org.apfloat.internal.LongKaratsubaConvolutionStrategy Maven / Gradle / Ivy
/*
* Apfloat arbitrary precision arithmetic library
* Copyright (C) 2002-2017 Mikko Tommila
*
* This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This library is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
package org.apfloat.internal;
import org.apfloat.ApfloatContext;
import org.apfloat.ApfloatRuntimeException;
import org.apfloat.spi.DataStorageBuilder;
import org.apfloat.spi.DataStorage;
/**
* Convolution strategy using the Karatsuba algorithm.
* The complexity of the algorithm is O(nlog(3)/log(2)) as
* the operands are split to two and multiplied using three multiplications
* (and five additions / subtractions). This splitting is done recursively
* until some cut-off point where the basic O(n2) algorithm is
* applied. The Karatsuba algorithm is faster than the basic O(n2)
* multiplication algorithm for medium size numbers larger than some certain
* size. For very large numbers, the transform-based convolution algorithms
* are faster.
*
* @since 1.4
* @version 1.4
* @author Mikko Tommila
*/
public class LongKaratsubaConvolutionStrategy
extends LongMediumConvolutionStrategy
{
/**
* Cut-off point for Karatsuba / basic convolution.
*
* Convolutions where the shorter number is at most this long
* are calculated using the basic O(n2) algorithm
* i.e. super.convolute().
*/
public static final int CUTOFF_POINT = 15;
/**
* Creates a convolution strategy using the specified radix.
*
* @param radix The radix that will be used.
*/
public LongKaratsubaConvolutionStrategy(int radix)
{
super(radix);
}
@Override
public DataStorage convolute(DataStorage x, DataStorage y, long resultSize)
throws ApfloatRuntimeException
{
if (Math.min(x.getSize(), y.getSize()) <= CUTOFF_POINT)
{
// The numbers are too short for Karatsuba to have any advantage, fall back to O(n^2) algorithm
return super.convolute(x, y, resultSize);
}
DataStorage shortStorage, longStorage;
if (x.getSize() > y.getSize())
{
shortStorage = y;
longStorage = x;
}
else
{
shortStorage = x;
longStorage = y;
}
long shortSize = shortStorage.getSize(),
longSize = longStorage.getSize(),
size = shortSize + longSize,
halfSize = longSize + 1 >> 1, // Split point for recursion, round up
x1size = longSize - halfSize,
x2size = halfSize,
y1size = shortSize - halfSize; // y2size = halfSize
ApfloatContext ctx = ApfloatContext.getContext();
DataStorageBuilder dataStorageBuilder = ctx.getBuilderFactory().getDataStorageBuilder();
DataStorage resultStorage = dataStorageBuilder.createDataStorage(size * 8);
resultStorage.setSize(size);
if (y1size <= 0)
{
// The shorter number is half of the longer number or less, use simplified algorithm
DataStorage.Iterator dst = resultStorage.iterator(DataStorage.WRITE, size, 0),
src1 = null;
long carry = 0;
long i = longSize,
xSize;
// Calculate sub-results in blocks of size shortSize
do
{
xSize = Math.min(i, shortSize);
x = longStorage.subsequence(i - xSize, xSize);
y = shortStorage;
// Calculate sub-convolutions recursively
DataStorage a = convolute(x, y, xSize + shortSize);
assert (a.getSize() == xSize + shortSize);
// Add the sub-results together
DataStorage.Iterator src2 = a.iterator(DataStorage.READ, xSize + shortSize, 0);
carry = baseAdd(src1, src2, carry, dst, shortSize);
src1 = src2;
i -= shortSize;
} while (i > 0);
// Propagate carry through the last sub-result and store to result data
carry = baseAdd(src1, null, carry, dst, xSize);
