 0; i < nVars; i++) Activity[i] *= 1e-100f;
            VarInc *= 1e-100f;
        }

        void ClaRescaleActivity()
        {
            for (int i = 0; i < Learnts.Count; i++) Learnts[i].Activity *= 1e-20f;
            ClaInc *= 1e-20;
        }

        public bool Solve()
        {
            SimplifyDB();
            Assert(DecisionLevel == 0);

            double nof_conflicts = 100;
            double nof_learnts = nClauses / 3;
            while (true)
            {
                if (Search((int)nof_conflicts, (int)nof_learnts) != l_Undef)
                {
                    CancelUntil(0);
                    return Ok;
                }
                nof_conflicts *= 1.5;
                nof_learnts *= 1.1;
            }
        }

        #endregion
    }
}

ottimalità

n=11n=12

n=12

--- MiniSAT.cs.old
+++ MiniSAT.cs
@@ -346,6 +346,7 @@ namespace MiniSAT
         const double RandomVarFreq = 0.02;

         List<List<Clause>> Watches = new List<List<Clause>>(); // 'watches[lit]' is a list of constraints watching 'lit' (will go there if literal becomes true).
+        List<List<Lit>> BinaryWatches = new List<List<Lit>>();
         public List<LBool> Assigns = new List<LBool>(); // The current assignments.
         public List<Lit> Trail = new List<Lit>(); // Assignment stack; stores all assigments made in the order they were made.
         List<int> TrailLim = new List<int>(); // Separator indices for different decision levels in 'trail'.
@@ -381,7 +382,9 @@ namespace MiniSAT
         void Remove(Clause c)
         {
             RemoveWatch(Watches[(~c[0]).Index], c);
+            if (Watches[(~c[0]).Index] != null && Watches[(~c[0]).Index].Count == 0) Watches[(~c[0]).Index] = null;
             RemoveWatch(Watches[(~c[1]).Index], c);
+            if (Watches[(~c[1]).Index] != null && Watches[(~c[1]).Index].Count == 0) Watches[(~c[1]).Index] = null;

             if (c.IsLearnt) LearntsLiterals -= c.Size;
             else ClausesLiterals -= c.Size;
@@ -408,6 +411,7 @@ namespace MiniSAT
             Level.Capacity = numVars;
             Reason.Capacity = numVars;
             Watches.Capacity = numVars << 1;
+            BinaryWatches.Capacity = numVars << 1;
             Order.Heap_.SetBounds(numVars + 1);
             Trail.Capacity = numVars;
             TrailPos.Capacity = numVars;
@@ -500,7 +504,7 @@ namespace MiniSAT

                 if (!learnt)
                 {
-                    Clauses.Add(c);
+                    if (c.Size > 2) Clauses.Add(c);
                     ClausesLiterals += c.Size;
                 }
                 else
@@ -526,8 +530,20 @@ namespace MiniSAT
                 }

                 // Watch clause:
-                Watches[(~c[0]).Index].Push(c);
-                Watches[(~c[1]).Index].Push(c);
+                if (c.Size == 2 && !learnt)
+                {
+                    if (BinaryWatches[(~c[0]).Index] == null) BinaryWatches[(~c[0]).Index] = new List<Lit>();
+                    BinaryWatches[(~c[0]).Index].Push(c[1]);
+                    if (BinaryWatches[(~c[1]).Index] == null) BinaryWatches[(~c[1]).Index] = new List<Lit>();
+                    BinaryWatches[(~c[1]).Index].Push(c[0]);
+                }
+                else
+                {
+                    if (Watches[(~c[0]).Index] == null) Watches[(~c[0]).Index] = new List<Clause>();
+                    Watches[(~c[0]).Index].Push(c);
+                    if (Watches[(~c[1]).Index] == null) Watches[(~c[1]).Index] = new List<Clause>();
+                    Watches[(~c[1]).Index].Push(c);
+                }
             }
         }

@@ -545,7 +561,7 @@ namespace MiniSAT

         static bool RemoveWatch(List<Clause> ws, Clause elem) // Pre-condition: 'elem' must exists in 'ws' OR 'ws' must be empty.
         {
-            if (ws.Count == 0) return false; // (skip lists that are already cleared)
+            if (ws == null || ws.Count == 0) return false; // (skip lists that are already cleared)
             int j = 0;
             for (; ws[j] != elem; j++) Assert(j < ws.Count - 1);
             for (; j < ws.Count - 1; j++) ws[j] = ws[j + 1];
@@ -556,8 +572,10 @@ namespace MiniSAT
         public Lit NewVar()
         {
             int index = nVars;
-            Watches.Add(new List<Clause>()); // (list for positive literal)
-            Watches.Add(new List<Clause>()); // (list for negative literal)
+            Watches.Add(null); // (list for positive literal)
+            Watches.Add(null); // (list for negative literal)
+            BinaryWatches.Add(null);
+            BinaryWatches.Add(null);
             Reason.Add(null);
             Assigns.Add(l_Undef);
             Level.Add(-1);
@@ -716,45 +734,85 @@ namespace MiniSAT
                 SimpDBProps--;

