546 lines
16 KiB
C#
546 lines
16 KiB
C#
using System;
|
|
using System.Runtime.CompilerServices;
|
|
|
|
namespace PerfectNumbers.Core.ByDivisor;
|
|
|
|
/// <summary>
|
|
/// Helper utilities for the BitContradiction divisor search. These methods implement the
|
|
/// interval-based carry propagation described in the BitContradictions plan document.
|
|
/// The solver itself will build on top of these primitives.
|
|
/// </summary>
|
|
internal static class BitContradictionSolver
|
|
{
|
|
private const int PingPongBlockColumns = 64;
|
|
private const int TailCarryBatchColumns = 1024;
|
|
private const int TailCarryBatchRepeats = 1;
|
|
|
|
[ThreadStatic]
|
|
private static Dictionary<long, sbyte>? _aBitsBuffer;
|
|
[ThreadStatic]
|
|
private static TopDownPruneFailure? _lastTopDownFailure;
|
|
|
|
internal readonly struct TopDownPruneFailure(long column, long carryMin, long carryMax, long unknown)
|
|
{
|
|
public readonly long Column = column;
|
|
public readonly long CarryMin = carryMin;
|
|
public readonly long CarryMax = carryMax;
|
|
public readonly long Unknown = unknown;
|
|
}
|
|
|
|
internal static TopDownPruneFailure? LastTopDownFailure => _lastTopDownFailure;
|
|
|
|
internal readonly struct ColumnBounds(long forcedOnes, long possibleOnes)
|
|
{
|
|
public readonly long ForcedOnes = forcedOnes;
|
|
public readonly long PossibleOnes = possibleOnes;
|
|
}
|
|
|
|
internal readonly struct CarryRange(long min, long max)
|
|
{
|
|
public readonly long Min = min;
|
|
public readonly long Max = max;
|
|
|
|
public static CarryRange Single(long value) => new(value, value);
|
|
public static readonly CarryRange Zero = Single(0);
|
|
}
|
|
|
|
internal enum ContradictionReason
|
|
{
|
|
None,
|
|
ParityUnreachable,
|
|
TruncatedLength,
|
|
TailNotZero,
|
|
}
|
|
|
|
internal static ColumnBounds ComputeColumnBounds(
|
|
ReadOnlySpan<bool> multiplicand,
|
|
ReadOnlySpan<bool> multiplier,
|
|
int columnIndex)
|
|
{
|
|
long forced = 0;
|
|
long possible = 0;
|
|
|
|
int multiplicandLength = multiplicand.Length;
|
|
int multiplierLength = multiplier.Length;
|
|
for (int i = 0; i <= columnIndex; i++)
|
|
{
|
|
bool a = i < multiplicandLength && multiplicand[i];
|
|
bool q = columnIndex - i < multiplierLength && multiplier[columnIndex - i];
|
|
if (!a && !q)
|
|
{
|
|
continue;
|
|
}
|
|
|
|
if (a && q)
|
|
{
|
|
forced++;
|
|
}
|
|
|
|
possible++;
|
|
}
|
|
|
|
return new ColumnBounds(forced, possible);
|
|
}
|
|
|
|
internal static bool TryPropagateCarry(
|
|
CarryRange currentCarry,
|
|
long forcedOnes,
|
|
long possibleOnes,
|
|
int requiredResultBit,
|
|
out CarryRange nextCarry,
|
|
out ContradictionReason reason)
|
|
{
|
|
long minSum = currentCarry.Min + forcedOnes;
|
|
long maxSum = currentCarry.Max + possibleOnes;
|
|
int parity = requiredResultBit & 1;
|
|
|
|
minSum = AlignUpToParity(minSum, parity);
|
|
maxSum = AlignDownToParity(maxSum, parity);
|
|
|
|
if (minSum > maxSum)
|
|
{
|
|
nextCarry = default;
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return false;
|
|
}
|
|
|
|
nextCarry = new CarryRange((minSum - requiredResultBit) >> 1, (maxSum - requiredResultBit) >> 1);
|
|
reason = ContradictionReason.None;
|
|
return true;
|
|
}
|
|
|
|
/// <summary>
|
|
/// Full column-wise propagation for a given set of q one-bit offsets (LSB=0) and exponent p.
