Count–min sketch

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In computing, the count–min sketch (CM sketch) is a probabilistic data structure that serves as a frequency table of events in a stream of data. It uses hash functions to map events to frequencies, but unlike a hash table uses only sub-linear space, at the expense of overcounting some events due to collisions. The count–min sketch was invented in 2003 by Graham Cormode and S. Muthu Muthukrishnan.[1]

Count–min sketches are somewhat similar to Bloom filters; the main distinction is that Bloom filters represent sets, while CM sketches represent multisets. Spectral Bloom filters with multi-set policy are conceptually isomorphic[clarification needed] to the count–min sketch.[2]

Data structure

The goal of the basic version of the count–min sketch is to consume a stream of events, one at a time, and count the frequency of the different types of events in the stream. At any time, the sketch can be queried for the frequency of a particular event type i (0 ≤ in for some n), and will return an estimate of this frequency that is within a certain distance of the true frequency, with a certain probability.[3]

The actual sketch data structure is a two-dimensional array of w columns and d rows. The parameters w and d are fixed when the sketch is created, and determine the time and space needs and the probability of error when the sketch is queried for a frequency or inner product. Associated with each of the d rows is a separate hash function; the hash functions must be pairwise independent. The parameters w and d can be chosen by setting w = ⌈e/ε and d = ⌈ln 1/δ, where the error in answering a query is within a factor of ε with probability δ.

When a new event of type i arrives we update as follows: for each row j of the table, apply the corresponding hash function to obtain a column index k = hj(i). Then increment the value in row j, column k by one.

Several types of queries are possible on the stream.

• The simplest is the point query, which asks for the count of an event type i. The estimated count is given by the least value in the table for i, namely $\hat a_i=\min_j \mathrm{count}[j, h_j(i)]$, where $\mathrm{count}$ is the table.

This estimate has the guarantee that $\hat a_i \leq a_i + \epsilon \sum_j^{n} |a_j|$ with probability $1-\delta$, where ai is the true frequency with which i occurred in the stream.

• An inner product query asks for the inner product between the histograms represented by two count–min sketches, $\mathrm{count}_a$ and $\mathrm{count}_b$.

Small modifications to the data structure can be used to sketch other different stream statistics.

Bias

One potential problem with count-min sketches is that they are biased estimators of the true frequency of events: they may overestimate, but never underestimate the true count in a point query. At least two suggested improvements to the sketch operations have been proposed to tackle this problem:[4]

• The count-mean-min sketch subtracts an estimate of the bias when doing a query, but is otherwise the same structure as a count-min sketch.
• Conservative updating changes the update, but not the query algorithms. To count c instances of event type i, one first computes an estimate $\hat a_i=\min_j \mathrm{count}[j, h_j(i)]$, then updates $\mathrm{count}[j, h_j(i)] \leftarrow \max\{\mathrm{count}[j, h_j(i)], \hat{c} + c\}$ for each row j.