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/*
* DSI utilities
*
* Copyright (C) 2010-2020 Paolo Boldi and Sebastiano Vigna
*
* 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 3 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 Lesser General Public License
* for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, see HyperLogLog counters represent the number of elements of a set in an approximate way. They have been
* introduced by Philippe Flajolet, Éric Fusy, Olivier Gandouet, and Freédeéric Meunier in
* “HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm”,
* Proceedings of the 13th conference on analysis of algorithm (AofA 07), pages
* 127−146, 2007. They are an improvement over the basic idea of loglog counting, introduced by
* Marianne Durand and Philippe Flajolet in “Loglog counting of large cardinalities”,
* ESA 2003, 11th Annual European Symposium, volume 2832 of Lecture Notes in Computer Science, pages 605−617, Springer, 2003.
*
*
Each counter is composed by {@link #m} registers, and each register is made of {@link #registerSize} bits.
* The first number depends on the desired relative standard deviation, and its logarithm can be computed using {@link #log2NumberOfRegisters(double)},
* whereas the second number depends on an upper bound on the number of distinct elements to be counted, and it can be computed
* using {@link #registerSize(long)}.
*
*
Actually, this class implements an array of counters. Each counter is completely independent, but they all use the same hash function.
* The reason for this design is that in our intended applications hundred of millions of counters are common, and the JVM overhead to create such a number of objects
* would be unbearable. This class allocates an array of {@link LongArrayBitVector}s, each containing {@link #CHUNK_SIZE} registers,
* and can thus handle billions of billions of registers efficiently (in turn, this means being able to
* handle an array of millions of billions of high-precision counters).
*
*
When creating an instance, you can choose the size of the array (i.e., the number of counters) and the desired relative standard deviation
* (either {@linkplain #HyperLogLogCounterArray(long, long, double) explicitly} or
* {@linkplain #HyperLogLogCounterArray(long, long, int) choosing the number of registers per counter}).
* Then, you can {@linkplain #add(long, long) add an element to a counter}. At any time, you can
* {@linkplain #count(long) count} count (approximately) the number of distinct elements that have been added to a counter.
*
*
If you need to reuse this class multiple times, you can {@linkplain #clear() clear all registers}, possibly {@linkplain #clear(long) setting a new seed}.
* The seed is used to compute the hash function used by the HyperLogLog counters.
*
*
*
Utility methods
*
* This class provides a number of utility methods that make it possible to {@linkplain #getCounter(long, long[]) extract a counter as an array of longs},
* {@linkplain #setCounter(long[], long) set the contents of a counter given an array of longs}, and
* {@linkplain #max(long[], long[]) maximize quickly two counters given as arrays of longs}.
*
* @author Paolo Boldi
* @author Sebastiano Vigna
*/
public class HyperLogLogCounterArray implements Serializable {
private static final long serialVersionUID = 1L;
private static final boolean ASSERTS = false;
private static final boolean DEBUG = false;
/** The logarithm of the maximum size in registers of a bit vector. */
public final static int CHUNK_SHIFT = 30;
/** The maximum size in registers of a bit vector. */
public final static long CHUNK_SIZE = 1L << CHUNK_SHIFT;
/** The mask used to obtain an register offset in a chunk. */
public final static long CHUNK_MASK = CHUNK_SIZE - 1;
/** The correct value for α, multiplied by {@link #m}2 (see the paper). */
private double alphaMM;
/** The number of registers minus one. */
protected final int mMinus1;
/** An array of arrays of longs containing all registers. */
protected final long bits[][];
/** {@link #registerSize}-bit views of {@link #bits}. */
protected final LongBigList registers[];
/** The shift that selects the chunk corresponding to a counter. */
protected final int counterShift;
/** A seed for hashing. */
protected long seed;
/** The mask OR'd with the output of the hash function so that {@link Long#numberOfTrailingZeros(long)} does not return too large a value. */
private final long sentinelMask;
/** Whether counters are aligned to longwords. */
protected final boolean longwordAligned;
/** A mask for the residual bits of a counter (the {@link #counterSize} % {@link Long#SIZE} lowest bits). */
protected final long counterResidualMask;
/** A mask containing a one in the most significant bit of each register (i.e., in positions of the form {@link #registerSize registerSize * (i + 1) - 1}). */
protected final long[] msbMask;
/** A mask containing a one in the least significant bit of each register (i.e., in positions of the form {@link #registerSize registerSize * i}). */
protected final long[] lsbMask;
/** The logarithm of the number of registers per counter (at most 30). */
public final int log2m;
/** The number of registers per counter. */
public final int m;
/** The size in bits of each register. */
public final int registerSize;
/** The size in bits of each counter ({@link #registerSize} * {@link #m}). */
public final int counterSize;
/** The size of a counter in longwords (ceiled if there are less then {@link Long#SIZE} registers per counter). */
public final int counterLongwords;
/**
* Returns the logarithm of the number of registers per counter that are necessary to attain a
* given relative standard deviation.
