using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.Json;

namespace DataStructures.Probabilistic
{
    public class BloomFilter<T> where T : notnull
    {
        private const uint FnvPrime = 16777619;
        private const uint FnvOffsetBasis = 2166136261;
        private readonly byte[] filter;
        private readonly int numHashes;
        private readonly int sizeBits;

        /// <summary>
        /// Initializes a new instance of the <see cref="BloomFilter{T}"/> class. This constructor will create a Bloom Filter
        /// of an optimal size with the optimal number of hashes to minimize the error rate.
        /// </summary>
        /// <param name="expectedNumElements">Expected number of unique elements that could be added to the filter.</param>
        public BloomFilter(int expectedNumElements)
        {
            numHashes = (int)Math.Ceiling(.693 * 8 * expectedNumElements / expectedNumElements); // compute optimal number of hashes
            filter = new byte[expectedNumElements]; // set up filter with 8 times as many bits as elements
            sizeBits = expectedNumElements * 8; // number of bit slots in the filter
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="BloomFilter{T}"/> class.
        /// This constructor let's you decide how large you want the filter to be as well as allowing you to specify
        /// how many hashes it will use. Only use if you don't care to optimize false positivity.
        /// </summary>
        /// <param name="sizeBits">size in bits you want the filter to be.</param>
        /// <param name="numHashes">number of hash functions to be used.</param>
        public BloomFilter(int sizeBits, int numHashes)
        {
            filter = new byte[sizeBits / 8 + 1];
            this.numHashes = numHashes;
            this.sizeBits = sizeBits;
        }

        /// <summary>
        /// Inserts an item into the bloom filter.
        /// </summary>
        /// <param name="item">The item being inserted into the Bloom Filter.</param>
        public void Insert(T item)
        {
            foreach (var slot in GetSlots(item))
            {
                filter[slot / 8] |= (byte)(1 << (slot % 8)); // set the filter at the decided slot to 1.
            }
        }

        /// <summary>
        /// Searches the Bloom Filter to determine if the item exists in the Bloom Filter.
        /// </summary>
        /// <param name="item">The item being searched for in the Bloom Filter.</param>
        /// <returns>true if the item has been added to the Bloom Filter, false otherwise.</returns>
        public bool Search(T item)
        {
            foreach (var slot in GetSlots(item))
            {
                var @byte = filter[slot / 8]; // Extract the byte in the filter.
                var mask = 1 << (slot % 8); // Build the mask for the slot number.
                if ((@byte & mask) != mask)
                {
                    return false;
                }
            }

            return true;
        }

        /// <summary>
        /// Yields the appropriate slots for the given item.
        /// </summary>
        /// <param name="item">The item to determine the slots for.</param>
        /// <returns>The slots of the filter to flip or check.</returns>
        private IEnumerable<int> GetSlots(T item)
        {
            var hash = item.GetHashCode();
            for (var i = 0; i < numHashes; i++)
            {
                yield return Math.Abs((i + 1) * hash) % sizeBits;
            }
        }
    }
}

Bloom Filters are one of a class of probabilistic data structures. The Bloom Filter uses hashes and probability to determine whether a particular item is present in a set. It can do so in constant time: O(1) and sub-linear space, though technically still O(n). An important feature of a Bloom Filter is that it is guaranteed never to provide a false negative, saying an element isn't present when it is. However, it has a probability (based on the tuning of its parameters) of providing a false positive, saying an element is present when it is not. The Bloom Filter uses a multi-hash scheme. On insertion, the inserted object is run through each hash, which produces a slot number. That slot number is flipped to 1 in the bit array. During a presence check, the object is run through the same set of hashes, and if each corresponding slot is 1, the filter reports the object has been added. If any of them are 0, it reports that the object has not been added. The hashes must be deterministic and uniformly distributed over the slots for the Bloom filter to operate effectively.

Complexity

Operation	Average
Initialize	O(1)
Insertion	O(1)
Query	O(1)
Space	O(n)

Steps

Initialization

1. Bloom Filter is Initialized, with a number of hash functions that will be run against it (henceforth known as k), and with an array of bits of size M with each bit set to 0. There are 3 distinct schemes to tune these parameters.
   1. M and k are explicitly set by the user
   2. k and M are calculated based off the expected number of elements to minimize false positives.
   3. k and M are calculated based off a desired error rate.

Insertion

1. Object is run through k hashes
2. For each result of the hash n determine the slot within the filter m by calculating n % M = m
3. Set slot m within the filter to 1

Query

1. Object is run through k hashes
2. For each result of the hash n determine the slot within the filter m by calculating n % M = m
3. Check slot m, if m is set to 0 return false
4. Return true

Example

Initialize

As an example, let us look at a Bloom Filter of Strings, we will initialize the Bloom Filter with 10 slots an we will use 3 hashes

slot	0	1	2	3	4	5	6	7	8	9
state	0	0	0	0	0	0	0	0	0	0

Insert

Let's try to insert foo, we will run foo through our three hash functions

h1(foo) = 2
h2(foo) = 5
h3(foo) = 6

With hashes run, we will flip the corresponding bits to 1

slot	0	1	2	3	4	5	6	7	8	9
state	0	0	1	0	0	1	1	0	0	0

Query

Query bar

Let's first try querying bar, to query bar we run bar through our three hash functions:

h1(bar) = 3
h2(bar) = 4
h3(bar) = 6

If we look at our bit array, bits 3 and 4 are both not set, if even just 1 bit is not set, we return false, so in this case we return false. bar has not been added

Query foo

Let's now try to query foo, when we run foo through our hashes we get:

h1(foo) = 2
h2(foo) = 5
h3(foo) = 6

Of course, since we already inserted foo, our table has each of the three bits our hashes produced set to 1, so we return true, foo is present

False Positive

Let's say we inserted bar and the current state of our table is:

slot	0	1	2	3	4	5	6	7	8	9
state	0	0	1	1	1	1	1	0	0	0

Let's now query baz, when we run baz through our hash functions we get:

h1(baz) = 3
h2(baz) = 5
h3(baz) = 6

Notice that this does not match either the result of foo or bar, however because slots 3, 5, and 6 are already set, we report true, that baz is in the set, and therefore produce a false positive.

Advantage Over HashSets

• Significantly more space-efficient, Both are technically O(n) space complexity, but since bloom filters will only take up several bits per item, hash sets must hold the entire item.
• Presence checks are guaranteed to be O(1) for Bloom Filters, for HashSets, the average is O(1), but worst case is O(n)

Disadvantage v.s. Hash Sets

• Bloom Filters can report false positives. Optimally there should be about a 1% false-positive rate.
• Bloom Filters do not store the objects inserted into it, so you cannot recover items inserted.

Optimizing

The probability of false positives increases with the probability of hash collisions within the filter. However, you can optimize the number of collisions if you have some sense of the cardinality of your set ahead of time. You can do this by optimizing k and M, M should be ~ 8-10 bits per expected item, and k should be (M/n) * ln2.

Examples

Implementations of the Bloom Filter are available for:

• C#

Video Explainer

Video Explainer by Narendra L