|
|
/// Returns true if the solver could decide locally (divides or contradiction).
|
|
/// </summary>
|
|
internal static bool TryCheckDivisibilityFromOneOffsets(
|
|
ReadOnlySpan<int> qOneOffsets,
|
|
ulong p,
|
|
out bool divides,
|
|
out ContradictionReason reason)
|
|
{
|
|
divides = false;
|
|
_lastTopDownFailure = null;
|
|
|
|
// These conditions will never be true on EvenPerfectBitScanner execution path.
|
|
// if (p == 0 || qOneOffsets.Length == 0)
|
|
// {
|
|
// reason = ContradictionReason.ParityUnreachable;
|
|
// return true;
|
|
// }
|
|
|
|
// While the scanner supports p >= 31, it's a tiny subset of all checked exponents. No need to support it.
|
|
// Exact small-case evaluation to short-circuit both success and failure deterministically.
|
|
// int highestBit = qOneOffsets[^1];
|
|
// if (p <= 64 && highestBit < 63)
|
|
// {
|
|
// ulong q = 0;
|
|
// foreach (int bit in qOneOffsets)
|
|
// {
|
|
// q |= 1UL << bit;
|
|
// }
|
|
|
|
// ulong mersenne = (1UL << (int)p) - 1UL;
|
|
// divides = q != 0 && (mersenne % q == 0);
|
|
// reason = divides ? ContradictionReason.None : ContradictionReason.ParityUnreachable;
|
|
// return true;
|
|
// }
|
|
|
|
int qOneOffsetsLength = qOneOffsets.Length;
|
|
int maxOffset = qOneOffsets[qOneOffsetsLength - 1];
|
|
if ((ulong)(maxOffset + 1) >= p)
|
|
{
|
|
reason = ContradictionReason.TruncatedLength;
|
|
return true;
|
|
}
|
|
|
|
// q from EvenPerfectBitScanner will always have the correct 2kp+1 form.
|
|
// if (qOneOffsets[0] != 0)
|
|
// {
|
|
// reason = ContradictionReason.ParityUnreachable;
|
|
// return true;
|
|
// }
|
|
|
|
Dictionary<long, sbyte> aBits = _aBitsBuffer ??= new Dictionary<long, sbyte>(capacity: qOneOffsetsLength << 2);
|
|
// aBits.EnsureCapacity(qOneOffsetsLength << 2);
|
|
aBits.Clear();
|
|
long maxKnownA = -1L;
|
|
long pLong = (long)p;
|
|
long maxAllowedA = pLong - (maxOffset + 1L);
|
|
if (maxAllowedA < 0)
|
|
{
|
|
reason = ContradictionReason.TruncatedLength;
|
|
return true;
|
|
}
|
|
|
|
if (!TryTopDownCarryPrune(qOneOffsets, qOneOffsetsLength, pLong, maxAllowedA, out _lastTopDownFailure))
|
|
{
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return true;
|
|
}
|
|
|
|
long lowColumn = 0;
|
|
long highColumn = pLong - 1;
|
|
CarryRange carryLow = CarryRange.Zero;
|
|
CarryRange carryHigh = CarryRange.Zero;
|
|
bool processTop = true;
|
|
|
|
while (lowColumn <= highColumn)
|
|
{
|
|
int remaining = (int)(highColumn - lowColumn + 1);
|
|
int blockSize = remaining < PingPongBlockColumns ? remaining : PingPongBlockColumns;
|
|
|
|
if (processTop)
|
|
{
|
|
if (!TryProcessTopDownBlock(qOneOffsets, qOneOffsetsLength, maxAllowedA, highColumn, blockSize, carryHigh, out carryHigh, out _lastTopDownFailure))
