In computer science, the count-distinct problem^[1] (also known in applied mathematics as the cardinality estimation 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

Instance: Consider a stream of elements ${\displaystyle x_{1},x_{2},\ldots ,x_{s}}$ with repetitions, and an integer ${\displaystyle m}$. Let ${\displaystyle n}$ denote the number of distinct elements in the stream, represented as ${\displaystyle \{e_{1},e_{2},\ldots ,e_{n}\}}$, where ${\displaystyle n=|\{e_{1},e_{2},\ldots ,e_{n}\}|}$.

Objective: Find an estimate ${\displaystyle {\widehat {n}}}$ of ${\displaystyle n}$ using only ${\displaystyle m}$ storage units, where ${\displaystyle m\ll n}$.

An example of an instance for the cardinality estimation problem is the stream: ${\displaystyle a,b,a,c,d,b,d}$. For this instance, ${\displaystyle n=|\left\{{a,b,c,d}\right\}|=4}$.

Naive solution

The naive solution to the problem is as follows:

 Initialize a counter, c, to zero, ${\displaystyle c\leftarrow 0}$.
 Initialize an efficient dictionary data structure, D, such as hash table or search tree in which insertion and membership can be performed quickly.  
 For each element ${\displaystyle x_{i}}$, a membership query is issued. 
     If ${\displaystyle x_{i}}$ is not a member of D (${\displaystyle x_{i}\notin D}$)
         Add ${\displaystyle x_{i}}$ to D
         Increase c by one, ${\displaystyle c\leftarrow c+1}$
     Otherwise (${\displaystyle x_{i}\in D}$) do nothing.
 Output ${\displaystyle n=c}$.

As long as the number of distinct elements is not too big, D 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 ${\displaystyle x_{i}}$ 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, ${\displaystyle n}$. State-of-the-art estimators hash every element ${\displaystyle e_{j}}$ into a low-dimensional data sketch using a hash function, ${\displaystyle h(e_{j})}$. The different techniques can be classified according to the data sketches they store.

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, however the bound on error is not currently known. Here is the algorithm:^[11]

 Initialize a counter, t, to zero, ${\displaystyle t\leftarrow 0}$.
 Initialize an empty buffer, B.  
 For each element ${\displaystyle a_{t}}$ in data stream ${\displaystyle A}$ of size ${\displaystyle n}$ do: 
   ${\displaystyle t\leftarrow t+1}$
   ${\displaystyle a\leftarrow a_{t}}$
   If ${\displaystyle a}$ is in B then
       Delete ${\displaystyle a}$ from B
   End If
   ${\displaystyle u\leftarrow }$ random number in ${\displaystyle [0,1)}$
   If ${\displaystyle u\geq p}$ then
       NEXT
   Else If ${\displaystyle |B|<s}$ then
       Insert ${\displaystyle (a,u)}$ into B
       NEXT
   Else
       ${\displaystyle (a',u')\leftarrow (a,u)}$ of max ${\displaystyle u}$ in B
       If ${\displaystyle u>u'}$ then
           ${\displaystyle p\leftarrow u}$
       Else
           Delete ${\displaystyle (a',u')}$ from B
           Insert ${\displaystyle (a,u)}$ into B
           ${\displaystyle p\leftarrow u'}$
       End If
   End If
 End For
return ${\displaystyle |B|/p}$.

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,

Instance: A stream of weighted elements ${\displaystyle x_{1},x_{2},\ldots ,x_{s}}$ with repetitions, and an integer ${\displaystyle m}$. Let ${\displaystyle n}$ be the number of distinct elements, namely ${\displaystyle n=|\left\{{x_{1},x_{2},\ldots ,x_{s}}\right\}|}$, and let these elements be ${\displaystyle \left\{{e_{1},e_{2},\ldots ,e_{n}}\right\}}$. Finally, let ${\displaystyle w_{j}}$ be the weight of ${\displaystyle e_{j}}$.

Objective: Find an estimate ${\displaystyle {\widehat {w}}}$ of ${\displaystyle w=\sum _{j=1}^{n}w_{j}}$ using only ${\displaystyle m}$ storage units, where ${\displaystyle m\ll n}$.

An example of an instance for the weighted problem is: ${\displaystyle a(3),b(4),a(3),c(2),d(3),b(4),d(3)}$. For this instance, ${\displaystyle e_{1}=a,e_{2}=b,e_{3}=c,e_{4}=d}$, the weights are ${\displaystyle w_{1}=3,w_{2}=4,w_{3}=2,w_{4}=3}$ and ${\displaystyle \sum {w_{j}}=12}$.

As an application example, ${\displaystyle x_{1},x_{2},\ldots ,x_{s}}$ could be IP packets received by a server. Each packet belongs to one of ${\displaystyle n}$ IP flows ${\displaystyle e_{1},e_{2},\ldots ,e_{n}}$. The weight ${\displaystyle w_{j}}$ can be the load imposed by flow ${\displaystyle e_{j}}$ on the server. Thus, ${\displaystyle \sum _{j=1}^{n}{w_{j}}}$ represents the total load imposed on the server by all the flows to which packets ${\displaystyle x_{1},x_{2},\ldots ,x_{s}}$ belong.

Solving the weighted count-distinct problem

Any extreme order statistics estimator (min/max sketches) for the unweighted problem can be generalized to an estimator for the weighted problem .^[12] For example, the weighted estimator proposed by Cohen et al.^[5] can be obtained when the continuous max sketches estimator is extended to solve the weighted problem. In particular, the HyperLogLog algorithm^[6] 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.

See also

• Count–min sketch
• Streaming algorithm
• Maximum likelihood
• Minimum-variance unbiased estimator

References