Bloom filter
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (November 2009) 
Part of a series on 
Probabilistic data structures 


Random trees 
Related 
Computer science portal 
A Bloom filter is a spaceefficient probabilistic data structure, conceived by Burton Howard Bloom in 1970, that is used to test whether an element is a member of a set. False positive matches are possible, but false negatives are not, thus a Bloom filter has a 100% recall rate. In other words, a query returns either "possibly in set" or "definitely not in set". Elements can be added to the set, but not removed (though this can be addressed with a "counting" filter). The more elements that are added to the set, the larger the probability of false positives.
Bloom proposed the technique for applications where the amount of source data would require an impractically large amount of memory if "conventional" errorfree hashing techniques were applied. He gave the example of a hyphenation algorithm for a dictionary of 500,000 words, out of which 90% follow simple hyphenation rules, but the remaining 10% require expensive disk accesses to retrieve specific hyphenation patterns. With sufficient core memory, an errorfree hash could be used to eliminate all unnecessary disk accesses; on the other hand, with limited core memory, Bloom's technique uses a smaller hash area but still eliminates most unnecessary accesses. For example, a hash area only 15% of the size needed by an ideal errorfree hash still eliminates 85% of the disk accesses, an 85–15 form of the Pareto principle (Bloom (1970)).
More generally, fewer than 10 bits per element are required for a 1% false positive probability, independent of the size or number of elements in the set (Bonomi et al. (2006)).
Contents
 1 Algorithm description
 2 Space and time advantages
 3 Probability of false positives
 4 Approximating the number of items in a Bloom filter
 5 The union and intersection of sets
 6 Interesting properties
 7 Examples
 8 Alternatives
 9 Extensions and applications
 10 See also
 11 Notes
 12 References
 13 External links
Algorithm description
An empty Bloom filter is a bit array of m bits, all set to 0. There must also be k different hash functions defined, each of which maps or hashes some set element to one of the m array positions with a uniform random distribution.
To add an element, feed it to each of the k hash functions to get k array positions. Set the bits at all these positions to 1.
To query for an element (test whether it is in the set), feed it to each of the k hash functions to get k array positions. If any of the bits at these positions is 0, the element is definitely not in the set – if it were, then all the bits would have been set to 1 when it was inserted. If all are 1, then either the element is in the set, or the bits have by chance been set to 1 during the insertion of other elements, resulting in a false positive. In a simple Bloom filter, there is no way to distinguish between the two cases, but more advanced techniques can address this problem.
The requirement of designing k different independent hash functions can be prohibitive for large k. For a good hash function with a wide output, there should be little if any correlation between different bitfields of such a hash, so this type of hash can be used to generate multiple "different" hash functions by slicing its output into multiple bit fields. Alternatively, one can pass k different initial values (such as 0, 1, ..., k − 1) to a hash function that takes an initial value; or add (or append) these values to the key. For larger m and/or k, independence among the hash functions can be relaxed with negligible increase in false positive rate (Dillinger & Manolios (2004a), Kirsch & Mitzenmacher (2006)). Specifically, Dillinger & Manolios (2004b) show the effectiveness of deriving the k indices using enhanced double hashing or triple hashing, variants of double hashing that are effectively simple random number generators seeded with the two or three hash values.
Removing an element from this simple Bloom filter is impossible because false negatives are not permitted. An element maps to k bits, and although setting any one of those k bits to zero suffices to remove the element, it also results in removing any other elements that happen to map onto that bit. Since there is no way of determining whether any other elements have been added that affect the bits for an element to be removed, clearing any of the bits would introduce the possibility for false negatives.
Onetime removal of an element from a Bloom filter can be simulated by having a second Bloom filter that contains items that have been removed. However, false positives in the second filter become false negatives in the composite filter, which may be undesirable. In this approach readding a previously removed item is not possible, as one would have to remove it from the "removed" filter.
It is often the case that all the keys are available but are expensive to enumerate (for example, requiring many disk reads). When the false positive rate gets too high, the filter can be regenerated; this should be a relatively rare event.
Space and time advantages
While risking false positives, Bloom filters have a strong space advantage over other data structures for representing sets, such as selfbalancing binary search trees, tries, hash tables, or simple arrays or linked lists of the entries. Most of these require storing at least the data items themselves, which can require anywhere from a small number of bits, for small integers, to an arbitrary number of bits, such as for strings (tries are an exception, since they can share storage between elements with equal prefixes). However, Bloom filters do not store the data items at all, and a separate solution must be provided for the actual storage. Linked structures incur an additional linear space overhead for pointers. A Bloom filter with 1% error and an optimal value of k, in contrast, requires only about 9.6 bits per element, regardless of the size of the elements. This advantage comes partly from its compactness, inherited from arrays, and partly from its probabilistic nature. The 1% falsepositive rate can be reduced by a factor of ten by adding only about 4.8 bits per element.