                 Lit p = Trail[QHead++]; // 'p' is enqueued fact to propagate.
-                List<Clause> ws = Watches[p.Index];
-                int i, j, end;
-                for (i = j = 0, end = ws.Count; i != end;)
                 {
-                    Clause c = ws[i++];
-                    // Make sure the false literal is data[1]
-                    Lit false_lit = ~p;
-                    if (c[0] == false_lit) { c[0] = c[1]; c[1] = false_lit; }
+                    List<Clause> ws = Watches[p.Index];
+                    if (ws != null)
+                    {
+                        int i, j, end;
+                        for (i = j = 0, end = ws.Count; i != end;)
+                        {
+                            Clause c = ws[i++];
+                            // Make sure the false literal is data[1]
+                            Lit false_lit = ~p;
+                            if (c[0] == false_lit) { c[0] = c[1]; c[1] = false_lit; }

-                    Assert(c[1] == false_lit);
+                            Assert(c[1] == false_lit);

-                    // If 0th watch is true, then clause is already satisfied.
-                    Lit first = c[0];
-                    LBool val = Value(first);
-                    if (val == l_True) ws[j++] = c;
-                    else
-                    {
-                        // Look for new watch
-                        for (int k = 2; k < c.Size; k++)
-                            if (Value(c[k]) != l_False)
+                            // If 0th watch is true, then clause is already satisfied.
+                            Lit first = c[0];
+                            LBool val = Value(first);
+                            if (val == l_True) ws[j++] = c;
+                            else
                             {
-                                c[1] = c[k]; c[k] = false_lit;
-                                Watches[(~c[1]).Index].Push(c);
-                                goto FoundWatch;
+                                // Look for new watch
+                                for (int k = 2; k < c.Size; k++)
+                                    if (Value(c[k]) != l_False)
+                                    {
+                                        c[1] = c[k]; c[k] = false_lit;
+                                        if (Watches[(~c[1]).Index] == null) Watches[(~c[1]).Index] = new List<Clause>();
+                                        Watches[(~c[1]).Index].Push(c);
+                                        goto FoundWatch;
+                                    }
+
+                                // Did not find watch -- clause is unit under assignment
+                                ws[j++] = c;
+                                if (!Enqueue(first, c))
+                                {
+                                    if (DecisionLevel == 0) Ok = false;
+                                    confl = c;
+                                    QHead = Trail.Count;
+                                    while (i < end) ws[j++] = ws[i++]; // Copy the remaining watches
+                                }
+                                FoundWatch:;
                             }
+                        }

-                        // Did not find watch -- clause is unit under assignment
-                        ws[j++] = c;
-                        if (!Enqueue(first, c))
+                        if (j == 0) Watches[p.Index] = null;
+                        else ws.ShrinkTo(j);
+                    }
+                }
+                // TODO BinaryWatches
+                {
+                    List<Lit> ws = BinaryWatches[p.Index];
+                    if (ws != null)
+                    {
+                        int i, j, end;
+                        for (i = j = 0, end = ws.Count; i != end;)
                         {
-                            if (DecisionLevel == 0) Ok = false;
-                            confl = c;
-                            QHead = Trail.Count;
-                            while (i < end) ws[j++] = ws[i++]; // Copy the remaining watches
+                            var first = ws[i++];
+
+                            // If 0th watch is true, then clause is already satisfied.
+                            LBool val = Value(first);
+                            if (val == l_True) ws[j++] = first;
+                            else
+                            {
+                                // Did not find watch -- clause is unit under assignment
+                                ws[j++] = first;
+                                var c = new BinaryClause(false, first, ~p); // Needed for consistency of interface
+                                if (!Enqueue(first, c))
+                                {
+                                    if (DecisionLevel == 0) Ok = false;
+                                    confl = c;
+                                    QHead = Trail.Count;
+                                    while (i < end) ws[j++] = ws[i++]; // Copy the remaining watches
+                                }
+                            }
                         }
-                        FoundWatch:;
+
+                        if (j == 0) Watches[p.Index] = null;
+                        else ws.ShrinkTo(j);
                     }
                 }
-                ws.ShrinkTo(j);
             }

             return confl;
@@ -792,8 +850,10 @@ namespace MiniSAT
             for (int i = SimpDBAssigns; i < nAssigns; i++)
             {
                 Lit p = Trail[i];
-                Watches[p.Index].Clear();
-                Watches[(~p).Index].Clear();
+                Watches[p.Index] = null;
+                Watches[(~p).Index] = null;
+                BinaryWatches[p.Index] = null;
+                BinaryWatches[(~p).Index] = null;
             }

             // Remove satisfied clauses:

Soluzioni distinte

Contare le soluzioni a un problema SAT è semplice, se a volte lento: trovi una soluzione, aggiungi una nuova clausola che la esclude direttamente ed esegui di nuovo. Qui è facile generare la classe di equivalenza delle soluzioni sotto le simmetrie del rettangolo, quindi il seguente codice è sufficiente per generare tutte le soluzioni distinte.