*
* @param rsd the relative standard deviation to be attained.
* @return the logarithm of the number of registers that are necessary to attain relative standard deviation rsd.
*/
public static int log2NumberOfRegisters(final double rsd) {
// 1.106 is valid for 16 registers or more.
return (int)Math.ceil(Fast.log2((1.106 / rsd) * (1.106 / rsd)));
}
/**
* Returns the relative standard deviation corresponding to a given logarithm of the number of registers per counter.
*
* @param log2m the logarithm of the number of registers per counter (at most 30).
* @return the resulting relative standard deviation.
*/
public static double relativeStandardDeviation(final int log2m) {
return (log2m == 4 ? 1.106 : log2m == 5 ? 1.070 : log2m == 6 ? 1.054 : log2m == 7 ? 1.046 : 1.04) / Math.sqrt(1 << log2m);
}
/**
* Returns the register size in bits, given an upper bound on the number of distinct elements.
*
* @param n an upper bound on the number of distinct elements.
* @return the register size in bits.
*/
public static int registerSize(final long n) {
return Math.max(5, (int)Math.ceil(Math.log(Math.log(n) / Math.log(2)) / Math.log(2)));
}
/** Returns the chunk of a given counter.
*
* @param counter a counter.
* @return its chunk.
*/
public int chunk(final long counter) {
return (int)(counter >>> counterShift);
}
/** Returns the bit offset of a given counter in its chunk.
*
* @param counter a counter.
* @return the starting bit of the given counter in its chunk.
*/
public long offset(final long counter) {
return (counter << log2m & CHUNK_MASK) * registerSize;
}
/**
* Creates a new array of counters.
*
* @param arraySize the number of counters.
* @param n the expected number of elements.
* @param rsd the relative standard deviation.
*/
public HyperLogLogCounterArray(final long arraySize, final long n, final double rsd) {
this(arraySize, n, log2NumberOfRegisters(rsd));
}
/**
* Creates a new array of counters.
*
* @param arraySize the number of counters.
* @param n the expected number of elements.
* @param log2m the logarithm of the number of registers per counter (at most 30).
*/
public HyperLogLogCounterArray(final long arraySize, final long n, final int log2m) {
this(arraySize, n, log2m, Util.randomSeed());
}
/**
* Creates a new array of counters.
*
* @param arraySize the number of counters.
* @param n the expected number of elements.
* @param log2m the logarithm of the number of registers per counter (at most 30).
* @param seed the seed used to compute the hash function.
*/
public HyperLogLogCounterArray(final long arraySize, final long n, final int log2m, final long seed) {
if (log2m > Integer.SIZE - 2) throw new IllegalArgumentException("The logarithm of the number of register per counter (" + log2m + ") is too large");
this.m = 1 << (this.log2m = log2m);
this.mMinus1 = m - 1;
this.registerSize = registerSize(n);
this.counterSize = registerSize << log2m;
counterShift = CHUNK_SHIFT - log2m;
sentinelMask = 1L << (1 << registerSize) - 2;
// System.err.println(arraySize + " " + m + " " + registerSize);
final long sizeInRegisters = arraySize * m;
final int numVectors = (int)((sizeInRegisters + CHUNK_MASK) >>> CHUNK_SHIFT);
bits = new long[numVectors][];
registers = new LongBigList[numVectors];
for(int i = 0; i < numVectors; i++) {
final LongArrayBitVector bitVector = LongArrayBitVector.ofLength(registerSize * Math.min(CHUNK_SIZE, sizeInRegisters - ((long)i << CHUNK_SHIFT)));
bits[i] = bitVector.bits();
registers[i] = bitVector.asLongBigList(registerSize);
}
counterLongwords = (counterSize + Long.SIZE - 1) / Long.SIZE;
counterResidualMask = (1L << counterSize % Long.SIZE) - 1;
longwordAligned = counterSize % Long.SIZE == 0;
// We initialise the masks for the broadword code in max().