However, if the number of potential values is small and many of them can be in the set, the Bloom filter is easily surpassed by the deterministic bit array, which requires only one bit for each potential element. Note also that hash tables gain a space and time advantage if they begin ignoring collisions and store only whether each bucket contains an entry; in this case, they have effectively become Bloom filters with k = 1.^{[1]}
Bloom filters also have the unusual property that the time needed either to add items or to check whether an item is in the set is a fixed constant, O(k), completely independent of the number of items already in the set. No other constantspace set data structure has this property, but the average access time of sparse hash tables can make them faster in practice than some Bloom filters. In a hardware implementation, however, the Bloom filter shines because its k lookups are independent and can be parallelized.
To understand its space efficiency, it is instructive to compare the general Bloom filter with its special case when k = 1. If k = 1, then in order to keep the false positive rate sufficiently low, a small fraction of bits should be set, which means the array must be very large and contain long runs of zeros. The information content of the array relative to its size is low. The generalized Bloom filter (k greater than 1) allows many more bits to be set while still maintaining a low false positive rate; if the parameters (k and m) are chosen well, about half of the bits will be set,^{[2]} and these will be apparently random, minimizing redundancy and maximizing information content.
Probability of false positives
Assume that a hash function selects each array position with equal probability. If m is the number of bits in the array, the probability that a certain bit is not set to 1 by a certain hash function during the insertion of an element is
If k is the number of hash functions, the probability that the bit is not set to 1 by any of the hash functions is
If we have inserted n elements, the probability that a certain bit is still 0 is
the probability that it is 1 is therefore
Now test membership of an element that is not in the set. Each of the k array positions computed by the hash functions is 1 with a probability as above. The probability of all of them being 1, which would cause the algorithm to erroneously claim that the element is in the set, is often given as
This is not strictly correct as it assumes independence for the probabilities of each bit being set. However, assuming it is a close approximation we have that the probability of false positives decreases as m (the number of bits in the array) increases, and increases as n (the number of inserted elements) increases.
An alternative analysis arriving at the same approximation without the assumption of independence is given by Mitzenmacher and Upfal.^{[3]} After all n items have been added to the Bloom filter, let q be the fraction of the m bits that are set to 0. (That is, the number of bits still set to 0 is qm.) Then, when testing membership of an element not in the set, for the array position given by any of the k hash functions, the probability that the bit is found set to 1 is . So the probability that all k hash functions find their bit set to 1 is . Further, the expected value of q is the probability that a given array position is left untouched by each of the k hash functions for each of the n items, which is (as above)
 .
It is possible to prove, without the independence assumption, that q is very strongly concentrated around its expected value. In particular, from the Azuma–Hoeffding inequality, they prove that^{[4]}
Because of this, we can say that the exact probability of false positives is
as before.
Optimal number of hash functions
For a given m and n, the value of k (the number of hash functions) that minimizes the false positive probability is
which gives
The required number of bits m, given n (the number of inserted elements) and a desired false positive probability p (and assuming the optimal value of k is used) can be computed by substituting the optimal value of k in the probability expression above:
which can be simplified to:
This results in:
This means that for a given false positive probability p, the length of a Bloom filter m is proportionate to the number of elements being filtered n.^{[5]} While the above formula is asymptotic (i.e. applicable as m,n → ∞), the agreement with finite values of m,n is also quite good; the false positive probability for a finite Bloom filter with m bits, n elements, and k hash functions is at most
So we can use the asymptotic formula if we pay a penalty for at most half an extra element and at most one fewer bit.^{[6]}
Approximating the number of items in a Bloom filter
Swamidass & Baldi (2007) showed that the number of items in a Bloom filter can be approximated with the following formula,
where is an estimate of the number of items in the filter, m is the length (size) of the filter, k is the number of hash functions, and X is the number of bits set to one.
The union and intersection of sets
Bloom filters are a way of compactly representing a set of items. It is common to try to compute the size of the intersection or union between two sets. Bloom filters can be used to approximate the size of the intersection and union of two sets. Swamidass & Baldi (2007) showed that for two Bloom filters of length , their counts, respectively can be estimated as
and
 .
The size of their union can be estimated as
 ,
where is the number of bits set to one in either of the two Bloom filters. Finally, the intersection can be estimated as
 ,
using the three formulas together.
