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BetterLattice.cs
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BetterLattice.cs
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
namespace Lattice
{
using Math;
public static class BetterLattice
{
public static List<Vector<BigInteger>> Enumerate(int dimensions, Matrix<BigRational> basis, Vector<BigRational> lower, Vector<BigRational> upper)
{
// step 1: convert to standard form
// add slack / surplus variables
// substitute original variables
Matrix<BigRational> transform;
Vector<BigRational> transformOffset;
Matrix<BigRational> A;
Vector<BigRational> b;
Setup(dimensions, basis, lower, upper, out transform, out transformOffset, out A, out b);
var solutions = new List<Vector<BigInteger>>();
Search(dimensions, basis, transform, transformOffset, A, b, solutions, new Stack<BigInteger>());
return solutions;
}
private static void Search(int dimensions, Matrix<BigRational> basis, Matrix<BigRational> transform, Vector<BigRational> transformOffset, Matrix<BigRational> A, Vector<BigRational> b, List<Vector<BigInteger>> solutions, Stack<BigInteger> set)
{
if (set.Count == dimensions)
{
Console.WriteLine("asdf");
solutions.Add(Vector.Create(set.AsEnumerable()));
}
else
{
var n = set.Count;
// Console.WriteLine("A:\n" + A);
// Console.WriteLine("b: " + b);
// Console.WriteLine("c: " + transform.GetRow(n));
// Console.WriteLine("result:");
// Console.WriteLine(Solve(A, b, transform.GetRow(n)));
var lower = transformOffset - transform * Solve(A, b, -transform.GetRow(n));
var upper = transformOffset - transform * Solve(A, b, transform.GetRow(n));
// Console.WriteLine(lower);
// Console.WriteLine(upper);
var min = BigRational.Ceiling(lower[n]);
var max = BigRational.Floor(upper[n]);
Console.WriteLine($"{set.Count}: {min} -> {max}");
for (var x = min; x <= max; ++x)
{
var A2 = Matrix.Create(A.Rows + 1, A.Columns, (row, col) => row < A.Rows ? A[row, col] : transform[n, col]);
var b2 = Vector.Create(b.Length + 1, row => row < b.Length ? b[row] : transformOffset[n] - x);
set.Push(x);
Search(dimensions, basis, transform, transformOffset, A2, b2, solutions, set);
set.Pop();
}
}
}
public static Vector<BigRational> Solve(Matrix<BigRational> A, Vector<BigRational> b, Vector<BigRational> c)
{
// permutation matrices on A
Matrix<BigRational> B;
Matrix<BigRational> N;
{
var BColumns = new List<int>();
var NColumns = new List<int>();
for (var i = 0; i < A.Columns; ++i)
{
BColumns.Add(i);
var rank = 0;
Matrix.Create(A.Rows, BColumns.Count, (row, col) => A[row, BColumns[col]]).GetRowReduce(out rank);
if (rank != BColumns.Count)
{
BColumns.Remove(i);
NColumns.Add(i);
}
}
B = Matrix.Create<BigRational>(A.Columns, BColumns.Count, (row, col) => row == BColumns[col] ? 1 : 0);
N = Matrix.Create<BigRational>(A.Columns, NColumns.Count, (row, col) => row == NColumns[col] ? 1 : 0);
}
var x = (A * B).GetInverse() * b;
while(true)
{
var λ = (A * B).GetTranspose().GetInverse() * (c * B);
var s = (c * N) - (A * N).GetTranspose() * λ;
var validEntering = Enumerable.Range(0, A.Columns - A.Rows).Where(i => s[i].Sign < 0);
// Console.WriteLine("B:\n" + (A * B));
// Console.WriteLine("N:\n" + (A * N));
// Console.WriteLine("λ: " + λ);
// Console.WriteLine("s: " + s);
// Console.WriteLine("x: " + x);
if (!validEntering.Any())
{
break;
}
var (_, q) = validEntering.Min(i => (s[i], i));
var d = (A * B).GetInverse() * (A * N.GetColumn(q));
// Console.WriteLine("q: " + q);
// Console.WriteLine("d: " + d);
if (d.All(x => x.Sign <= 0))
{
throw new ArithmeticException("Solution is unbounded");
}
var (xq, p) = x.Select((x, i) => (x: x, i: i)).Where(a => d[a.i].Sign > 0).Min(a => (a.x / d[a.i], a.i));
// Console.WriteLine("q: " + p);
x -= xq * d;
// Console.WriteLine("X: " + x);
x += xq * Vector.CreateBasis<BigRational>(A.Rows, p);
// Console.WriteLine("X: " + x);
(B, N) = (Matrix.Create(B.Rows, B.Columns, (row, col) => col == p ? N[row, q] : B[row, col]), Matrix.Create(N.Rows, N.Columns, (row, col) => col == q ? B[row, p] : N[row, col]));
// Console.WriteLine();
}
return x * B.GetTranspose();
}
private static void Setup(int dimensions, Matrix<BigRational> basis, Vector<BigRational> lower, Vector<BigRational> upper, out Matrix<BigRational> transform, out Vector<BigRational> transformOffset, out Matrix<BigRational> A, out Vector<BigRational> b)
{
var constraints = new BigRational[2 * dimensions, 3 * dimensions + 1];
for (var row = 0; row < 2 * dimensions; ++row)
{
for (var col = 0; col < 3 * dimensions + 1; ++col)
{
constraints[row, col] = 0;
}
}
for (var row = 0; row < dimensions; ++row)
{
// lhs <= upper
// lhs + s = upper
constraints[2 * row + 0, 2 * row + 0 + dimensions] = 1;
constraints[2 * row + 0, 3 * dimensions] = upper[row];
// lhs >= lower
// lhs - s = lower
constraints[2 * row + 1, 2 * row + 1 + dimensions] = -1;
constraints[2 * row + 1, 3 * dimensions] = lower[row];
for (var col = 0; col < dimensions; ++col)
{
constraints[2 * row + 0, col] = basis[row, col];
constraints[2 * row + 1, col] = basis[row, col];
}
}
var count = dimensions;
constraints = Matrix.Create(constraints).GetRowReduce(count).ToArray();
if (count != dimensions)
{
throw new ArithmeticException("Unable to reduce constraint matrix appropriately");
}
b = Vector.Create(dimensions, row => constraints[row, 3 * dimensions]);
for (var row = dimensions; row < dimensions * 2; ++row)
{
if (constraints[row, 3 * dimensions].Sign < 0)
{
for (var col = 0; col < 3 * dimensions + 1; ++col)
{
constraints[row, col] = -constraints[row, col];
}
}
}
transform = Matrix.Create(dimensions, 2 * dimensions, (row, col) => constraints[row, dimensions + col]);
transformOffset = Vector.Create(dimensions, row => constraints[row, 3 * dimensions]);
A = Matrix.Create(dimensions, 2 * dimensions, (row, col) => constraints[dimensions + row, dimensions + col]);
b = Vector.Create(dimensions, row => constraints[dimensions + row, 3 * dimensions]);
var v = Vector.Create(70368744177664, 0, 0, 70368744177664, 70368744177664, 0, 0, 70368744177664, 70368744177664, 0, 70368744177664, 0, 70368744177664, 0, 70368744177664, 0, 0, 70368744177664, 70368744177664, 0, 0, 70368744177664, 70368744177664, 0, 0, 70368744177664, 0, 70368744177664, 0, 2199023255552, 0, 17592186044416, 0, 17592186044416);
// Console.WriteLine(A);
// Console.WriteLine(b);
// Console.WriteLine(transform.GetRow(0));
// Console.WriteLine(transformOffset[0]);
}
}
}