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{{Short description|Problem in computer science}}
In computer science, the '''count-distinct problem'''<ref>{{cite journal | last1=Ullman | first1=Jeff |author1-link=Jeffrey Ullman| last2 = Rajaraman | first2 = Anand | last3=Leskovec | first3=Jure |author3-link=Jure Leskovec| title=Mining data streams | url=http://infolab.stanford.edu/~ullman/mmds/ch4.pdf}}
</ref>
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==Formal definition==
: '''Instance''': Consider a stream of elements <math>x_1, x_2, \ldots, x_s
: '''Objective''': Find an estimate <math> \widehat{n} </math> of <math> n </math> using only <math> m </math> storage units, where <math> m \ll n </math>.
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The intuition behind such estimators is that each sketch carries information about the desired quantity. For example, when every element <math> e_j </math> is associated with a uniform [[Random variable|RV]], <math> h(e_j) \sim U(0,1) </math>, the expected minimum value of <math> h(e_1),h(e_2), \ldots, h(e_n) </math> is <math> 1/(n+1) </math>. The hash function guarantees that <math> h(e_j) </math> is identical for all the appearances of <math> e_j </math>. 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<ref>{{cite journal | last1=Flajolet | first1=Philippe|author1-link=Philippe Flajolet | last2 = Martin | first2 = G. Nigel | year=1985 | title=Probabilistic counting algorithms for data base applications | journal=J. Comput. Syst. Sci. | volume=31| issue=2| pages=182–209| doi=10.1016/0022-0000(85)90041-8| url=https://hal.inria.fr/inria-00076244/file/RR-0313.pdf}} </ref> 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 Kane (mathematician)|Daniel M. Kane]], [[Jelani Nelson]], and David P. Woodruff.<ref name=optimalf0>{{cite journal | last1=Kane | first1=Daniel M. | last2 = Nelson | first2 = Jelani | last3=Woodruff | first3=David P. | year=2010 | authorlink1=
===Bottom-''m'' sketches===
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for a practical overview with comparative simulation results.
=== Python implementation of Knuth's CVM
<syntaxhighlight lang="python3" line="1">
Compared to other approximation algorithms for the count-distinct problem the CVM Algorithm<ref>{{Cite journal |last=Chakraborty |first=Sourav |last2=Vinodchandran |first2=N. V. |last3=Meel |first3=Kuldeep S. |date=2022 |title=Distinct Elements in Streams: An Algorithm for the (Text) Book |url=http://arxiv.org/abs/2301.10191 |pages=6 pages, 727571 bytes |doi=10.4230/LIPIcs.ESA.2022.34 |issn=1868-8969}}</ref> (named by [[Donald Knuth]] after the initials of Sourav Chakraborty, N. V. Vinodchandran, and Kuldeep S. Meel) uses sampling instead of hashing. The CVM Algorithm provides an unbiased estimator for the number of distinct elements in a stream,<ref name=":0" /> in addition to the standard (ε-δ) guarantees. Below is the CVM algorithm, including the slight modification by Donald Knuth, that adds the while loop to ensure B is reduced. <ref name=":0">{{cite journal |last1=Knuth |first1=Donald |date=May 2023 |title=The CVM Algorithm for Estimating Distinct Elements in Streams |url=https://cs.stanford.edu/~knuth/papers/cvm-note.pdf |journal=}}</ref>▼
def algorithm_d(stream, s: int):
m = len(stream) # We assume that this is given to us in advance.
t = -1 # Note that Knuth indexes the stream from 1.
p = 1
a = 0
buffer = []
while t < (m - 1):
t += 1
a = stream[t]
u = uniform(0, 1)
buffer = list(filter(lambda x: x[1] != a, buffer))
if u < p:
if len(buffer) < s:
buffer.append([u, a])
else:
buffer = sorted(buffer)
p = max(buffer[-1][0], u)
buffer.pop()
buffer.append([u, a])
return len(buffer) / p
</syntaxhighlight>
=== CVM algorithm ===
▲Compared to other approximation algorithms for the count-distinct problem the CVM Algorithm<ref>{{Cite
{{nowrap|Initialize <math> p \leftarrow 1 </math>}}
Initialize max buffer size <math> s </math>, where <math> s \geq 1 </math>
Initialize an empty buffer, {{mvar|B}}
{{nowrap|For each element <math> a_t </math>}} in data stream <math> A </math> of size <math> n </math> do:
{{nowrap|If <math> (a_t, u), \forall u</math> is in {{mvar|B}} then}}
{{nowrap|Delete <math> (a_t, u) </math> from {{mvar|B}}}}
{{nowrap|<math> u \leftarrow </math> random number in <math> [0, 1) </math>}}
{{nowrap|If <math> u < p </math> then}}
{{nowrap|If <math> |B| < s </math> then}}
insert <math> (a_t, u) </math> in {{mvar|B}}
else
<math>(a',u')</math> such that <math>u' = \max\{u'':(a'',u'')\in B, \forall a''\}</math> /* <math>(a',u')</math> whose <math>u'</math> is maximum in {{mvar|B}} */
If <math> u > u' </math> then
<math>p\leftarrow u</math>
else
Replace <math>(a',u')</math> with <math> (a_t, u) </math>
<math>p\leftarrow u'</math>
{{nowrap|return <math> |B| / p </math>.}}
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. <ref name=":0">{{cite journal |last1=Knuth |first1=Donald |date=May 2023 |title=The CVM Algorithm for Estimating Distinct Elements in Streams |url=https://cs.stanford.edu/~knuth/papers/cvm-note.pdf |journal=}}</ref>
{{nowrap|Initialize <math> p \leftarrow 1 </math>}}
Initialize max buffer size <math> s </math>, where <math> s \geq 1 </math>
Initialize an empty buffer, {{mvar|B}}
{{nowrap|For each element <math> a_t </math>}} in data stream <math> A </math> of size <math> n </math> do:
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{{nowrap|If <math> u \leq p </math> then}}
Insert <math> (a_t, u) </math> into {{mvar|B}}
{{nowrap|While <math> |B| = s \wedge u < p </math> then}}
Remove every element of <math>(a',
{{nowrap|<math> p \leftarrow \frac{p}{2} </math>}}
If <math> u < p </math> then
▲ {{nowrap|End For}}
{{nowrap|End For}}
{{nowrap|return <math> |B| / p </math>.}}
==Weighted count-distinct problem==
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[[Category:Statistical algorithms]]
[[Category:Articles with example Python (programming language) code]]
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