Interesting properties
 Unlike a standard hash table, a Bloom filter of a fixed size can represent a set with an arbitrarily large number of elements; adding an element never fails due to the data structure "filling up." However, the false positive rate increases steadily as elements are added until all bits in the filter are set to 1, at which point all queries yield a positive result.
 Union and intersection of Bloom filters with the same size and set of hash functions can be implemented with bitwise OR and AND operations, respectively. The union operation on Bloom filters is lossless in the sense that the resulting Bloom filter is the same as the Bloom filter created from scratch using the union of the two sets. The intersect operation satisfies a weaker property: the false positive probability in the resulting Bloom filter is at most the falsepositive probability in one of the constituent Bloom filters, but may be larger than the false positive probability in the Bloom filter created from scratch using the intersection of the two sets.
 Some kinds of superimposed code can be seen as a Bloom filter implemented with physical edgenotched cards. An example is Zatocoding, invented by Calvin Mooers in 1947, in which the set of categories associated with a piece of information is represented by notches on a card, with a random pattern of four notches for each category.
Examples
 Akamai's web servers use Bloom filters to prevent "onehitwonders" from being stored in its disk caches.^{[7]} Onehitwonders are web objects requested by users just once.^{[7]} Nearly threequarters of urls accessed from a typical web cache are onehitwonders! Using a Bloom filter to detect the second request for a web object and caching that object only on its second request prevents onehit wonders from entering the disk cache, significantly reducing disk workload and increasing disk cache hit rates.^{[7]}
 Google BigTable, Apache HBase and Apache Cassandra use Bloom filters to reduce the disk lookups for nonexistent rows or columns. Avoiding costly disk lookups considerably increases the performance of a database query operation.^{[8]}^{[9]}
 The Google Chrome web browser used to use a Bloom filter to identify malicious URLs. Any URL was first checked against a local Bloom filter, and only if the Bloom filter returned a positive result was a full check of the URL performed (and the user warned, if that too returned a positive result).^{[10]}^{[11]}
 The Squid Web Proxy Cache uses Bloom filters for cache digests.^{[12]}
 Bitcoin uses Bloom filters to speed up wallet synchronization.^{[13]}^{[14]}
 The Venti archival storage system uses Bloom filters to detect previously stored data.^{[15]}
 The SPIN model checker uses Bloom filters to track the reachable state space for large verification problems.^{[16]}
 The Cascading analytics framework uses Bloom filters to speed up asymmetric joins, where one of the joined data sets is significantly larger than the other (often called Bloom join^{[17]} in the database literature).^{[18]}
 The Exim mail transfer agent (MTA) uses Bloom filters in its ratelimit feature.^{[19]}
 Medium uses Bloom filters to avoid recommending articles a user has previously read.^{[20]}
Alternatives
Classic Bloom filters use bits of space per inserted key, where is the false positive rate of the Bloom filter. However, the space that is strictly necessary for any data structure playing the same role as a Bloom filter is only per key (Pagh, Pagh & Rao 2005). Hence Bloom filters use 44% more space than a hypothetical equivalent optimal data structure. The number of hash functions used to achieve a given false positive rate is proportional to which is not optimal as it has been proved that an optimal data structure would need only a constant number of hash functions independent of the false positive rate.
Stern & Dill (1996) describe a probabilistic structure based on hash tables, hash compaction, which Dillinger & Manolios (2004b) identify as significantly more accurate than a Bloom filter when each is configured optimally. Dillinger and Manolios, however, point out that the reasonable accuracy of any given Bloom filter over a wide range of numbers of additions makes it attractive for probabilistic enumeration of state spaces of unknown size. Hash compaction is, therefore, attractive when the number of additions can be predicted accurately; however, despite being very fast in software, hash compaction is poorly suited for hardware because of worstcase linear access time.
Putze, Sanders & Singler (2007) have studied some variants of Bloom filters that are either faster or use less space than classic Bloom filters. The basic idea of the fast variant is to locate the k hash values associated with each key into one or two blocks having the same size as processor's memory cache blocks (usually 64 bytes). This will presumably improve performance by reducing the number of potential memory cache misses. The proposed variants have however the drawback of using about 32% more space than classic Bloom filters.
The space efficient variant relies on using a single hash function that generates for each key a value in the range where is the requested false positive rate. The sequence of values is then sorted and compressed using Golomb coding (or some other compression technique) to occupy a space close to bits. To query the Bloom filter for a given key, it will suffice to check if its corresponding value is stored in the Bloom filter. Decompressing the whole Bloom filter for each query would make this variant totally unusable. To overcome this problem the sequence of values is divided into small blocks of equal size that are compressed separately. At query time only half a block will need to be decompressed on average. Because of decompression overhead, this variant may be slower than classic Bloom filters but this may be compensated by the fact that a single hash function need to be computed.
