text: /*-------------------------------------------------------------------------
   *
   * bloomfilter.c
   *		Space-efficient set membership testing
   *
   * A Bloom filter is a probabilistic data structure that is used to test an
   * element's membership of a set.  False positives are possible, but false
   * negatives are not; a test of membership of the set returns either "possibly
   * in set" or "definitely not in set".  This is typically very space efficient,
   * which can be a decisive advantage.
   *
   * Elements can be added to the set, but not removed.  The more elements that
   * are added, the larger the probability of false positives.  Caller must hint
   * an estimated total size of the set when the Bloom filter is initialized.
   * This is used to balance the use of memory against the final false positive
   * rate.
   *
   * The implementation is well suited to data synchronization problems between
   * unordered sets, especially where predictable performance is important and
   * some false positives are acceptable.  It's also well suited to cache
   * filtering problems where a relatively small and/or low cardinality set is
   * fingerprinted, especially when many subsequent membership tests end up
   * indicating that values of interest are not present.  That should save the
   * caller many authoritative lookups, such as expensive probes of a much larger
   * on-disk structure.
   *
   * Copyright (c) 2018, PostgreSQL Global Development Group
   *
   * IDENTIFICATION
   *	  src/backend/lib/bloomfilter.c
   *
   *-------------------------------------------------------------------------
   */
  #include "postgres.h"
  
  #include <math.h>
  
  #include "access/hash.h"
  #include "lib/bloomfilter.h"
  
  #define MAX_HASH_FUNCS		10
  
  struct bloom_filter
  {
  	/* K hash functions are used, seeded by caller's seed */
  	int			k_hash_funcs;
  	uint64		seed;
  	/* m is bitset size, in bits.  Must be a power of two <= 2^32.  */
  	uint64		m;
  	unsigned char bitset[FLEXIBLE_ARRAY_MEMBER];
  };
  
  static int	my_bloom_power(uint64 target_bitset_bits);
  static int	optimal_k(uint64 bitset_bits, int64 total_elems);
  static void k_hashes(bloom_filter *filter, uint32 *hashes, unsigned char *elem,
  		 size_t len);
  static inline uint32 mod_m(uint32 a, uint64 m);
  
  /*
   * Create Bloom filter in caller's memory context.  We aim for a false positive
   * rate of between 1% and 2% when bitset size is not constrained by memory
   * availability.
   *
   * total_elems is an estimate of the final size of the set.  It should be
   * approximately correct, but the implementation can cope well with it being
   * off by perhaps a factor of five or more.  See "Bloom Filters in
   * Probabilistic Verification" (Dillinger & Manolios, 2004) for details of why
   * this is the case.
   *
   * bloom_work_mem is sized in KB, in line with the general work_mem convention.
   * This determines the size of the underlying bitset (trivial bookkeeping space
   * isn't counted).  The bitset is always sized as a power of two number of
   * bits, and the largest possible bitset is 512MB (2^32 bits).  The
   * implementation allocates only enough memory to target its standard false
   * positive rate, using a simple formula with caller's total_elems estimate as
   * an input.  The bitset might be as small as 1MB, even when bloom_work_mem is
   * much higher.
   *
   * The Bloom filter is seeded using a value provided by the caller.  Using a
   * distinct seed value on every call makes it unlikely that the same false
   * positives will reoccur when the same set is fingerprinted a second time.
   * Callers that don't care about this pass a constant as their seed, typically
   * 0.  Callers can use a pseudo-random seed in the range of 0 - INT_MAX by
   * calling random().
   */
  bloom_filter *
  bloom_create(int64 total_elems, int bloom_work_mem, uint64 seed)
  {
  	bloom_filter *filter;
  	int			bloom_power;
  	uint64		bitset_bytes;
  	uint64		bitset_bits;
  
  	/*
  	 * Aim for two bytes per element; this is sufficient to get a false
  	 * positive rate below 1%, independent of the size of the bitset or total
  	 * number of elements.  Also, if rounding down the size of the bitset to
  	 * the next lowest power of two turns out to be a significant drop, the
  	 * false positive rate still won't exceed 2% in almost all cases.
  	 */
  	bitset_bytes = Min(bloom_work_mem * UINT64CONST(1024), total_elems * 2);
  	bitset_bytes = Max(1024 * 1024, bitset_bytes);
  
  	/*
  	 * Size in bits should be the highest power of two <= target.  bitset_bits
  	 * is uint64 because PG_UINT32_MAX is 2^32 - 1, not 2^32
  	 */
  	bloom_power = my_bloom_power(bitset_bytes * BITS_PER_BYTE);
  	bitset_bits = UINT64CONST(1) << bloom_power;
  	bitset_bytes = bitset_bits / BITS_PER_BYTE;
  
  	/* Allocate bloom filter with unset bitset */
  	filter = palloc0(offsetof(bloom_filter, bitset) +
  					 sizeof(unsigned char) * bitset_bytes);
  	filter->k_hash_funcs = optimal_k(bitset_bits, total_elems);
  	filter->seed = seed;
  	filter->m = bitset_bits;
  
  	return filter;
  }
  
  /*
   * Free Bloom filter
   */
  void
  bloom_free(bloom_filter *filter)
  {
  	pfree(filter);
  }
  
  /*
   * Add element to Bloom filter
   */
  void
  bloom_add_element(bloom_filter *filter, unsigned char *elem, size_t len)
  {
  	uint32		hashes[MAX_HASH_FUNCS];
  	int			i;
  
  	k_hashes(filter, hashes, elem, len);
  