msbMask = new long[counterLongwords];
lsbMask = new long[counterLongwords];
for(int i = registerSize - 1; i < msbMask.length * Long.SIZE; i += registerSize) msbMask[i / Long.SIZE] |= 1L << i % Long.SIZE;
for(int i = 0; i < lsbMask.length * Long.SIZE; i += registerSize) lsbMask[i / Long.SIZE] |= 1L << i % Long.SIZE;
this.seed = seed;
if (DEBUG) System.err.println("Register size: " + registerSize + " log2m (b): " + log2m + " m: " + m);
// See the paper.
switch (log2m) {
case 4:
alphaMM = 0.673 * m * m; break;
case 5:
alphaMM = 0.697 * m * m; break;
case 6:
alphaMM = 0.709 * m * m; break;
default:
alphaMM = (0.7213 / (1 + 1.079 / m)) * m * m;
}
}
/** Clears all registers and sets a new seed (e.g., using {@link Util#randomSeed()}).
*
* @param seed the new seed used to compute the hash function
*/
public void clear(final long seed) {
clear();
this.seed = seed;
}
/** Clears all registers. */
public void clear() {
for(final long[] a: bits) Arrays.fill(a, 0);
}
private final static long jenkins(final long x, final long seed) {
long a, b, c;
/* Set up the internal state */
a = seed + x;
b = seed;
c = 0x9e3779b97f4a7c13L; /* the golden ratio; an arbitrary value */
a -= b; a -= c; a ^= (c >>> 43);
b -= c; b -= a; b ^= (a << 9);
c -= a; c -= b; c ^= (b >>> 8);
a -= b; a -= c; a ^= (c >>> 38);
b -= c; b -= a; b ^= (a << 23);
c -= a; c -= b; c ^= (b >>> 5);
a -= b; a -= c; a ^= (c >>> 35);
b -= c; b -= a; b ^= (a << 49);
c -= a; c -= b; c ^= (b >>> 11);
a -= b; a -= c; a ^= (c >>> 12);
b -= c; b -= a; b ^= (a << 18);
c -= a; c -= b; c ^= (b >>> 22);
return c;
}
/** Adds an element to a counter.
*
* @param k the index of the counter.
* @param v the element to be added.
*/
public void add(final long k, final long v) {
final long x = jenkins(v, seed);
final int j = (int)(x & mMinus1);
final int r = Long.numberOfTrailingZeros(x >>> log2m | sentinelMask);
if (ASSERTS) assert r < (1 << registerSize) - 1;
if (ASSERTS) assert r >= 0;
final LongBigList l = registers[chunk(k)];
final long offset = ((k << log2m) + j) & CHUNK_MASK;
l.set(offset, Math.max(r + 1, l.getLong(offset)));
}
/** Returns the array of big lists of registers underlying this array of counters.
*
*
The main purpose of this method is debugging, as it makes comparing
* the evolution of the state of two implementations easy.
*
* @return the array of big lists of registers underlying this array of counters.
*/
public LongBigList[] registers() {
return registers;
}
/** Estimates the number of distinct elements that have been added to a given counter so far.
*
*
This is an low-level method that should be used only after having understood in detail
* the inner workings of this class.
*
* @param bits the bit array containing the counter.
* @param offset the starting bit position of the counter in bits.
* @return an approximation of the number of distinct elements that have been added to counter so far.