                // Force it to the known optimal weight
                for (int i = optimal[n]; i < unaryWeights.Length; i++) solver.AddClause(~unaryWeights[i]);
                while (solver.Solve())
                {
                    var rows = new uint[height];
                    int cell = 0;
                    for (int y = 0; y < height; y++)
                    {
                        for (int x = 0; x < width; x++)
                        {
                            if (solver.Model[cell++] == Solver.l_True) rows[y] |= 1U << x;
                        }
                    }
                    var poly = new FreePolyomino(new OrientedPolyomino(rows));
                    Console.WriteLine(poly.CanonicalOrientation);

                    foreach (var orientation in poly.OrientedPolyominos)
                    {
                        if (orientation.BBWidth != width || orientation.BBHeight != height) continue;

                        // Exclude it
                        List<Solver.Lit> soln = new List<Solver.Lit>(previous);
                        cell = 0;
                        for (int y = 0; y < height; y++)
                        {
                            uint row = orientation.Rows[y];
                            for (int x = 0; x < width; x++, cell++)
                            {
                                if ((row & 1) == 1) soln.Add(~new Solver.Lit(cell));
                                row >>= 1;
                            }
                        }
                        solver.AddClause(soln.ToArray());
                    }
                }

n

100_111
010_111

0110_1111
1100_1111

01000_11111_01110
00100_11111_11100

011000_111111_011110
110000_111111_111100
011000_011110_111111
001100_111100_111111
110000_111100_111111
001100_111111_111100

0010000_0111100_0111110_1111111
0001000_1111000_1111111_1111100
0001000_0111000_1111111_1111110
0100000_1111000_1111111_1111100
0100000_1111000_1111100_1111111
0001000_0111000_1111110_1111111
0001000_0111110_1111111_1111000
0100000_1111000_0111110_1111111
0001000_1111100_1111111_1111000
1100000_1110000_1111100_1111111
0100000_1111111_1111100_0111100
0011000_0111000_0111110_1111111
1010000_1110000_1111100_1111111
0011000_1110000_1111100_1111111
0010100_0011100_1111100_1111111
0011000_1011100_1111111_1111000
0100000_1111111_0111110_0111100
1100000_1110000_1111111_1111100
0100000_1111100_1111111_0111100
1010000_1110000_1111111_1111100
1110000_0110000_1111111_0111110
0110000_1110000_1111100_1111111
0110000_1110000_1111111_1011110
0111000_0011000_1111111_0011111
0011100_0011000_1111111_0011111
1000100_1111111_0011110_0111100
0010000_1111111_0011111_0011110
0011000_0111000_1111111_0101111
0011000_0011101_1111111_0001111
0101000_0111000_0111110_1111111
0001000_0111100_1111100_1111111
1000100_1111111_0111100_0111100
0110000_1110000_1111111_1111100
1010000_1110000_1111111_0111110
0101000_0111000_1111111_0011111
0001000_1111111_1111100_1111000
0110000_0111010_1111111_0011110
0011000_0001110_0111110_1111111
0010000_1111111_0111110_0011110
0101000_0111000_1111111_0111110
0010000_1111100_1111111_0011110
0010000_0011100_1111111_1111110
0110000_0111000_1111111_0111110
0010000_0011100_1111111_0111111
0011000_0111010_1111111_0011110
0110000_1110100_1111111_0111100
0110000_0011100_1111100_1111111
0001000_0111100_1111111_1111100
0010000_0111100_1111111_1011110
0011000_0011100_1111111_1111100
0110000_0111000_0111110_1111111
0011000_0011100_1111100_1111111
0011000_0011100_0011111_1111111
0010000_0111100_0011111_1111111
0010000_0011100_1111110_1111111
0011000_0111000_1111111_0111110
0010000_0111100_1111111_0111110
0010000_0011110_1111111_0111110
0010000_0111100_1111111_0011111
0011000_0011100_1111111_0011111
0010100_0011100_1111111_1111100
0010000_0111101_1111111_0011110
0010000_0111110_1111111_0011110

01110000_01110000_11111111_01111110
11100000_11100000_11111100_11111111
00111000_00111000_00111111_11111111
11100000_11100000_11111111_11111100
00111000_00111000_11111111_00111111
01110000_01110000_01111110_11111111

