# Count-Min Sketch

October 26, 2016

A lesser known probabalistic data structure (alongside bloom filters and the hyperloglog) includes the Count-Min Sketch. To cite Cormode and Muthukrishnan in their most recent paper “Approximating Data with the Count-Min Data Structure” [1] to illustrate the reason behind sketches

Consider a popular website which wants to keep track of statistics on the queries used to search the site. One could keep track of the full log of queries [but the log will] become very large. This problem is an instance of the count tracking problem. [..] Notice that in this scenario, we can tolerate a little imprecision. In general, we are interested only in the queries that are asked frequently. So it is acceptable if there is some fuzziness in the counts [which yields] more efficient and lightweight solutions. This tradeoff is at the heart of sketches.

The API for count tracking includes two methods: update (item, counts) and estimate (item). This could be done with a hashmap, but the size may become unweildy. The Count-Min Sketch data structure primarily consists of a fixed array of counters, of width w and depth d. all counters are initialized to zero, and each array is associated with a different hashing function. The hashes don’t need to be particularly strong (ie, they don’t have to be cryptographic).

Update operations are straight forward: each row runs its corresponding hash function to find the counter, then the counter is updated with the number of counts in the function call. Estimates are similar: each hash is applied and the counters are found. The minimum of all counters found then becomes the returned value. This works because the hash functions spread out collisions.

The more generic verion of Count-Min Sketches is to identify “Heavy-Hitters.” This adds another function to our API: HH (k) which returns the set of items which have a frequency as large as 1/k of the overall frequency. This can already be addressed with our current information, however we can speed up this query by additionally storing the frequencies of groups of items.

Outstanding questions:

• Why is taking the minimum acceptable as opposed to, say, the max or average?

Other Sketches:

• Bloom Filter (simple membership)
• massive database queries
• AMS Sketch (represents high dimensional vectors in small space)
• machine learning
• Distinct Sketches (estimate the number of different elements within the set)
• unique visitors

Also, also, also.