Count–min sketch

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 ≤ i ≤ n 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.[lower-alpha 1]

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 an additive factor of ε with probability 1 − δ (see below), and e is Euler's number.

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 = h_j(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 ${\displaystyle {\hat {a}}_{i}=\min _{j}\mathrm {count} [j,h_{j}(i)]}$, where ${\displaystyle \mathrm {count} }$ is the table.

Obviously, for each i, one has ${\displaystyle a_{i}\leq {\hat {a}}_{i}}$, where ${\displaystyle a_{i}}$ is the true frequency with which i occurred in the stream.

Additionally, this estimate has the guarantee that ${\displaystyle {\hat {a}}_{i}\leq a_{i}+\varepsilon N}$ with probability ${\displaystyle 1-\delta }$, where ${\displaystyle N=\sum _{i=0}^{n}a_{i}}$ is the stream size, i.e. the total number of items seen by the sketch.

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

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

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. The estimated count is then the median (rather than the minimum) of

${\displaystyle \mathrm {count} [j,h_{j}(i)]-{\frac {N-\mathrm {count} [j,h_{j}(i)]}{w-1}}}$

for all rows j.[5] Since this estimate may overestimate and could be worse than the regular count–min estimate, Goyal et al. recommend taking the minimum of both estimates.[4]

Conservative updating changes the update, but not the query algorithms. To count c instances of event type i, one first computes an estimate ${\displaystyle {\hat {a}}_{i}=\min _{j}\mathrm {count} [j,h_{j}(i)]}$, then updates ${\displaystyle \mathrm {count} [j,h_{j}(i)]\leftarrow \max\{\mathrm {count} [j,h_{j}(i)],{\hat {a_{i}}}+c\}}$ for each row j.

• Feature hashing
• Locality-sensitive hashing
• MinHash

• Dwork, Cynthia; Naor, Moni; Pitassi, Toniann; Rothblum, Guy N.; Yekhanin, Sergey (2010). Pan-private streaming algorithms. Proc. ICS. CiteSeerX 10.1.1.165.5923.
• Schechter, Stuart; Herley, Cormac; Mitzenmacher, Michael (2010). Popularity is everything: A new approach to protecting passwords from statistical-guessing attacks. USENIX Workshop on Hot Topics in Security. CiteSeerX 10.1.1.170.9356.

• Count–min FAQ