011000000_111000000_111111111_111111100_011111000
001110000_000110000_001111100_001111111_111111111
000110000_001110000_111111111_001111111_000111110
001100000_011100000_111111111_011111110_001111100
001100000_011100000_111111100_111111111_001111100
011100000_001100000_011111000_011111110_111111111
000110000_000111000_001111111_111111111_000011111
000110000_000111000_111111100_111111111_111110000
000110000_001110000_011111110_111111111_000111110
001100000_001110000_011111110_111111111_000111110

1110000000_1111000000_1111110000_1111111100_1111111111
1110000000_1111000000_1111110000_1111111111_1111111100
1111000000_0111000000_0111111000_1111111111_0111111110
0011100000_0011110000_0011111100_0011111111_1111111111
0111000000_0111100000_0111111000_0111111110_1111111111
0111000000_0111100000_0111111000_1111111111_0111111110
0111000000_1111000000_0111111000_1111111111_1111111001
0111000000_1111000000_0111111000_1111111111_1111111010
0011100000_0011110000_0011111100_1111111111_0011111111
0111100000_0011100000_0011111100_1111111111_0011111111
0011100000_0111100000_0011111100_1111111111_0111111101

2
Finalmente riuscito a ottenere una ricerca di <43 con n = 12 da completare senza superare la memoria che può essere allocata ad un singolo processo; ci sono voluti 7,5 giorni e non ho riscontrato alcun miglioramento, quindi a meno che non abbia rotto il solutore SAT il risultato per n = 12 è anche ottimale sotto il presupposto di bbox.
Peter Taylor,

1
Tonnellate di sforzo proprio qui! Molto impressionante.
Jonathan Allan,

9

Nell'interesse di avviare il processo, ecco una risposta rapida (ma non molto ottimale).

Modello:

n = 8:
########
 ######
  #####
   ####
    ###
     ##

Prendi un triangolo con lunghezza n - 1, attacca un quadrato extra sull'angolo e taglia il quadrato inferiore.

Prova che tutti gli n-ominos si adattano:

Nota che qualsiasi n-omino può stare in un rettangolo con lunghezza + larghezza al massimo n + 1.

Se un n-omino si inserisce in un rettangolo con lunghezza + larghezza al massimo n, si adatta perfettamente al triangolo (che è l'unione di tutti questi rettangoli). Se capita di usare il quadrato tagliato, la trasposizione si adatterà al triangolo.

Altrimenti, abbiamo una catena con al massimo un ramo. Possiamo sempre adattare un'estremità della catena al quadrato extra (provalo con la cassa), e il resto si adatta a un rettangolo con lunghezza + larghezza al massimo n, riducendo al caso sopra.

L'unico caso in cui quanto sopra non funziona è il caso in cui utilizziamo sia il quadrato extra sia il quadrato di taglio. C'è solo uno di questi n-omino (la lunga L), e quello si adatta al triangolo trasposto.

Codice (Python 2):

def f(n):
    if n < 7:
        return [0, 1, 2, 4, 6, 9, 12][n]
    return n * (n - 1) / 2

Tavolo:

 1:   1
 2:   2
 3:   4
 4:   6
 5:   9
 6:  12
 7:  21
 8:  28
 9:  36
10:  45
11:  55
12:  66
13:  78
14:  91
15: 105
16: 120
17: 136
18: 153
19: 171
20: 190
... more cases can be generated if necessary.

n

Il limite del personaggio mi ha preso. Se hai bisogno di ulteriori spiegazioni, posso chiarire.
Mago del grano,

@CatWizard Capisco cosa intendi. Non sarei sorpreso se potessi estenderlo alla rimozione di circa la metà delle file (per n grandi).

8

C #, punteggio: 1, 2, 4, 6, 9, 12, 17, 20, 26, 31, 38, 44

#

##

#..
###

.##.
####

..#..
#####
###..

##....
######
####..

..##...
.###...
#######
.#####.

..###...
..###...
..######
########

..##.....
.###.....
#######..
#########
..#####..

.###......
.####.....
.######...
##########
.########.

.###.......
.####......
.####......
.#######...
###########
.#########.

.####.......
#####.......
.#####......
############
.##########.
.########...

Il formato di output del programma è un po 'più compatto.