assert (carry == 0);
}
else
{
// The numbers are roughly equal size (shorter is more than half of the longer), use Karatsuba algorithm
DataStorage x1 = longStorage.subsequence(0, x1size),
x2 = longStorage.subsequence(x1size, x2size),
y1 = shortStorage.subsequence(0, y1size),
y2 = shortStorage.subsequence(y1size, halfSize);
// Calculate a = x1 + x2
DataStorage a = add(x1, x2);
// Calculate b = y1 + y2
DataStorage b = add(y1, y2);
// Calculate sub-convolutions recursively
DataStorage c = convolute(a, b, a.getSize() + b.getSize());
a = convolute(x1, y1, x1size + y1size);
b = convolute(x2, y2, 2 * halfSize);
// Calculate c = c - a - b
subtract(c, a);
subtract(c, b);
long cSize = c.getSize(),
c1size = cSize - halfSize;
if (c1size > x1size + y1size)
{
// We know that the top one or two words of c are zero
// Omit them to avoid later having c1size > x1size + y1size
long zeros = c1size - x1size - y1size;
assert (isZero(c, 0));
assert (zeros == 1 || isZero(c, 1));
assert (zeros <= 2);
cSize -= zeros;
c1size -= zeros;
c = c.subsequence(zeros, cSize);
}
assert (a.getSize() == x1size + y1size);
assert (b.getSize() == 2 * halfSize);
assert (cSize >= 2 * halfSize && cSize <= 2 * halfSize + 2);
assert (c1size <= x1size + y1size);
// Add the sub-results a + b + c together
DataStorage.Iterator src1 = a.iterator(DataStorage.READ, x1size + y1size, 0),
src2 = b.iterator(DataStorage.READ, 2 * halfSize, 0),
src3 = c.iterator(DataStorage.READ, cSize, 0),
dst = resultStorage.iterator(DataStorage.WRITE, size, 0);
long carry = 0;
carry = baseAdd(src2, null, carry, dst, halfSize);
carry = baseAdd(src2, src3, carry, dst, halfSize);
carry = baseAdd(src1, src3, carry, dst, c1size);
carry = baseAdd(src1, null, carry, dst, x1size + y1size - c1size);
assert (carry == 0);
}
return resultStorage;
}
// Return x1 + x2
private DataStorage add(DataStorage x1, DataStorage x2)
{
long x1size = x1.getSize(),
x2size = x2.getSize();
assert (x1size <= x2size);
long size = x2size + 1;
ApfloatContext ctx = ApfloatContext.getContext();
DataStorageBuilder dataStorageBuilder = ctx.getBuilderFactory().getDataStorageBuilder();
DataStorage resultStorage = dataStorageBuilder.createDataStorage(size * 8);
resultStorage.setSize(size);
// Calculate x1 + x2
DataStorage.Iterator src1 = x1.iterator(DataStorage.READ, x1size, 0),
src2 = x2.iterator(DataStorage.READ, x2size, 0),
dst = resultStorage.iterator(DataStorage.WRITE, size, 0);
long carry = 0;
carry = baseAdd(src1, src2, carry, dst, x1size);
carry = baseAdd(src2, null, carry, dst, x2size - x1size);
baseAdd(null, null, carry, dst, 1); // Set carry digit to the top word
if (carry == 0)
{
resultStorage = resultStorage.subsequence(1, size - 1); // Omit zero top word
}
return resultStorage;
}
// x1 -= x2
private void subtract(DataStorage x1, DataStorage x2)
{
long x1size = x1.getSize(),
x2size = x2.getSize();
assert (x1size >= x2size);
DataStorage.Iterator src1 = x1.iterator(DataStorage.READ_WRITE, x1size, 0),
src2 = x2.iterator(DataStorage.READ, x2size, 0),
dst = src1;
long carry = 0;
carry = baseSubtract(src1, src2, carry, dst, x2size);
carry = baseSubtract(src1, null, carry, dst, x1size - x2size);
assert (carry == 0);
}
private boolean isZero(DataStorage x, long index)
{
DataStorage.Iterator i = x.iterator(DataStorage.READ, index, index + 1);
long data = i.getLong();
i.next();
return data == 0;
}
private static final long serialVersionUID = -4812398042499004749L;
}