A Scalable, Counting, Bloom Filter
This project aims to demonstrate a novel bloom filter implementation that can scale, and provide not only the addition of new members, but reliable removal of existing members.
Bloom filters are a probabilistic data structure that provide space-efficient storage of elements at the cost of possible false positive on membership queries.
This project aims to demonstrate a novel construction that can scale, and provide not only the addition of new members, but reliable removal of existing members.
dablooms implements such a structure that takes additional metadata to classify elements in order to make an intelligent decision as to which bloom filter an element should belong.
dablooms, in addition to the above, has several features.
- Implemented as a static C library
- Memory mapped
- 4 bit counters
- Sequence counters for clean/dirty checks
- Python wrapper
For performance, the low-level operations are implemented in C. It is also memory mapped which provides async flushing and persistence at low cost. In an effort to maintain memory efficiency, rather than using integers, or even whole bytes as counters, we use only four bit counters. These four bit counters allow up to 15 items to share a counter in the map. If more than a small handful are sharing said counter, the bloom filter would be overloaded (resulting in excessive false positives anyway) at any sane error rate, so there is no benefit in supporting larger counters.
The bloom filter also employs change sequence numbers to track operations performed on the bloom filter. These allow the application to determine failed writes, aka a 'dirty' filter where an element is only partially written. Upon restart, this information allows the application to determine if a filter is clean or dirty and make an appropriate decision. The counters also provide a means for us to identify a position at which the bloom filter is valid in order to replay operations to "catch up" to a current state.
Also included is a wrapper to easily leverage the library in Python.
>>> import pydablooms >>> bloom = pydablooms.Dablooms(capacity=1000, ... error_rate=.05, ... filepath='/tmp/bloom.bin', ... id=1) >>> bloom.add('foo', 2) 1 >>> bloom.check('bar') 0 >>> bloom.delete('foo', 2) 0 >>> bloom.check('foo') 0
After you have cloned the repo, type
make install (
sudo maybe needed).
To use a specific version of Python, build directory, or destination
directory, use the
Look at the output of
make help for more options.
An example build might be
make install PYTHON=python2.7 BLDDIR=/tmp/dablooms/bld DESTDIR=/tmp/dablooms/pkg
To run a quick and dirty test, type
make test. This test files uses
and defaults to
/usr/share/dict/words. If your path differs, you can use the
WORDS flag to specific its location, such as
make test WORDS=/usr/dict/words.
This will run a simple test script that iterates through a word dictionary and adds each word to dablooms. It iterates again, removing every fifth element. Lastly, it saves the file, opens a new filter, and iterates a third time checking the existence of each word. It prints results of the true negatives, false positives, true positives, and false negatives.
The maximum error rate for the filter is, by default, set to .05 (5%) and the initial capacity is set to 100k. Since the dictionary is near 500k, we should have created 4 new filters in order to scale to size.
Check out the performance yourself, and checkout the size of the resulting file!
Bloom Filter Basics
Bloom filters are probabilistic data structures that provide
space-efficient storage of elements at the cost of occasional false positives on
membership queries, i.e. a bloom filter may state true on query when it in fact does
not contain said element. A bloom filter is traditionally implemented as an array of
M bits, where
M is the size of the bloom filter. On initialization all bits are
set to zero. A filter is also parameterized by a constant
k that defines the number
of hash functions used to set and test bits in the filter. Each hash function should
output one index in
M. When inserting an element
x into the filter, the bits
h1(x), h2(x), ..., hk(X) are set.
In order to query a bloom filter, say for element
x, it suffices to verify if
all bits in indices
h1(x), h2(x), ..., hk(x) are set. If one or more of these
bits is not set then the queried element is definitely not present in the
filter. However, if all these bits are set, then the element is considered to
be in the filter. Given this procedure, an error probability exists for positive
matches, since the tested indices might have been set by the insertion of other
Counting Bloom Filters: Solving Removals
The same property that results in false positives also makes it difficult to remove an element from the filter as there is no easy means of discerning if another element is hashed to the same bit. Unsetting a bit that is hashed by multiple elements can cause false negatives. Using a counter, instead of a bit, can circumvent this issue. The bit can be incremented when an element is hashed to a given location, and decremented upon removal. Membership queries rely on whether a given counter is greater than zero. This reduces the exceptional space-efficiency provided by the standard bloom filter.