*/
public double count(final long[] bits, final long offset) {
int remaining = (int)(Long.SIZE - offset % Long.SIZE);
int word = (int)(offset / Long.SIZE);
long curr = bits[word] >>> offset % Long.SIZE;
final int registerSize = this.registerSize;
final int mask = (1 << registerSize) - 1;
double s = 0;
int zeroes = 0;
long r;
for (int j = m; j-- != 0;) {
if (remaining >= registerSize) {
r = curr & mask;
curr >>>= registerSize;
remaining -= registerSize;
}
else {
r = (curr | bits[++word] << remaining) & mask;
curr = bits[word] >>> registerSize - remaining;
remaining += Long.SIZE - registerSize;
}
// if (ASSERTS) assert r == registers[chunk(k)].getLong(offset(k) + j) : "[" + j + "] " + r + "!=" + registers[chunk(k)].getLong(offset(k) + j);
if (r == 0) zeroes++;
s += 1. / (1L << r);
}
s = alphaMM / s;
if (DEBUG) System.err.println("Zeroes: " + zeroes);
if (zeroes != 0 && s < 5. * m / 2) {
if (DEBUG) System.err.println("Small range correction");
return m * Math.log((double)m / zeroes);
}
else return s;
}
/** Estimates the number of distinct elements that have been added to a given counter so far.
*
* @param k the index of the counter.
* @return an approximation of the number of distinct elements that have been added to counter k so far.
*/
public double count(final long k) {
return count(bits[chunk(k)], offset(k));
}
/** Sets the contents of a counter of this {@link HyperLogLogCounterArray} using a provided array of longs.
*
*
Warning: this is a low-level method. You must know what you're doing.
*
* @param source an array of at least {@link #counterLongwords} longs containing a counter.
* @param chunkBits the array where the counter will be stored.
* @param index the index of the counter that will be filled with the provided array.
* @see #getCounter(long[], long, long[])
*/
public final void setCounter(final long[] source, final long[] chunkBits, final long index) {
if (longwordAligned) System.arraycopy(source, 0, chunkBits, (int)(offset(index) / Long.SIZE), counterLongwords);
else {
// Offset in bits
final long offset = offset(index);
// Offsets in elements in the array
final int longwordOffset = (int)(offset / Long.SIZE);
// Offset in bits in the word of index longwordOffset
final int bitOffset = (int)(offset % Long.SIZE);
final int last = counterLongwords - 1;
if (bitOffset == 0) {
for(int i = last; i-- != 0;) chunkBits[longwordOffset + i] = source[i];
chunkBits[longwordOffset + last] &= ~counterResidualMask;
chunkBits[longwordOffset + last] |= source[last] & counterResidualMask;
}
else {
chunkBits[longwordOffset] &= (1L << bitOffset) - 1;
chunkBits[longwordOffset] |= source[0] << bitOffset;
for(int i = 1; i < last; i++) chunkBits[longwordOffset + i] = source[i - 1] >>> Long.SIZE - bitOffset | source[i] << bitOffset;
final int remaining = counterSize % Long.SIZE + bitOffset;
final long mask = -1L >>> (Long.SIZE - Math.min(Long.SIZE, remaining));
chunkBits[longwordOffset + last] &= ~mask;
chunkBits[longwordOffset + last] |= mask & (source[last - 1] >>> Long.SIZE - bitOffset | source[last] << bitOffset);
// Note that it is impossible to enter in this conditional unless you use 7 or more bits per register, which is unlikely.
if (remaining > Long.SIZE) {
final long mask2 = (1L << remaining - Long.SIZE) - 1;
chunkBits[longwordOffset + last + 1] &= ~mask2;
chunkBits[longwordOffset + last + 1] |= mask2 & (source[last] >>> Long.SIZE - bitOffset);
}
}
if (ASSERTS) {
final LongArrayBitVector l = LongArrayBitVector.wrap(chunkBits);
for(int i = 0; i < counterSize; i++) assert l.getBoolean(offset + i) == ((source[i / Long.SIZE] & (1L << i % Long.SIZE)) != 0);
}
}
}
/** Sets the contents of a counter of this {@link HyperLogLogCounterArray} using a provided array of longs.
*
*
Warning: this is a low-level method. You must know what you're doing.
*
* @param source an array of at least {@link #counterLongwords} longs containing a counter.
* @param index the index of the counter that will be filled with the provided array.
* @see #setCounter(long[], long[], long)
*/
public void setCounter(final long[] source, final long index) {
setCounter(source, bits[chunk(index)], index);
}
/** Extracts a counter from this {@link HyperLogLogCounterArray} and writes it into an array of longs.
*
*
Warning: this is a low-level method. You must know what you're doing.