Questo utilizza un approccio casuale seminato e ho ottimizzato i semi. Faccio valere un vincolo delimitatore che è sia plausibile che coerente con i dati noti per piccoli valori di n. Se quel vincolo è effettivamente valido allora

  1. L'output è ottimale fino a n = 8 (mediante validazione della forza bruta, non inclusa).
  2. Inizia il numero di soluzioni ottimali (distinte fino alla simmetria) 1, 1, 2, 2, 2, 6, 63, 6.
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;

namespace Sandbox
{
    class FreePolyomino : IEquatable<FreePolyomino>
    {
        public static void Main()
        {
            for (int i = 1; i < 12; i++)
            {
                int seed;
                switch (i) {
                    default: seed = 1103199029; break;
                    case 9: seed = 693534956; break; // 26
                    case 10: seed = 2005746461; break; // 31
                    case 11: seed = 377218946; break; // 38
                    case 12: seed = 1281379414; break; // 44
                }

                var rnd = new Random(seed);
                var polys = FreePolyomino.All(i);
                var minUnion = FreePolyomino.RandomMinimalUnion2(polys, rnd, i, (i + 1) >> 1);
                Console.WriteLine($"{i}\t{minUnion.Weight}\t{minUnion}");
            }
        }

        internal FreePolyomino(OrientedPolyomino orientation)
        {
            var orientations = new HashSet<OrientedPolyomino>();
            orientations.Add(orientation);
            var tmp = orientation.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.FlipV(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);
            tmp = tmp.Rot90(); orientations.Add(tmp);

            OrientedPolyominos = orientations.OrderBy(x => x).ToArray();
        }

        public IReadOnlyList<OrientedPolyomino> OrientedPolyominos { get; private set; }

        public OrientedPolyomino CanonicalOrientation => OrientedPolyominos[0];

        public static IEnumerable<FreePolyomino> All(int numCells)
        {
            if (numCells < 1) throw new ArgumentOutOfRangeException(nameof(numCells));
            if (numCells == 1) return new FreePolyomino[] { new FreePolyomino(OrientedPolyomino.Unit) };

            // We do this in two phases because identifying two equal oriented polyominos is faster than first building
            // free polyominos and then identifying that they're equal.
            var oriented = new HashSet<OrientedPolyomino>();
            foreach (var smaller in All(numCells - 1))
            {
                // We can add a cell to a side. The easiest way to do this is to add to the bottom of one of the rotations.
                // TODO Optimise by distinguishing the symmetries.
                foreach (var orientation in smaller.OrientedPolyominos)
                {
                    int h = orientation.BBHeight;
                    var bottomRow = orientation.Rows[h - 1];
                    for (int deltax = 0; deltax < orientation.BBWidth; deltax++)
                    {
                        if (((bottomRow >> deltax) & 1) == 1) oriented.Add(orientation.Union(OrientedPolyomino.Unit, deltax, h));
                    }
                }

                // We can add a cell in the middle, provided it connects up.
                var canon = smaller.CanonicalOrientation;
                uint prev = 0, curr = 0, next = canon.Rows[0];
                for (int y = 0; y < canon.BBHeight; y++)
                {
                    (prev, curr, next ) = (curr, next, y + 1 < canon.BBHeight ? canon.Rows[y + 1] : 0);
                    uint valid = (prev | next | (curr << 1) | (curr >> 1)) & ~curr;
                    for (int x = 0; x < canon.BBWidth; x++)
                    {
                        if (((valid >> x) & 1) == 1) oriented.Add(canon.Union(OrientedPolyomino.Unit, x, y));
                    }
                }
            }

            // Now cluster the oriented polyominos into equivalence classes under dihedral symmetry.
            return new HashSet<FreePolyomino>(oriented.Select(orientation => new FreePolyomino(orientation)));
        }

        internal static OrientedPolyomino RandomMinimalUnion2(IEnumerable<FreePolyomino> polys, Random rnd, int maxWidth, int maxHeight, int target = int.MaxValue)
        {
            var union = OrientedPolyomino.Unit;
            foreach (var poly in polys.Shuffle(rnd).ToList())
            {
                union = poly.MinimalUnion(union, rnd, maxWidth, maxHeight);
                if (union.Weight > target) throw new Exception("Too heavy");
            }

            return new FreePolyomino(union).CanonicalOrientation;
        }

        private OrientedPolyomino MinimalUnion(FreePolyomino other, Random rnd, int maxWidth, int maxHeight)
        {
            // Choose the option which does least work.
            return OrientedPolyominos.Count <= other.OrientedPolyominos.Count ?
                MinimalUnion(other.CanonicalOrientation, rnd, maxWidth, maxHeight) :
                other.MinimalUnion(CanonicalOrientation, rnd, maxWidth, maxHeight);
        }

        private OrientedPolyomino MinimalUnion(OrientedPolyomino other, Random rnd, int maxWidth, int maxHeight)
        {
            OrientedPolyomino best = default(OrientedPolyomino);
            int containsWeight = Math.Min(CanonicalOrientation.Weight, other.Weight);
            int bestWeight = int.MaxValue;
            int ties = 0;
            foreach (var orientation in OrientedPolyominos)
            {
                // Bounding boxes must overlap, but otherwise we brute force
                for (int dx = 1 - orientation.BBWidth; dx < other.BBWidth; dx++)
                {
                    for (int dy = 1 - orientation.BBHeight; dy < other.BBHeight; dy++)
                    {
                        var union = other.Union(orientation, dx, dy, maxWidth, maxHeight);
                        if (union.Rows == null) continue;