Another alternative to classic Bloom filter is the one based on space efficient variants of cuckoo hashing. In this case once the hash table is constructed, the keys stored in the hash table are replaced with short signatures of the keys. Those signatures are strings of bits computed using a hash function applied on the keys.
Extensions and applications
Cache Filtering
Content delivery networks deploy web caches around the world to cache and serve web content to users with greater performance and reliability. A key application of Bloom filters is their use in efficiently determining what web objects to store in these web caches.^{[7]} Nearly threequarters of the URLs accessed from a typical web cache are "onehitwonders" that are accessed by users only once and never again! It is clearly wasteful of disk resources to store onehitwonders in a web cache, since they will never be accessed again. To prevent caching onehitwonders, a Bloom filter is used to keep track of all URLs that are accessed by users. A web object is cached only when it has been accessed at least once before, i.e., the object is cached on its second request. The use of a Bloom filter in this fashion significantly reduces the disk write workload, since onehitwonders are never written to the disk cache. Further, filtering out the onehitwonders also saves cache space on disk, increasing the cache hit rates.^{[7]}
Counting filters
Counting filters provide a way to implement a delete operation on a Bloom filter without recreating the filter afresh. In a counting filter the array positions (buckets) are extended from being a single bit to being an nbit counter. In fact, regular Bloom filters can be considered as counting filters with a bucket size of one bit. Counting filters were introduced by Fan et al. (1998).
The insert operation is extended to increment the value of the buckets, and the lookup operation checks that each of the required buckets is nonzero. The delete operation then consists of decrementing the value of each of the respective buckets.
Arithmetic overflow of the buckets is a problem and the buckets should be sufficiently large to make this case rare. If it does occur then the increment and decrement operations must leave the bucket set to the maximum possible value in order to retain the properties of a Bloom filter.
The size of counters is usually 3 or 4 bits. Hence counting Bloom filters use 3 to 4 times more space than static Bloom filters. In theory, an optimal data structure equivalent to a counting Bloom filter should not use more space than a static Bloom filter.
Another issue with counting filters is limited scalability. Because the counting Bloom filter table cannot be expanded, the maximal number of keys to be stored simultaneously in the filter must be known in advance. Once the designed capacity of the table is exceeded, the false positive rate will grow rapidly as more keys are inserted.
Bonomi et al. (2006) introduced a data structure based on dleft hashing that is functionally equivalent but uses approximately half as much space as counting Bloom filters. The scalability issue does not occur in this data structure. Once the designed capacity is exceeded, the keys could be reinserted in a new hash table of double size.
The space efficient variant by Putze, Sanders & Singler (2007) could also be used to implement counting filters by supporting insertions and deletions.
Rottenstreich, Kanizo & Keslassy (2012) introduced a new general method based on variable increments that significantly improves the false positive probability of counting Bloom filters and their variants, while still supporting deletions. Unlike counting Bloom filters, at each element insertion, the hashed counters are incremented by a hashed variable increment instead of a unit increment. To query an element, the exact values of the counters are considered and not just their positiveness. If a sum represented by a counter value cannot be composed of the corresponding variable increment for the queried element, a negative answer can be returned to the query.
Decentralized aggregation
Bloom filters can be organized in distributed data structures to perform fully decentralized computations of aggregate functions. Decentralized aggregation makes collective measurements locally available in every node of a distributed network without involving a centralized computational entity for this purpose.^{[22]}^{[23]}
Data synchronization
Bloom filters can be used for approximate data synchronization as in Byers et al. (2004). Counting Bloom filters can be used to approximate the number of differences between two sets and this approach is described in Agarwal & Trachtenberg (2006).
Bloomier filters
Chazelle et al. (2004) designed a generalization of Bloom filters that could associate a value with each element that had been inserted, implementing an associative array. Like Bloom filters, these structures achieve a small space overhead by accepting a small probability of false positives. In the case of "Bloomier filters", a false positive is defined as returning a result when the key is not in the map. The map will never return the wrong value for a key that is in the map.
Compact approximators
Boldi & Vigna (2005) proposed a latticebased generalization of Bloom filters. A compact approximator associates to each key an element of a lattice (the standard Bloom filters being the case of the Boolean twoelement lattice). Instead of a bit array, they have an array of lattice elements. When adding a new association between a key and an element of the lattice, they compute the maximum of the current contents of the k array locations associated to the key with the lattice element. When reading the value associated to a key, they compute the minimum of the values found in the k locations associated to the key. The resulting value approximates from above the original value.