|
|
{
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return true;
|
|
}
|
|
|
|
highColumn -= blockSize;
|
|
}
|
|
else
|
|
{
|
|
if (!TryProcessBottomUpBlock(qOneOffsets, aBits, maxAllowedA, lowColumn, blockSize, ref carryLow, ref maxKnownA, out reason))
|
|
{
|
|
return true;
|
|
}
|
|
|
|
lowColumn += blockSize;
|
|
}
|
|
|
|
processTop = !processTop;
|
|
}
|
|
|
|
long carryMin = carryLow.Min > carryHigh.Min ? carryLow.Min : carryHigh.Min;
|
|
long carryMax = carryLow.Max < carryHigh.Max ? carryLow.Max : carryHigh.Max;
|
|
if (carryMin > carryMax)
|
|
{
|
|
divides = false;
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return true;
|
|
}
|
|
|
|
divides = true;
|
|
reason = ContradictionReason.None;
|
|
return true;
|
|
}
|
|
|
|
internal static bool TryCheckDivisibilityFromOneOffsets(
|
|
ReadOnlySpan<int> qOneOffsets,
|
|
ulong p,
|
|
out bool divides) => TryCheckDivisibilityFromOneOffsets(qOneOffsets, p, out divides, out _);
|
|
|
|
[MethodImpl(MethodImplOptions.AggressiveInlining | MethodImplOptions.AggressiveOptimization)]
|
|
private static bool TryAdvanceZeroColumns(ref CarryRange carry, int columnCount)
|
|
{
|
|
long carryMin = carry.Min;
|
|
long carryMax = carry.Max;
|
|
while (columnCount > 0)
|
|
{
|
|
int step = columnCount > 62 ? 62 : columnCount;
|
|
long add = (1L << step) - 1L;
|
|
long nextMin = (carryMin + add) >> step;
|
|
long nextMax = carryMax >> step;
|
|
if (nextMin > nextMax)
|
|
{
|
|
return false;
|
|
}
|
|
|
|
carryMin = nextMin;
|
|
carryMax = nextMax;
|
|
columnCount -= step;
|
|
}
|
|
|
|
carry = new CarryRange(carryMin, carryMax);
|
|
return true;
|
|
}
|
|
|
|
private static bool TryProcessBottomUpBlock(
|
|
ReadOnlySpan<int> qOneOffsets,
|
|
Dictionary<long, sbyte> aBits,
|
|
long maxAllowedA,
|
|
long startColumn,
|
|
int columnCount,
|
|
ref CarryRange carry,
|
|
ref long maxKnownA,
|
|
out ContradictionReason reason)
|
|
{
|
|
long endColumn = startColumn + columnCount;
|
|
int qOneOffsetsLength = qOneOffsets.Length;
|
|
for (long column = startColumn; column < endColumn; column++)
|
|
{
|
|
long forced = 0L;
|
|
long unknown = 0L;
|
|
long chosenIndex = -1L;
|
|
|
|
for (int offsetIndex = 0; offsetIndex < qOneOffsetsLength; offsetIndex++)
|
|
{
|
|
int offset = qOneOffsets[offsetIndex];
|
|
long aIndex = column - offset;
|
|
if (aIndex < 0)
|
|
{
|
|
break;
|
|
}
|
|
|
|
if (aIndex > maxAllowedA)
|
|
{
|
|
continue; // a must be zero here
|
|
}
|
|
|
|
if (aBits.TryGetValue(aIndex, out sbyte aVal))
|
|
{
|
|
if (aVal == 1)
|
|
{
|
|
forced++;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