                        if (union.Weight == containsWeight) return union;

                        if (union.Weight < bestWeight)
                        {
                            best = union;
                            bestWeight = union.Weight;
                            ties = 1;
                        }
                        else if (union.Weight == bestWeight)
                        {
                            ties++;
                            if (rnd.Next(ties) == 0) best = union;
                        }
                    }
                }
            }

            if (best.Rows == null) throw new Exception();

            return best;
        }

        public bool Equals(FreePolyomino other) => other != null && CanonicalOrientation.Equals(other.CanonicalOrientation);
        public override bool Equals(object obj) => Equals(obj as FreePolyomino);
        public override int GetHashCode() => CanonicalOrientation.GetHashCode();
    }

    [DebuggerDisplay("{ToString()}")]
    struct OrientedPolyomino : IComparable<OrientedPolyomino>, IEquatable<OrientedPolyomino>
    {
        public static readonly OrientedPolyomino Unit = new OrientedPolyomino(1);

        public OrientedPolyomino(params uint[] rows)
        {
            if (rows.Length == 0) throw new ArgumentException("We don't support the empty polyomino", nameof(rows));
            if (rows.Any(row => row == 0) || rows.All(row => (row & 1) == 0)) throw new ArgumentException("Polyomino is not packed into the corner", nameof(rows));
            var colsUsed = rows.Aggregate(0U, (accum, row) => accum | row);
            BBWidth = Helper.Width(colsUsed);
            if (colsUsed != ((1U << BBWidth) - 1)) throw new ArgumentException("Polyomino has empty columns", nameof(rows));
            Rows = rows;
        }

        public IReadOnlyList<uint> Rows { get; private set; }
        public int BBWidth { get; private set; }
        public int BBHeight => Rows.Count;

        #region Dihedral symmetries

        public OrientedPolyomino FlipH()
        {
            int width = BBWidth;
            return new OrientedPolyomino(Rows.Select(x => Helper.Reverse(x, width)).ToArray());
        }

        public OrientedPolyomino FlipV() => new OrientedPolyomino(Rows.Reverse().ToArray());

        public OrientedPolyomino Rot90()
        {
            uint[] rot = new uint[BBWidth];
            for (int y = 0; y < BBHeight; y++)
            {
                for (int x = 0; x < BBWidth; x++)
                {
                    rot[x] |= ((Rows[y] >> x) & 1) << (BBHeight - 1 - y);
                }
            }
            return new OrientedPolyomino(rot);
        }

        #endregion

        #region Conglomeration

        public OrientedPolyomino Union(OrientedPolyomino other, int deltax, int deltay, int maxWidth = int.MaxValue, int maxHeight = int.MaxValue)
        {
            // NB deltax or deltay could be negative
            int minCol = Math.Min(0, deltax);
            int maxCol = Math.Max(BBWidth - 1, other.BBWidth - 1 + deltax);
            int width = maxCol + 1 - minCol; if (width > maxWidth) return default(OrientedPolyomino);

            int minRow = Math.Min(0, deltay);
            int maxRow = Math.Max(BBHeight - 1, other.BBHeight - 1 + deltay);
            int height = maxRow + 1 - minRow; if (height > maxHeight) return default(OrientedPolyomino);
            uint[] unionRows = new uint[height];

            for (int y = 0; y < BBHeight; y++)
            {
                unionRows[(deltay < 0 ? -deltay : 0) + y] |= Rows[y] << (deltax < 0 ? -deltax : 0);
            }
            for (int y = 0; y < other.BBHeight; y++)
            {
                unionRows[(deltay < 0 ? 0 : deltay) + y] |= other.Rows[y] << (deltax < 0 ? 0 : deltax);
            }

            return new OrientedPolyomino(unionRows);
        }

        #endregion

        #region Identity

        public int CompareTo(OrientedPolyomino other)
        {
            // Favour wide-and-short orientations for the canonical one.
            if (BBHeight != other.BBHeight) return BBHeight.CompareTo(other.BBHeight);

            for (int i = 0; i < BBHeight; i++)
            {
                if (Rows[i] != other.Rows[i]) return Rows[i].CompareTo(other.Rows[i]);
            }

            return 0;
        }
        public bool Equals(OrientedPolyomino other) => CompareTo(other) == 0;
        public override int GetHashCode() => Rows.Aggregate(0, (h, row) => h * 37 + (int)row);
        public override bool Equals(object obj) => (obj is OrientedPolyomino other) && Equals(other);
        public override string ToString()
        {
            var width = BBWidth;
            return string.Join("_", Rows.Select(row => Helper.ToString(row, width)));
        }