*
* @param chunkBits the array storing the counter.
* @param index the index of the counter to be extracted.
* @param dest an array of at least {@link #counterLongwords} longs where the counter of given index will be written.
* @see #setCounter(long[], long[], long)
*/
public final void getCounter(final long[] chunkBits, final long index, final long[] dest) {
if (longwordAligned) System.arraycopy(chunkBits, (int)(offset(index) / Long.SIZE), dest, 0, counterLongwords);
else {
// Offset in bits
final long offset = offset(index);
// Offsets in elements in the array
final int longwordOffset = (int)(offset / Long.SIZE);
// Offset in bits in the word of index longwordOffset
final int bitOffset = (int)(offset % Long.SIZE);
final int last = counterLongwords - 1;
if (bitOffset == 0) {
for(int i = last; i-- != 0;) dest[i] = chunkBits[longwordOffset + i];
dest[last] = chunkBits[longwordOffset + last] & counterResidualMask;
}
else {
for(int i = 0; i < last; i++) dest[i] = chunkBits[longwordOffset + i] >>> bitOffset | chunkBits[longwordOffset + i + 1] << Long.SIZE - bitOffset;
dest[last] = chunkBits[longwordOffset + last] >>> bitOffset & counterResidualMask;
}
}
}
/** Extracts a counter from this {@link HyperLogLogCounterArray} and writes it into an array of longs.
*
*
Warning: this is a low-level method. You must know what you're doing.
*
* @param index the index of the counter to be extracted.
* @param dest an array of at least {@link #counterLongwords} longs where the counter of given index will be written.
* @see #getCounter(long[], long, long[])
*/
public final void getCounter(final long index, final long[] dest) {
getCounter(bits[chunk(index)], index, dest);
}
/** Transfers the content of a counter between two parallel array of longwords.
*
*
Warning: this is a low-level method. You must know what you're doing.
*
* @param source the source array.
* @param dest the destination array.
* @param node the node number.
*/
public final void transfer(final long[] source, final long[] dest, final long node) {
if (longwordAligned) {
final int longwordOffset = (int)(offset(node) / Long.SIZE);
System.arraycopy(source, longwordOffset, dest, longwordOffset, counterLongwords);
}
else {
// Offset in bits in the array
final long offset = offset(node);
// Offsets in elements in the array
final int longwordOffset = (int)(offset / Long.SIZE);
// Offset in bits in the word of index longwordOffset
final int bitOffset = (int)(offset % Long.SIZE);
final int last = counterLongwords - 1;
if (bitOffset == 0) {
for(int i = last; i-- != 0;) dest[longwordOffset + i] = source[longwordOffset + i];
dest[longwordOffset + last] &= ~counterResidualMask;
dest[longwordOffset + last] |= source[longwordOffset + last] & counterResidualMask;
}
else {
final long mask = -1L << bitOffset;
dest[longwordOffset] &= ~mask;
dest[longwordOffset] |= source[longwordOffset] & mask;
for(int i = 1; i < last; i++) dest[longwordOffset + i] = source[longwordOffset + i];
final int remaining = (counterSize + bitOffset) % Long.SIZE;
if (remaining == 0) dest[longwordOffset + last] = source[longwordOffset + last];
else {
final long mask2 = (1L << remaining) - 1;
dest[longwordOffset + last] &= ~mask2;
dest[longwordOffset + last] |= mask2 & source[longwordOffset + last];
}
}
if (ASSERTS) {
final LongArrayBitVector aa = LongArrayBitVector.wrap(source);
final LongArrayBitVector bb = LongArrayBitVector.wrap(dest);
for(int i = 0; i < counterSize; i++) assert aa.getBoolean(offset + i) == bb.getBoolean(offset + i);
}
}
}
/** Performs a multiple precision subtraction, leaving the result in the first operand.
*
* @param x an array of longs.
* @param y an array of longs that will be subtracted from x.
* @param l the length of x and y.
*/
private static final void subtract(final long[] x, final long[] y, final int l) {
boolean borrow = false;
for(int i = 0; i < l; i++) {
if (! borrow || x[i]-- != 0) borrow = x[i] < y[i] ^ x[i] < 0 ^ y[i] < 0; // This expression returns the result of an unsigned strict comparison.
x[i] -= y[i];
}
}
/** Computes the register-by-register maximum of two counters.