Scalable Bloom Filters: Solving Scale
Another important property of a bloom filter is its linear relationship between size and storage capacity. If the maximum allowable error probability and the number of elements to store are both known, it is relatively straightforward to dimension an appropriate filter. However, it is not always possible to know how many elements will need to be stored a priori. There is a trade off between over-dimensioning filters or suffering from a ballooning error probability as it fills.
Almeida, Baquero, Preguiça, Hutchison published a paper in 2006, on Scalable Bloom Filters, which suggested a means of scalable bloom filters by creating essentially a list of bloom filters that act as one large bloom filter. When greater capacity is desired, a new filter is added to the list.
Membership queries are conducted on each filter with the positives evaluated if the element is found in any one of the filters. Naively, this leads to an increasing compounding error probability since the probability of the given structure evaluates to:
1 - 𝚺(1 - P)
It is possible to bound this error probability by adding a reducing tightening
r. As a result, the bounded error probability is represented as:
1 - 𝚺(1 - P0 * r^i) where r is chosen as 0 < r < 1
Since size is simply a function of an error probability and capacity, any
array of growth functions can be applied to scale the size of the bloom filter
as necessary. We found it sufficient to pick .9 for
Problems with Mixing Scalable and Counting Bloom Filters
Scalable bloom filters do not allow for the removal of elements from the filter. In addition, simply converting each bloom filter in a scalable bloom filter into a counting filter also poses problems. Since an element can be in any filter, and bloom filters inherently allow for false positives, a given element may appear to be in two or more filters. If an element is inadvertently removed from a filter which did not contain it, it would introduce the possibility of false negatives.
If however, an element can be removed from the correct filter, it maintains the integrity of said filter, i.e. prevents the possibility of false negatives. Thus, a scaling, counting, bloom filter is possible if upon additions and deletions one can correctly decide which bloom filter contains the element.
There are several advantages to using a bloom filters. A bloom filter gives the application cheap, memory efficient set operations, with no actual data stored about the given element. Rather, bloom filters allow the application to test, with some given error probability, the membership of an item. This leads to the conclusion that the majority of operations performed on bloom filters are the queries of membership, rather than the addition and removal of elements. Thus, for a scaling, counting, bloom filter, we can optimize for membership queries at the expense of additions and removals. This expense comes not in performance, but in the addition of more metadata concerning an element and its relation to the bloom filter. With the addition of some sort of identification of an element, which does not need to be unique as long as it is fairly distributed, it is possible to correctly determine which filter an element belongs to, thereby able to maintain the integrity of a given bloom filter with accurate additions and removals.
dablooms is one such implementation of a scaling, counting, bloom filter that takes additional metadata during additions and deletions in the form of a monotonically increasing integer to classify elements such as a timestamp. This is used during additions/removals to easily classify an element into the correct bloom filter (essentially a comparison against a range).
dablooms is designed to scale itself using these monotonically increasing identifiers and the given capacity. When a bloom filter is at capacity, dablooms will create a new bloom filter using the to-be-added elements identifier as the beginning identifier for the new bloom filter. Given the fact that the identifiers monotonically increase, new elements will be added to the newest bloom filter. Note, in theory and as implemented, nothing prevents one from adding an element to any "older" filter. You just run the increasing risk of the error probability growing beyond the bound as it becomes "overfilled".
You can then remove any element from any bloom filter using the identifier to intelligently pick which bloom filter to remove from. Consequently, as you continue to remove elements from bloom filters that you are not continuing to add to, these bloom filters will become more accurate.