        #endregion

        public int Weight => Rows.Sum(row => (int)Helper.Weight(row));
    }

    static class Helper
    {
        public static int Width(uint x)
        {
            int w = 0;
            if ((x >> 16) != 0) { w += 16; x >>= 16; }
            if ((x >> 8) != 0) { w += 8; x >>= 8; }
            if ((x >> 4) != 0) { w += 4; x >>= 4; }
            if ((x >> 2) != 0) { w += 2; x >>= 2; }
            switch (x)
            {
                case 0: break;
                case 1: w++; break;
                case 2:
                case 3: w += 2; break;
                default: throw new Exception("Unreachable code");
            }

            return w;
        }

        public static uint Reverse(uint x, int width)
        {
            uint rev = 0;
            while (width-- > 0)
            {
                rev = (rev << 1) | (x & 1);
                x >>= 1;
            }
            return rev;
        }

        internal static string ToString(uint x, int width)
        {
            char[] chs = new char[width];
            for (int i = 0; i < width; i++)
            {
                chs[i] = (char)('0' + (x & 1));
                x >>= 1;
            }
            return new string(chs);
        }

        internal static uint Weight(uint v)
        {
            // https://graphics.stanford.edu/~seander/bithacks.html
            v = v - ((v >> 1) & 0x55555555);
            v = (v & 0x33333333) + ((v >> 2) & 0x33333333);
            return ((v + (v >> 4) & 0xF0F0F0F) * 0x1010101) >> 24;
        }

        public static IEnumerable<T> Shuffle<T>(this IEnumerable<T> elts, Random rnd = null)
        {
            rnd = rnd ?? new Random();
            T[] arr = elts.ToArray();
            int n = arr.Length;
            while (n > 0)
            {
                int idx = rnd.Next(n - 1);
                yield return arr[idx];
                arr[idx] = arr[n - 1];
                arr[n - 1] = default(T); // Help GC if T is a class
                n--;
            }
        }
    }
}

Demo online


3

Posizionamento avido in ordine casuale

[1, 2, 4, 6, 9, 12, 17, 21, 27, 32]

Le regioni trovate sono riportate di seguito, così come il programma ruggine che le ha generate. Chiamalo con un parametro della riga di comando uguale a n che desideri cercare. Finora l'ho eseguito fino a n = 10. Nota che non è ancora ottimizzato per la velocità, lo farò più tardi e probabilmente accelererò molto le cose.

L'algoritmo è semplice, mischio i poliomino in un ordine casuale (seminato), quindi li posiziono uno alla volta nella posizione con la massima sovrapposizione possibile con la regione finora. Lo faccio 100 volte e produco la migliore regione risultante.

Regioni

Size 1: 1
#

Size 2: 2
##

Size 3: 4
###
 # 

Size 4: 6
####
##  

Size 5: 9
 ### 
#####
 #   

Size 6: 12
######
####  
##    

Size 7: 17
 ####  
#####  
#######
    #  

Size 8: 21
 #   
 ### 
#####
#### 
 ##  
###  
 ##  
 #   

Size 9: 27
#########
#######  
   ##### 
   ###   
   ###   

Size 10: 32
 ##    
 ##    
 ###   
 ###   
 ####  
#####  
 ##### 
 ######
 #     
 #     

Programma

Nota: richiede di notte, ma basta cambiare la semina per liberarsene, se ti interessa.

#![feature(int_to_from_bytes)]
extern crate rand;
use rand::{ChaChaRng, Rng, SeedableRng};

use std::fmt;
use std::collections::HashSet;

#[derive(Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
struct Poly(Vec<(isize, isize)>);