  	/* Map a bit-wise address to a byte-wise address + bit offset */
  	for (i = 0; i < filter->k_hash_funcs; i++)
  	{
  		filter->bitset[hashes[i] >> 3] |= 1 << (hashes[i] & 7);
  	}
  }
  
  /*
   * Test if Bloom filter definitely lacks element.
   *
   * Returns true if the element is definitely not in the set of elements
   * observed by bloom_add_element().  Otherwise, returns false, indicating that
   * element is probably present in set.
   */
  bool
  bloom_lacks_element(bloom_filter *filter, unsigned char *elem, size_t len)
  {
  	uint32		hashes[MAX_HASH_FUNCS];
  	int			i;
  
  	k_hashes(filter, hashes, elem, len);
  
  	/* Map a bit-wise address to a byte-wise address + bit offset */
  	for (i = 0; i < filter->k_hash_funcs; i++)
  	{
  		if (!(filter->bitset[hashes[i] >> 3] & (1 << (hashes[i] & 7))))
  			return true;
  	}
  
  	return false;
  }
  
  /*
   * What proportion of bits are currently set?
   *
   * Returns proportion, expressed as a multiplier of filter size.  That should
   * generally be close to 0.5, even when we have more than enough memory to
   * ensure a false positive rate within target 1% to 2% band, since more hash
   * functions are used as more memory is available per element.
   *
   * This is the only instrumentation that is low overhead enough to appear in
   * debug traces.  When debugging Bloom filter code, it's likely to be far more
   * interesting to directly test the false positive rate.
   */
  double
  bloom_prop_bits_set(bloom_filter *filter)
  {
  	int			bitset_bytes = filter->m / BITS_PER_BYTE;
  	uint64		bits_set = 0;
  	int			i;
  
  	for (i = 0; i < bitset_bytes; i++)
  	{
  		unsigned char byte = filter->bitset[i];
  
  		while (byte)
  		{
  			bits_set++;
  			byte &= (byte - 1);
  		}
  	}
  
  	return bits_set / (double) filter->m;
  }
  
  /*
   * Which element in the sequence of powers of two is less than or equal to
   * target_bitset_bits?
   *
   * Value returned here must be generally safe as the basis for actual bitset
   * size.
   *
   * Bitset is never allowed to exceed 2 ^ 32 bits (512MB).  This is sufficient
   * for the needs of all current callers, and allows us to use 32-bit hash
   * functions.  It also makes it easy to stay under the MaxAllocSize restriction
   * (caller needs to leave room for non-bitset fields that appear before
   * flexible array member, so a 1GB bitset would use an allocation that just
   * exceeds MaxAllocSize).
   */
  static int
  my_bloom_power(uint64 target_bitset_bits)
  {
  	int			bloom_power = -1;
  
  	while (target_bitset_bits > 0 && bloom_power < 32)
  	{
  		bloom_power++;
  		target_bitset_bits >>= 1;
  	}
  
  	return bloom_power;
  }
  
  /*
   * Determine optimal number of hash functions based on size of filter in bits,
   * and projected total number of elements.  The optimal number is the number
   * that minimizes the false positive rate.
   */
  static int
  optimal_k(uint64 bitset_bits, int64 total_elems)
  {
  	int			k = rint(log(2.0) * bitset_bits / total_elems);
  
  	return Max(1, Min(k, MAX_HASH_FUNCS));
  }
  
  /*
   * Generate k hash values for element.
   *
   * Caller passes array, which is filled-in with k values determined by hashing
   * caller's element.
   *
   * Only 2 real independent hash functions are actually used to support an
   * interface of up to MAX_HASH_FUNCS hash functions; enhanced double hashing is
   * used to make this work.  The main reason we prefer enhanced double hashing
   * to classic double hashing is that the latter has an issue with collisions
   * when using power of two sized bitsets.  See Dillinger & Manolios for full
   * details.
   */
  static void
  k_hashes(bloom_filter *filter, uint32 *hashes, unsigned char *elem, size_t len)
  {
  	uint64		hash;
  	uint32		x, y;
  	uint64		m;
  	int			i;
  
  	/* Use 64-bit hashing to get two independent 32-bit hashes */
  	hash = DatumGetUInt64(hash_any_extended(elem, len, filter->seed));
  	x = (uint32) hash;
  	y = (uint32) (hash >> 32);
  	m = filter->m;
  
  	x = mod_m(x, m);
  	y = mod_m(y, m);
  
  	/* Accumulate hashes */
  	hashes[0] = x;
  	for (i = 1; i < filter->k_hash_funcs; i++)
  	{
  		x = mod_m(x + y, m);
  		y = mod_m(y + i, m);
  
  		hashes[i] = x;
  	}
  }
  
  /*
   * Calculate "val MOD m" inexpensively.
   *
   * Assumes that m (which is bitset size) is a power of two.
   *
   * Using a power of two number of bits for bitset size allows us to use bitwise
   * AND operations to calculate the modulo of a hash value.  It's also a simple
   * way of avoiding the modulo bias effect.
   */
  static inline uint32
  mod_m(uint32 val, uint64 m)
  {
  	Assert(m <= PG_UINT32_MAX + UINT64CONST(1));
  	Assert(((m - 1) & m) == 0);
  
  	return val & (m - 1);
  }
  