Stable Bloom filters
Deng & Rafiei (2006) proposed Stable Bloom filters as a variant of Bloom filters for streaming data. The idea is that since there is no way to store the entire history of a stream (which can be infinite), Stable Bloom filters continuously evict stale information to make room for more recent elements. Since stale information is evicted, the Stable Bloom filter introduces false negatives, which do not appear in traditional Bloom filters. The authors show that a tight upper bound of false positive rates is guaranteed, and the method is superior to standard Bloom filters in terms of false positive rates and time efficiency when a small space and an acceptable false positive rate are given.
Scalable Bloom filters
Almeida et al. (2007) proposed a variant of Bloom filters that can adapt dynamically to the number of elements stored, while assuring a minimum false positive probability. The technique is based on sequences of standard bloom filters with increasing capacity and tighter false positive probabilities, so as to ensure that a maximum false positive probability can be set beforehand, regardless of the number of elements to be inserted.
Layered Bloom filters
A layered Bloom filter consists of multiple Bloom filter layers. Layered Bloom filters allow keeping track of how many times an item was added to the Bloom filter by checking how many layers contain the item. With a layered Bloom filter a check operation will normally return the deepest layer number the item was found in.^{[24]}^{[25]}
Attenuated Bloom filters
An attenuated Bloom filter of depth D can be viewed as an array of D normal Bloom filters. In the context of service discovery in a network, each node stores regular and attenuated Bloom filters locally. The regular or local Bloom filter indicates which services are offered by the node itself. The attenuated filter of level i indicates which services can be found on nodes that are ihops away from the current node. The ith value is constructed by taking a union of local Bloom filters for nodes ihops away from the node.^{[26]}
Let's take a small network shown on the graph below as an example. Say we are searching for a service A whose id hashes to bits 0,1, and 3 (pattern 11010). Let n1 node to be the starting point. First, we check whether service A is offered by n1 by checking its local filter. Since the patterns don't match, we check the attenuated Bloom filter in order to determine which node should be the next hop. We see that n2 doesn't offer service A but lies on the path to nodes that do. Hence, we move to n2 and repeat the same procedure. We quickly find that n3 offers the service, and hence the destination is located.^{[27]}
By using attenuated Bloom filters consisting of multiple layers, services at more than one hop distance can be discovered while avoiding saturation of the Bloom filter by attenuating (shifting out) bits set by sources further away.^{[26]}
Chemical structure searching
Bloom filters are often used to search large chemical structure databases (see chemical similarity). In the simplest case, the elements added to the filter (called a fingerprint in this field) are just the atomic numbers present in the molecule, or a hash based on the atomic number of each atom and the number and type of its bonds. This case is too simple to be useful. More advanced filters also encode atom counts, larger substructure features like carboxyl groups, and graph properties like the number of rings. In hashbased fingerprints, a hash function based on atom and bond properties is used to turn a subgraph into a PRNG seed, and the first output values used to set bits in the Bloom filter.
Molecular fingerprints arose as a way to screen out obvious rejects in molecular subgraph searches. They are more often used in calculating molecular similarity, computed as the Tanimoto similarity between their respective fingerprints. The Daylight and Indigo fingerprints, use a variable length (which are restricted to be powers of two) that adapts to the number of items to ensure the final filter is not oversaturated. A length for the filter is chosen to keep the saturation of the filter approximately at a fixed value (for example 30%).
Strictly speaking, the MACCS keys and CACTVS keys are not Bloom filters, rather they are deterministic bitarrays. Similarly, they are often incorrectly considered to be molecular fingerprints, but they are actually "structural keys". They do not use hash functions, but use a dictionary to map specific substructures to specific indices in the filter. In contrast with Bloom filters, this is a onetoone mapping that does not use hash functions at all.
See also
Notes
 ↑ Mitzenmacher & Upfal (2005).
 ↑ Blustein & ElMaazawi (2002), pp. 21–22
 ↑ Mitzenmacher & Upfal (2005), pp. 109–111, 308.
 ↑ Mitzenmacher & Upfal (2005), p. 308.
 ↑ Starobinski, Trachtenberg & Agarwal (2003)
 ↑ Goel & Gupta (2010).
 ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} ^{7.4} "Bruce M. Maggs and Ramesh K. Sitaraman, Algorithmic nuggets in content delivery, ACM SIGCOMM Computer Communication Review (CCR), July 2015." (PDF).
 ↑ (Chang et al. 2006).
 ↑ "11.6. Schema Design". apache.org.