unknown++;
|
|
if (chosenIndex == -1L)
|
|
{
|
|
chosenIndex = aIndex;
|
|
}
|
|
}
|
|
}
|
|
|
|
long minSum = carry.Min;
|
|
long maxSum = carry.Max;
|
|
if (unknown == 1 && ((minSum ^ maxSum) & 1) == 0)
|
|
{
|
|
int requiredUnknownParity = 1 ^ ((int)minSum & 1) ^ ((int)forced & 1);
|
|
if (requiredUnknownParity != 0)
|
|
{
|
|
if (chosenIndex >= 0)
|
|
{
|
|
aBits[chosenIndex] = 1;
|
|
forced++;
|
|
unknown = 0;
|
|
if (chosenIndex > maxKnownA)
|
|
{
|
|
maxKnownA = chosenIndex;
|
|
}
|
|
}
|
|
}
|
|
else
|
|
{
|
|
unknown = 0;
|
|
}
|
|
}
|
|
|
|
minSum += forced;
|
|
maxSum += forced + unknown;
|
|
const int requiredParity = 1;
|
|
|
|
if (minSum == maxSum)
|
|
{
|
|
if ((minSum & 1) != requiredParity)
|
|
{
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return false;
|
|
}
|
|
|
|
long nextCarry = (minSum - requiredParity) >> 1;
|
|
carry = CarryRange.Single(nextCarry);
|
|
}
|
|
else
|
|
{
|
|
if ((minSum & 1) != requiredParity)
|
|
{
|
|
if (unknown == 0)
|
|
{
|
|
reason = ContradictionReason.ParityUnreachable;
|
|
return false;
|
|
}
|
|
|
|
if (chosenIndex >= 0)
|
|
{
|
|
aBits[chosenIndex] = 1;
|
|
forced++;
|
|
unknown--;
|
|
if (chosenIndex > maxKnownA)
|
|
{
|
|
maxKnownA = chosenIndex;
|
|
}
|
|
}
|
|
}
|
|
|
|
if (!TryPropagateCarry(
|
|
carry,
|
|
forced,
|
|
forced + unknown,
|
|
requiredParity,
|
|
out var next,
|
|
out var propagateReason))
|
|
{
|
|
reason = propagateReason;
|
|
return false;
|
|
}
|
|
|
|
carry = next;
|
|
}
|
|
|
|
if (chosenIndex >= 0 && !aBits.TryAdd(chosenIndex, 1))
|
|
{
|
|
if (chosenIndex > maxKnownA)
|
|
{
|
|
maxKnownA = chosenIndex;
|
|
}
|
|
}
|
|
}
|
|
|
|
reason = ContradictionReason.None;
|
|
return true;
|
|
}
|
|
|
|
private static bool TryProcessTopDownBlock(
|
|
ReadOnlySpan<int> qOneOffsets,
|
|
int qOneOffsetsLength,
|
|
long maxAllowedA,
|
|
long startHighColumn,
|
|
int columnCount,
|
|
CarryRange carryOut,
|
|
out CarryRange nextCarryOut,
|
|
out TopDownPruneFailure? failure)
|
|
{
|
|
failure = null;
|
|
long carryOutMin = carryOut.Min,
|
|
carryOutMax = carryOut.Max,
|
|
value;
|
|
int windowStart = qOneOffsetsLength;
|
|
int windowEnd = qOneOffsetsLength;
|
|
long endColumn = startHighColumn - columnCount + 1;
|
|
|
|
for (long column = startHighColumn; column >= endColumn; column--)
|
|
{
|
|
// value is used as high column index
|
|
value = column;
|
|
while (windowEnd > 0 && qOneOffsets[windowEnd - 1] > value)
|
|
{
|
|
windowEnd--;
|
|
}
|
|
|
|
// value is used as low column index
|
|
value = column - maxAllowedA;
|
|
if (value < 0)
|
|
{
|
|
value = 0;
|
|
}
|
|
|
|
while (windowStart > 0 && qOneOffsets[windowStart - 1] >= value)
|
|
{
|
|