impl Poly {
    fn new(mut v: Vec<(isize, isize)>) -> Self {
        v.sort();
        Poly(v)
    }
    fn flip_hor(&self) -> Self {
        Poly::new(self.0.iter().map(|&(a, b)| (-a, b)).collect())
    }
    fn flip_vert(&self) -> Self {
        Poly::new(self.0.iter().map(|&(a, b)| (a, -b)).collect())
    }
    fn transpose(&self) -> Self {
        Poly::new(self.0.iter().map(|&(a, b)| (b, a)).collect())
    }
    fn offset_by(&self, x: isize, y: isize) -> Self {
        Poly::new(self.0.iter().map(|&(a, b)| (a+x, b+y)).collect())
    }
    fn offset_canon(&self) -> Self {
        let (mut min_x, mut min_y) = self.0[0];
        for &(x, y) in &self.0 {
            if x < min_x {
                min_x = x;
            }
            if y < min_y {
                min_y = y;
            }
        }
        self.offset_by(-min_x, -min_y)
    }
    fn transformations(&self) -> Vec<Self> {
        vec!(
            self.offset_canon(),
            self.flip_hor().offset_canon(),
            self.flip_vert().offset_canon(),
            self.flip_vert().flip_hor().offset_canon(),
            self.transpose().offset_canon(),
            self.transpose().flip_hor().offset_canon(),
            self.transpose().flip_vert().offset_canon(),
            self.transpose().flip_vert().flip_hor().offset_canon(),
        )
    }
    fn canonicalize(&self) -> Self {
        self.transformations().into_iter().min().unwrap().transpose()
    }
    fn max_box(&self) -> (isize, isize) {
        let (mut max_x, mut max_y) = self.0[0];
        for &(x, y) in &self.0 {
            if x > max_x {
                max_x = x;
            }
            if y > max_y {
                max_y = y;
            }
        }
        (max_x, max_y)
    }
    fn extend(&self) -> HashSet<Self> {
        let elems: HashSet<(isize, isize)> = self.0.iter().cloned().collect();
        let mut perim: HashSet<(isize, isize)> = HashSet::new();
        let mut neighbors: HashSet<Self> = HashSet::new();
        for &(x, y) in &self.0 {
            for (dx, dy) in vec!((0, 1), (1, 0), (-1, 0), (0, -1)) {
                let p = (x + dx, y + dy);
                if !elems.contains(&p) {
                    if !perim.contains(&p) {
                        let mut poly_points = self.0.clone();
                        poly_points.push(p);
                        let new_poly = Poly::new(poly_points).canonicalize();
                        neighbors.insert(new_poly);
                        perim.insert(p);
                    }
                }
            }
        }
        neighbors
    }
}

impl fmt::Display for Poly {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        let p = self.offset_canon();
        let (max_x, max_y) = p.max_box();
        let (max_x, max_y) = (max_x as usize, max_y as usize);
        let mut grid = vec![vec![" "; max_x+1]; max_y+1];
        for &(x, y) in &p.0 {
            grid[y as usize][x as usize] = "#"
        }
        let s = grid.into_iter().map(|r| r.concat()).collect::<Vec<String>>().join("\n");
        write!(f, "{}", s)
    }
}

fn all_polys(n: usize) -> HashSet<Poly> {
    let mut polys = HashSet::new();
    polys.insert(Poly::new(vec!((0, 0))));
    for _ in 0..n-1 {
        let mut next_polys = HashSet::new();
        for poly in polys {
            next_polys.extend(poly.extend());
        }
        polys = next_polys;
    }
    polys
}

fn overlap_polys(polys: &Vec<Poly>, seed: u64) -> Poly {
    let mut seq = polys.clone();
    let mut seed_array = [0; 32];
    for i in 0..32 {
        seed_array[i] = seed.to_be().to_bytes()[i%8];
    }
    let mut rng = ChaChaRng::from_seed(seed_array);
    rng.shuffle(&mut seq);
    let mut points: HashSet<(isize, isize)> = seq[0].0.iter().cloned().collect();
    for poly in seq {
        let max_x = points.iter().map(|a| a.0).max().unwrap();
        let max_y = points.iter().map(|a| a.1).max().unwrap();
        let mut best_overlap_amount = 0;
        let mut best_placement = poly.clone();
        for t_poly in poly.transformations() {
            let (t_max_x, t_max_y) = t_poly.max_box();
            for x in -t_max_x ..= max_x {
                for y in -t_max_y ..= max_y {
                    let mut overlap_amount = 0;
                    let st_poly = t_poly.offset_by(x, y);
                    for point in &st_poly.0 {
                        if points.contains(point) {
                            overlap_amount += 1;
                        }
                    }
                    if overlap_amount > best_overlap_amount {
                        best_overlap_amount = overlap_amount;
                        best_placement = st_poly;
                    }
                }
            }
        }
        for &point in &best_placement.0 {
            points.insert(point);
        }
    }
    Poly::new(points.into_iter().collect()).canonicalize()
}
fn main() {
    let seed_start = 123456789;
    let iters = 100;
    let max_poly = std::env::args().nth(1).unwrap().parse().unwrap();
    let mut sizes = vec!();
    for i in 1..=max_poly {
        let mut ap: Vec<Poly> = all_polys(i).into_iter().collect();
        ap.sort();
        let mut best_result = overlap_polys(&ap, seed_start);
        let mut best_score = best_result.0.len();
        for i in 1..iters {
            let result = overlap_polys(&ap, seed_start + i);
            if result.0.len() < best_score {
                best_score = result.0.len();
                best_result = result;
            }
        }
        println!("Size {}: {}\n{}\n", i, best_result.0.len(), best_result); 
        sizes.push(best_result.0.len());
    }
    println!("{:?}", sizes); 
}
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