 ↑ Yakunin, Alex (20100325). "Alex Yakunin's blog: Nice Bloom filter application". Blog.alexyakunin.com. Retrieved 20140531.
 ↑ "Issue 10896048: Transition safe browsing from bloom filter to prefix set.  Code Review". Chromiumcodereview.appspot.com. Retrieved 20140703.
 ↑ Wessels, Duane (January 2004), "10.7 Cache Digests", Squid: The Definitive Guide (1st ed.), O'Reilly Media, p. 172, ISBN 0596001622,
Cache Digests are based on a technique first published by Pei Cao, called Summary Cache. The fundamental idea is to use a Bloom filter to represent the cache contents.
 ↑ Bitcoin 0.8.0
 ↑ "The Bitcoin Foundation  Supporting the development of Bitcoin". bitcoinfoundation.org.
 ↑ "Plan 9 /sys/man/8/venti". Plan9.belllabs.com. Retrieved 20140531.
 ↑ http://spinroot.com/
 ↑ Mullin (1990)
 ↑ "BloomJoin: BloomFilter + CoGroup  LiveRamp Blog". Blog.liveramp.com. Retrieved 20140531.
 ↑ "Exim source code". github. Retrieved 20140303.
 ↑ "What are Bloom filters?". Medium. Retrieved 20151101.
 ↑ "Bruce M. Maggs and Ramesh K. Sitaraman, Algorithmic nuggets in content delivery, ACM SIGCOMM Computer Communication Review (CCR), July 2015." (PDF).
 ↑ Pournaras, Warnier & Brazier (2013)
 ↑ "DIAS  Dynamic Intelligent Aggregation Service".
 ↑ "pmylund/gobloom". Github.com. Retrieved 20140613.
 ↑ First Name Middle Name Last Name (20100822). "IEEE Xplore Abstract  A multilayer bloom filter for duplicated URL detection". Ieeexplore.ieee.org. doi:10.1109/ICACTE.2010.5578947. Retrieved 20140613.
 ↑ ^{26.0} ^{26.1} Koucheryavy et al. (2009)
 ↑ Kubiatowicz et al. (2000)
References
 Koucheryavy, Y.; Giambene, G.; Staehle, D.; BarceloArroyo, F.; Braun, T.; Siris, V. (2009), "Traffic and QoS Management in Wireless Multimedia Networks", COST 290 Final Report, USA: 111
 Kubiatowicz, J.; Bindel, D.; Czerwinski, Y.; Geels, S.; Eaton, D.; Gummadi, R.; Rhea, S.; Weatherspoon, H.; et al. (2000), "Oceanstore: An architecture for globalscale persistent storage" (PDF), ACM SIGPLAN Notices, USA: 190–201
 Agarwal, Sachin; Trachtenberg, Ari (2006), "Approximating the number of differences between remote sets" (PDF), IEEE Information Theory Workshop, Punta del Este, Uruguay: 217, ISBN 142440035X, doi:10.1109/ITW.2006.1633815
 Ahmadi, Mahmood; Wong, Stephan (2007), "A Cache Architecture for Counting Bloom Filters", 15th international Conference on Networks (ICON2007), p. 218, ISBN 9781424412297, doi:10.1109/ICON.2007.4444089
 Almeida, Paulo; Baquero, Carlos; Preguica, Nuno; Hutchison, David (2007), "Scalable Bloom Filters" (PDF), Information Processing Letters, 101 (6): 255–261, doi:10.1016/j.ipl.2006.10.007
 Byers, John W.; Considine, Jeffrey; Mitzenmacher, Michael; Rost, Stanislav (2004), "Informed content delivery across adaptive overlay networks", IEEE/ACM Transactions on Networking, 12 (5): 767, doi:10.1109/TNET.2004.836103
 Bloom, Burton H. (1970), "Space/Time Tradeoffs in Hash Coding with Allowable Errors", Communications of the ACM, 13 (7): 422–426, doi:10.1145/362686.362692
 Boldi, Paolo; Vigna, Sebastiano (2005), "Mutable strings in Java: design, implementation and lightweight textsearch algorithms", Science of Computer Programming, 54 (1): 3–23, doi:10.1016/j.scico.2004.05.003
 Bonomi, Flavio; Mitzenmacher, Michael; Panigrahy, Rina; Singh, Sushil; Varghese, George (2006), "An Improved Construction for Counting Bloom Filters", Algorithms – ESA 2006, 14th Annual European Symposium (PDF), Lecture Notes in Computer Science, 4168, pp. 684–695, ISBN 9783540388753, doi:10.1007/11841036_61