windowStart--;
|
|
}
|
|
|
|
if (windowStart > windowEnd)
|
|
{
|
|
windowStart = windowEnd;
|
|
}
|
|
|
|
// value is unknown bits count here
|
|
value = windowEnd - windowStart;
|
|
// carryOutMin now becomes nextCarryMin to limit registry pressure
|
|
carryOutMin = (carryOutMin << 1) + 1 - value;
|
|
// carryOutMax now becomes nextCarryMax to limit registry pressure
|
|
carryOutMax = (carryOutMax << 1) + 1;
|
|
if (carryOutMin < 0)
|
|
{
|
|
carryOutMin = 0;
|
|
}
|
|
|
|
if (carryOutMax < 0)
|
|
{
|
|
carryOutMax = 0;
|
|
}
|
|
|
|
if (carryOutMin > carryOutMax)
|
|
{
|
|
failure = new TopDownPruneFailure(column, carryOutMin, carryOutMax, value);
|
|
nextCarryOut = default;
|
|
return false;
|
|
}
|
|
}
|
|
|
|
nextCarryOut = new CarryRange(carryOutMin, carryOutMax);
|
|
return true;
|
|
}
|
|
|
|
private static long AlignUpToParity(long value, int parity) => (value & 1) == parity ? value : value + 1;
|
|
|
|
private static long AlignDownToParity(long value, int parity) => (value & 1) == parity ? value : value - 1;
|
|
|
|
private static bool TryTopDownCarryPrune(
|
|
ReadOnlySpan<int> qOneOffsets,
|
|
int qOneOffsetsLength,
|
|
long pLong,
|
|
long maxAllowedA,
|
|
out TopDownPruneFailure? failure)
|
|
{
|
|
failure = null;
|
|
long carryOutMin = 0,
|
|
carryOutMax = 0,
|
|
value;
|
|
int windowStart = qOneOffsetsLength;
|
|
int windowEnd = qOneOffsetsLength;
|
|
|
|
for (long column = pLong - 1; column >= 0; column--)
|
|
{
|
|
// value is used as high column index
|
|
value = column;
|
|
while (windowEnd > 0 && qOneOffsets[windowEnd - 1] > value)
|
|
{
|
|
windowEnd--;
|
|
}
|
|
|
|
// value is used as low column index
|
|
value = column - maxAllowedA;
|
|
if (value < 0)
|
|
{
|
|
value = 0;
|
|
}
|
|
|
|
while (windowStart > 0 && qOneOffsets[windowStart - 1] >= value)
|
|
{
|
|
windowStart--;
|
|
}
|
|
|
|
if (windowStart > windowEnd)
|
|
{
|
|
windowStart = windowEnd;
|
|
}
|
|
|
|
// value is used as unknown bits count here
|
|
value = windowEnd - windowStart;
|
|
// carryOutMin now becomes nextCarryMin to limit registry pressure
|
|
carryOutMin = (carryOutMin << 1) + 1 - value;
|
|
// carryOutMax now becomes nextCarryMax to limit registry pressure
|
|
carryOutMax = (carryOutMax << 1) + 1;
|
|
if (carryOutMin < 0)
|
|
{
|
|
carryOutMin = 0;
|
|
}
|
|
|
|
if (carryOutMax < 0)
|
|
{
|
|
carryOutMax = 0;
|
|
}
|
|
|
|
if (carryOutMin > carryOutMax)
|
|
{
|
|
failure = new TopDownPruneFailure(column, carryOutMin, carryOutMax, value);
|
|
return false;
|
|
}
|
|
}
|
|
|
|
if (carryOutMin > 0)
|
|
{
|
|
failure = new TopDownPruneFailure(0, carryOutMin, carryOutMax, 0);
|
|
return false;
|
|
}
|
|
|
|
return true;
|
|
}
|
|
}
|