The HyperLogLog algorithm [1] is a space efficient method to estimate the cardinality of extraordinarily large datasets. This module is written in C for Python >= 3.6. It implements a 64 bit version of HyperLogLog [2] using a Murmur64A hash.

Install Python development libraries. This step will depend on your OS. On Ubuntu 24.0.4 LTS:

sudo apt install python-dev-is-python3

Install HLL:

pip install HLL

Example usage:

from HLL import HyperLogLog

hll = HyperLogLog(10) # use 2^10 registers
hll.add('some data')

estimate = hll.cardinality()
print(estimate)

New features:

• Added intersection_cardinality() for estimating the intersection of two HyperLogLogs using Ertl's Joint Maximum Likelihood Estimation (JMLE) method.
• Added registers() which returns all register values as a bytes object of length 2^p.
• Added add_range(start, count) for bulk insertion of sequential integers without Python-to-C overhead.
• Added type checking to merge() — passing non-HyperLogLog objects now raises TypeError.

Sparse representation rewrite:

• Replaced the sparse linked list with a sorted dynamic array of packed entries. Each entry is 4 bytes (down from 16+ bytes per linked list node), improving cache locality and reducing memory overhead.
• Sparse-to-dense conversion is now based on allocated capacity vs dense representation size, instead of a fixed count threshold. The HyperLogLog switches to dense when the sparse array allocation would exceed the equivalent dense storage.
• Sparse-sparse merge() now uses a two-pointer merge in O(n) time instead of iterating all 2^p registers.

Bug fixes:

• Fixed transformToDense not returning an error on allocation failure, leading to NULL pointer dereference.
• Fixed memory leaks in error paths where malloc'd error messages were never freed.
• Fixed double-counting in add_range() where added was incremented twice per element.
• Fixed incorrect Py_ssize_t types for PyArg_ParseTuple s# format in add() and hash().
• Fixed _get_meta() reporting incorrect max_buffer_size.

Cleanup:

• Removed stale setMemoryErrorMsg declaration from header.
• Removed unused .travis.yml.
• Synced version strings across hll.c and setup.py.

• Use compiler built-ins for leading zero counts if using GCC or Clang.

• Fix bug causing cardinalities on the order of $2^{45}$ to be under-estimated.

• Remove support for Python 2.7.

Deprecation notice: this is the last supported version for Python 2.7.x.

• Fixed bug where HyperLogLogs of unequal sizes could be merged.
• Fixed bug causing cardinality estimates to be off when repeatedly merging sparse HyperLogLogs loaded from a pickle dump.

• Algorithm has been updated to a 64 bit version [2]. This fixes the spike in relative error when switching from linear counting in the original HyperLogLog algorithm.
• Hash function has been updated to the 64 bit Murmur64A function.
• More efficiently store registers using a combination of sparse and dense representations.
• Improved method for counting the number of leading zeroes.
• Changed the return type of cardinality() from float to integer.
• Changed the return logic of add(). This method no longer always indicates if a register was updated using its return value. This behavior is only preserved in dense representation. In sparse representation, add() always returns False.
• HyperLogLog objects pickled in 1.x and 2.x are not compatible.
• Added get_register()
• Added hash()
• Added _get_meta()
• Deprecated murmur2_hash()
• Deprecated registers()
• Deprecated set_register()

HyperLogLog objects implement a 64 bit HyperLogLog algorithm [2]. They can be used to estimate the cardinality of very large datasets. The estimation accuracy is proportional to the number of registers. Using more registers increases the accuracy and using less registers decreases the accuracy. The number of registers is set in powers of 2 using the parameter p and defaults to p=12 or $2^{12}$ registers.

>>> from HLL import HyperLogLog
>>> hll = HyperLogLog() # Default to 2^12 registers
>>> hll.size()
4096
>>> hll = HyperLogLog(3) # Use 2^3 registers
>>> hll.size()
8
>>> for data in ['one', 'two', 'three', 'four',]:
...     hll.add(data)
>>> hll.cardinality()
4

HyperLogLogs use a Murmur64A hash. This function is fast and has a good uniform distribution of bits which is necessary for accurate estimations. The seed to this hash function can be set in the HyperLogLog constructor:

>>> hll = HyperLogLog(p=2, seed=123456789)
>>> hll.seed()
123456789

The hash function can also be called directly:

>>> hll.hash('something')
393810339

Individual registers can be printed:

>>> for i in range(2**4):
...     print(hll.get_register(i))
0
0
3
0
4

HyperLogLog objects can be merged. This is done by taking the maximum value of their respective registers:

>>> A = HyperLogLog(p=4)
>>> A.add('hello')
>>> B = HyperLogLog(p=4)
>>> B.add('world')
>>> A.merge(B)
>>> A.cardinality()
2

The intersection cardinality of two HyperLogLog objects can be estimated using Ertl's Joint Maximum Likelihood Estimation (JMLE) method [2]. This estimation degrades in accuracy when the Jaccard, J=|A∩B|/|A∪B|, is less than .05 and severely degrades when J < .01.

Note that objects objects must have the same p value:

>>> A = HyperLogLog(p=12)
>>> B = HyperLogLog(p=12)
>>> for i in range(50000):
...     A.add(str(i))
...     B.add(str(i))
>>> for i in range(50000, 100000):
...     A.add(str(i))
>>> A.intersection_cardinality(B)
50164

Registers are stored using both sparse and dense representation. Originally all registers are initialized to zero. However storing all these zeroes individually is wasteful. Inspired by the sorted linked list approach in [3], a sorted dynamic array of packed 4-byte entries is used to store only registers that have been set (e.g. have a non-zero value). When the array's allocated memory would exceed the equivalent dense storage, the HyperLogLog switches to dense representation where registers are stored individually using 6 bits.

Sparse representation can be disabled using the sparse flag:

>>> HyperLogLog(p=2, sparse=False)

Adding an element to the HyperLogLog does not immediately update the sorted array. A temporary buffer is used to defer this operation. When the buffer is full the items are sorted and merged into the array in one pass.

The buffer size can be set using max_buffer_size:

>>> HyperLogLog(p=15, max_buffer_size=10**5)

This software is released under the MIT License.

[1] P. Flajolet, E. Fusy, O. Gandouet, F. Meunier. "HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm," Conference on the Analysis of Algorithms 2007.

[2] O. Ertl, "New Cardinality Estimation Methods for HyperLogLog Sketches," arXiv:1706.07290 [cs], June 2017.

[3] S. Heule, M. Nunkesser, A. Hall. "HyperLogLog in Practice: Algorithmic Engineering of a State of the Art Cardinality Estimation Algorithm," Proceedings of the EDBT 2013 Conference, ACM, Genoa March 2013.