 Broder, Andrei; Mitzenmacher, Michael (2005), "Network Applications of Bloom Filters: A Survey" (PDF), Internet Mathematics, 1 (4): 485–509, doi:10.1080/15427951.2004.10129096
 Chang, Fay; Dean, Jeffrey; Ghemawat, Sanjay; Hsieh, Wilson; Wallach, Deborah; Burrows, Mike; Chandra, Tushar; Fikes, Andrew; Gruber, Robert (2006), "Bigtable: A Distributed Storage System for Structured Data", Seventh Symposium on Operating System Design and Implementation
 Charles, Denis; Chellapilla, Kumar (2008), "Bloomier Filters: A second look", The Computing Research Repository (CoRR), arXiv:0807.0928
 Chazelle, Bernard; Kilian, Joe; Rubinfeld, Ronitt; Tal, Ayellet (2004), "The Bloomier filter: an efficient data structure for static support lookup tables", Proceedings of the Fifteenth Annual ACMSIAM Symposium on Discrete Algorithms (PDF), pp. 30–39
 Cohen, Saar; Matias, Yossi (2003), "Spectral Bloom Filters", Proceedings of the 2003 ACM SIGMOD International Conference on Management of Data (PDF), pp. 241–252, ISBN 158113634X, doi:10.1145/872757.872787^{[dead link]}
 Deng, Fan; Rafiei, Davood (2006), "Approximately Detecting Duplicates for Streaming Data using Stable Bloom Filters", Proceedings of the ACM SIGMOD Conference (PDF), pp. 25–36
 Dharmapurikar, Sarang; Song, Haoyu; Turner, Jonathan; Lockwood, John (2006), "Fast packet classification using Bloom filters", Proceedings of the 2006 ACM/IEEE Symposium on Architecture for Networking and Communications Systems (PDF), pp. 61–70, ISBN 1595935800, doi:10.1145/1185347.1185356
 Dietzfelbinger, Martin; Pagh, Rasmus (2008), "Succinct Data Structures for Retrieval and Approximate Membership", The Computing Research Repository (CoRR), arXiv:0803.3693
 Swamidass, S. Joshua; Baldi, Pierre (2007), "Mathematical correction for fingerprint similarity measures to improve chemical retrieval", Journal of chemical information and modeling, ACS Publications, 47 (3): 952–964, PMID 17444629, doi:10.1021/ci600526a
 Dillinger, Peter C.; Manolios, Panagiotis (2004a), "Fast and Accurate Bitstate Verification for SPIN", Proceedings of the 11th International Spin Workshop on Model Checking Software, SpringerVerlag, Lecture Notes in Computer Science 2989
 Dillinger, Peter C.; Manolios, Panagiotis (2004b), "Bloom Filters in Probabilistic Verification", Proceedings of the 5th International Conference on Formal Methods in ComputerAided Design, SpringerVerlag, Lecture Notes in Computer Science 3312
 Donnet, Benoit; Baynat, Bruno; Friedman, Timur (2006), "Retouched Bloom Filters: Allowing Networked Applications to Flexibly Trade Off False Positives Against False Negatives", CoNEXT 06 – 2nd Conference on Future Networking Technologies
 Eppstein, David; Goodrich, Michael T. (2007), "Spaceefficient straggler identification in roundtrip data streams via Newton's identities and invertible Bloom filters", Algorithms and Data Structures, 10th International Workshop, WADS 2007, SpringerVerlag, Lecture Notes in Computer Science 4619, pp. 637–648, arXiv:0704.3313
 Fan, Li; Cao, Pei; Almeida, Jussara; Broder, Andrei (2000), "Summary Cache: A Scalable WideArea Web Cache Sharing Protocol", IEEE/ACM Transactions on Networking, 8 (3): 281–293, doi:10.1109/90.851975. A preliminary version appeared at SIGCOMM '98.