*
*
This method will allocate two temporary arrays. To reduce object creation, use {@link #max(long[], long[], long[], long[])}.
*
* @param x a first array of at least {@link #counterLongwords} longs containing a counter.
* @param y a second array of at least {@link #counterLongwords} longs containing a counter.
*/
public final void max(final long[] x, final long[] y) {
max(x, y, new long[x.length], new long[y.length]);
}
/** Computes the register-by-register maximum of two counters.
*
* @param x a first array of at least {@link #counterLongwords} longs containing a counter.
* @param y a second array of at least {@link #counterLongwords} longs containing a counter.
* @param accumulator a support array of at least {@link #counterLongwords} longs.
* @param mask a support array of at least {@link #counterLongwords} longs.
*/
public final void max(final long[] x, final long[] y, final long[] accumulator, final long[] mask) {
final int l = x.length;
final long[] msbMask = this.msbMask;
/* We work in two phases. Let H_r (msbMask) by the mask with the
* highest bit of each register (of size r) set, and L_r (lsbMask)
* be the mask with the lowest bit of each register set.
* We describe the algorithm on a single word.
*
* If the first phase we perform an unsigned strict register-by-register
* comparison of x and y, using the formula
*
* z = ((((y | H_r) - (x & ~H_r)) | (y ^ x))^ (y | ~x)) & H_r
*
* Then, we generate a register-by-register mask of all ones or
* all zeroes, depending on the result of the comparison, using the
* formula
*
* (((z >> r-1 | H_r) - L_r) | H_r) ^ z
*
* At that point, it is trivial to select from x and y the right values.
*/
// We load y | H_r into the accumulator.
for(int i = l; i-- != 0;) accumulator[i] = y[i] | msbMask[i];
// We subtract x & ~H_r, using mask as temporary storage
for(int i = l; i-- != 0;) mask[i] = x[i] & ~msbMask[i];
subtract(accumulator, mask, l);
// We OR with x ^ y, XOR with (x | ~y), and finally AND with H_r.
for(int i = l; i-- != 0;) accumulator[i] = ((accumulator[i] | (y[i] ^ x[i])) ^ (y[i] | ~x[i])) & msbMask[i];
if (ASSERTS) {
final LongBigList a = LongArrayBitVector.wrap(x).asLongBigList(registerSize);
final LongBigList b = LongArrayBitVector.wrap(y).asLongBigList(registerSize);
for(int i = 0; i < m; i++) {
final long pos = (i + 1) * (long)registerSize - 1;
assert (b.getLong(i) < a.getLong(i)) == ((accumulator[(int)(pos / Long.SIZE)] & 1L << pos % Long.SIZE) != 0);
}
}
// We shift by registerSize - 1 places and put the result into mask.
final int rMinus1 = registerSize - 1, longSizeMinusRMinus1 = Long.SIZE - rMinus1;
for(int i = l - 1; i-- != 0;) mask[i] = accumulator[i] >>> rMinus1 | accumulator[i + 1] << longSizeMinusRMinus1 | msbMask[i];
mask[l - 1] = accumulator[l - 1] >>> rMinus1 | msbMask[l - 1];
// We subtract L_r from mask.
subtract(mask, lsbMask, l);
// We OR with H_r and XOR with the accumulator.
for(int i = l; i-- != 0;) mask[i] = (mask[i] | msbMask[i]) ^ accumulator[i];
if (ASSERTS) {
final long[] t = x.clone();
final LongBigList a = LongArrayBitVector.wrap(t).asLongBigList(registerSize);
final LongBigList b = LongArrayBitVector.wrap(y).asLongBigList(registerSize);
for(int i = 0; i < Long.SIZE * l / registerSize; i++) a.set(i, Math.max(a.getLong(i), b.getLong(i)));
// Note: this must be kept in sync with the line computing the result.
for(int i = l; i-- != 0;) assert t[i] == (~mask[i] & x[i] | mask[i] & y[i]);
}
// Finally, we use mask to select the right bits from x and y and store the result.
for(int i = l; i-- != 0;) x[i] ^= (x[i] ^ y[i]) & mask[i];
}
}