Count-distinct problem

In computer science, the count-distinct problem
is the problem of finding the number of distinct elements in a data stream with repeated elements.
This is a well-known problem with numerous applications. The elements might represent IP addresses of packets passing through a router, unique visitors to a web site, elements in a large database, motifs in a DNA sequence, or elements of RFID/sensor networks.

Formal definition

An example of an instance for the cardinality estimation problem is the stream:. For this instance,.

Naive solution

The naive solution to the problem is as follows:
Initialize a counter,, to zero,
Initialize an efficient dictionary data structure,, such as hash table or search tree in which insertion and membership can be performed quickly.
, a membership query is issued.

Increase by one,
Otherwise do nothing.

As long as the number of distinct elements is not too big, fits in main memory and an exact answer can be retrieved.
However, this approach does not scale for bounded storage, or if the computation performed for each element should be minimized. In such a case, several streaming algorithms have been proposed that use a fixed number of storage units.

HyperLogLog algorithm

Streaming algorithms

To handle the bounded storage constraint, streaming algorithms use a randomization to produce a non-exact estimation of the distinct number of elements,.
State-of-the-art estimators hash every element into a low-dimensional data sketch using a hash function,.
The different techniques can be classified according to the data sketches they store.

Min/max sketches

Min/max sketches store only the minimum/maximum hashed values. Examples of known min/max sketch estimators: Chassaing et al. presents max sketch which is the minimum-variance unbiased estimator for the problem. The continuous max sketches estimator is the maximum likelihood estimator. The estimator of choice in practice is the HyperLogLog algorithm.
The intuition behind such estimators is that each sketch carries information about the desired quantity. For example, when every element is associated with a uniform RV,, the expected minimum value of is. The hash function guarantees that is identical for all the appearances of. Thus, the existence of duplicates does not affect the value of the extreme order statistics.
There are other estimation techniques other than min/max sketches. The first paper on count-distinct estimation describes the Flajolet–Martin algorithm, a bit pattern sketch. In this case, the elements are hashed into a bit vector and the sketch holds the logical OR of all hashed values. The first asymptotically space- and time-optimal algorithm for this problem was given by Daniel M. Kane, Jelani Nelson, and David P. Woodruff.

Bottom-''m'' sketches

Bottom-m sketches
are a generalization of min sketches, which maintain the minimal values, where.
See Cosma et al. for a theoretical overview of count-distinct estimation algorithms, and Metwally
for a practical overview with comparative simulation results.

Python implementation of Knuth's CVM algorithm

def algorithm_d:
p = 1.0
buffer =
for a in stream:
if a in buffer:
buffer.pop
u = uniform
if u < p:
if len < s:
buffer = u
else:
a_p, u_p = max
if u > u_p:
p = u
else:
buffer.pop
buffer = u
p = u_p
return len / p

CVM algorithm

Compared to other approximation algorithms for the count-distinct problem the CVM Algorithm uses sampling instead of hashing. The CVM Algorithm provides an unbiased estimator for the number of distinct elements in a stream, in addition to the standard guarantees. Below is the CVM algorithm, including the slight modification by Donald Knuth.

Initialize max buffer size, where
Initialize an empty buffer,
in data stream of size do:

insert in
else
such that /* whose is maximum in */
If then

else
Replace with

The previous version of the CVM algorithm is improved with the following modification by Donald Knuth, that adds the while loop to ensure B is reduced.

Initialize max buffer size, where
Initialize an empty buffer,
in data stream of size do:

Insert into

Remove every element of of with

If then
Insert into

Weighted count-distinct problem

In its weighted version, each element is associated with a weight and the goal is to estimate the total sum of weights.
Formally,
An example of an instance for the weighted problem is:. For this instance,, the weights are and.
As an application example, could be IP packets received by a server. Each packet belongs to one of IP flows. The weight can be the load imposed by flow on the server. Thus, represents the total load imposed on the server by all the flows to which packets belong.

Solving the weighted count-distinct problem

Any extreme order statistics estimator for the unweighted problem can be generalized to an estimator for the weighted problem
For example, the weighted estimator proposed by Cohen et al. can be obtained when the continuous max sketches estimator is extended to solve the weighted problem.
In particular, the HyperLogLog algorithm can be extended to solve the weighted problem. The extended HyperLogLog algorithm offers the best performance, in terms of statistical accuracy and memory usage, among all the other known algorithms for the weighted problem.