August 4, 2020

There are 19,055 distinct words in the Bible:

$ cat bible.txt | tr -cs A-Za-z ‘
‘ | sort -u | wc -w

It’s easy enough to count the number of distinct items in a set (its “cardinality”) when the set is small, but when the set is large, the intermediate storage required for the distinct items can be overwhelming.

Phillipe Flajolet and various co-authors wrote a series of papers in which they developed methods of estimating the cardinality of a set with only a small amount of auxiliary storage, using randomization; Flajolet’s algorithms can be seen as an improvement on Robert Morris’ counting algorithm that we studied in a previous exercise. We will study Flajolet’s loglog algorithm in today’s exercise and perhaps have a look at his other algorithms in future exercises.

The basic idea is to apply a hash function to each element of the set. The first bit of the hash value will be zero about half the time, the first two bits of the hash value will be zero about a quarter of the time, the first three bits of the hash value will be zero about an eighth of the time, and so on; by looking at the maximum number of leading zero-bits, we can estimate the cardinality of the set. Flajolet extends this algorithm by splitting the counts among 2k buckets and averaging the estimated cardinalities; the bucket is selected randomly by looking at the last k bits of the hash value.

Your task is to implement Flajolet’s loglog algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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