 Goel, Ashish; Gupta, Pankaj (2010), "Small subset queries and bloom filters using ternary associative memories, with applications", ACM Sigmetrics 2010, 38: 143, doi:10.1145/1811099.1811056
 Kirsch, Adam; Mitzenmacher, Michael (2006), "Less Hashing, Same Performance: Building a Better Bloom Filter", in Azar, Yossi; Erlebach, Thomas, Algorithms – ESA 2006, 14th Annual European Symposium (PDF), Lecture Notes in Computer Science, 4168, SpringerVerlag, Lecture Notes in Computer Science 4168, pp. 456–467, ISBN 9783540388753, doi:10.1007/11841036
 Mortensen, Christian Worm; Pagh, Rasmus; Pătraşcu, Mihai (2005), "On dynamic range reporting in one dimension", Proceedings of the Thirtyseventh Annual ACM Symposium on Theory of Computing, pp. 104–111, ISBN 1581139608, doi:10.1145/1060590.1060606
 Pagh, Anna; Pagh, Rasmus; Rao, S. Srinivasa (2005), "An optimal Bloom filter replacement", Proceedings of the Sixteenth Annual ACMSIAM Symposium on Discrete Algorithms (PDF), pp. 823–829
 Porat, Ely (2008), "An Optimal Bloom Filter Replacement Based on Matrix Solving", The Computing Research Repository (CoRR), arXiv:0804.1845
 Putze, F.; Sanders, P.; Singler, J. (2007), "Cache, Hash and SpaceEfficient Bloom Filters", in Demetrescu, Camil, Experimental Algorithms, 6th International Workshop, WEA 2007 (PDF), Lecture Notes in Computer Science, 4525, SpringerVerlag, Lecture Notes in Computer Science 4525, pp. 108–121, ISBN 9783540728443, doi:10.1007/9783540728450
 Sethumadhavan, Simha; Desikan, Rajagopalan; Burger, Doug; Moore, Charles R.; Keckler, Stephen W. (2003), "Scalable hardware memory disambiguation for high ILP processors", 36th Annual IEEE/ACM International Symposium on Microarchitecture, 2003, MICRO36 (PDF), pp. 399–410, ISBN 076952043X, doi:10.1109/MICRO.2003.1253244
 Shanmugasundaram, Kulesh; Brönnimann, Hervé; Memon, Nasir (2004), "Payload attribution via hierarchical Bloom filters", Proceedings of the 11th ACM Conference on Computer and Communications Security, pp. 31–41, ISBN 1581139616, doi:10.1145/1030083.1030089
 Starobinski, David; Trachtenberg, Ari; Agarwal, Sachin (2003), "Efficient PDA Synchronization", IEEE Transactions on Mobile Computing, 2 (1): 40, doi:10.1109/TMC.2003.1195150
 Stern, Ulrich; Dill, David L. (1996), "A New Scheme for MemoryEfficient Probabilistic Verification", Proceedings of Formal Description Techniques for Distributed Systems and Communication Protocols, and Protocol Specification, Testing, and Verification: IFIP TC6/WG6.1 Joint International Conference, Chapman & Hall, IFIP Conference Proceedings, pp. 333–348, CiteSeerX: 10.1.1.47.4101
 Haghighat, Mohammad Hashem; Tavakoli, Mehdi; Kharrazi, Mehdi (2013), "Payload Attribution via Character Dependent MultiBloom Filters", Transaction on Information Forensics and Security, IEEE, 99 (5): 705, doi:10.1109/TIFS.2013.2252341
 Mitzenmacher, Michael; Upfal, Eli (2005), Probability and computing: Randomized algorithms and probabilistic analysis, Cambridge University Press, pp. 107–112, ISBN 9780521835404
 Mullin, James K. (1990), "Optimal semijoins for distributed database systems", Software Engineering, IEEE Transactions on, 16 (5): 558–560, doi:10.1109/32.52778
 Rottenstreich, Ori; Kanizo, Yossi; Keslassy, Isaac (2012), "The VariableIncrement Counting Bloom Filter", 31st Annual IEEE International Conference on Computer Communications, 2012, Infocom 2012 (PDF), pp. 1880–1888, ISBN 9781467307734, doi:10.1109/INFCOM.2012.6195563
 Blustein, James; ElMaazawi, Amal (2002), "optimal case for general Bloom filters", Bloom Filters — A Tutorial, Analysis, and Survey, Dalhousie University Faculty of Computer Science, pp. 1–31
 Pournaras, E.; Warnier, M.; Brazier, F.M.T.. (2013), "A generic and adaptive aggregation service for largescale decentralized networks", Complex Adaptive Systems Modeling, 1:19, doi:10.1186/21943206119
External links
Wikimedia Commons has media related to Bloom filter. 
 Why Bloom filters work the way they do (Michael Nielsen, 2012)
 Bloom Filters — A Tutorial, Analysis, and Survey (Blustein & ElMaazawi, 2002) at Dalhousie University
 Table of falsepositive rates for different configurations from a University of Wisconsin–Madison website
 Interactive Processing demonstration from ashcan.org
 "More Optimal Bloom Filters," Ely Porat (Nov/2007) Google TechTalk video on YouTube
 "Using Bloom Filters" Detailed Bloom Filter explanation using Perl
 "A Garden Variety of Bloom Filters  Explanation and Analysis of Bloom filter variants
 "Bloom filters, fast and simple"  Explanation and example implementation in Python