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Explain Flajolet Martin Algorithm with example.

Thanks for excellent explanation. But I have one question. When x=4, Hash function's output is 5 and there are 5 zeros in tail. Therefor our max R is 5 and distinct elements will be 32. But in input stream we have 4 distinct elements. I think, we should not count all-zero element. May you explain more, please?


2 Answers
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Flajolet-Martin algorithm approximates the number of unique objects in a stream or a database in one pass. If the stream contains n elements with m of them unique, this algorithm runs in $O(n)$ time and needs $O(log(m))$ memory.

Algorithm:

  1. Create a bit vector (bit array) of sufficient length L, such that $2^L \gt n$, the number of elements in the stream. Usually a 64-bit vector is sufficient since $2^64$ is quite large for most purposes.

  2. The i-th bit in this vector/array represents whether we have seen a hash function value whose binary representation ends in $0^i$. So initialize each bit to 0.

  3. The i-th bit in this vector/array represents whether we have seen a hash function value whose binary representation ends in 0i. So initialize each bit to 0.

  4. The i-th bit in this vector/array represents whether we have seen a hash function value whose binary representation ends in 0i. So initialize each bit to 0.

  5. Once input is exhausted, get the index of the first 0 in the bit array (call this R). By the way, this is just the number of consecutive 1s (i.e. we have seen $0, 00, ..., 0^{R-1}$ as the output of the hash function) plus one.

  6. Calculate the number of unique words as $2^R/ \phi$, where $\phi$ is 0.77351. A proof for this can be found in the original paper listed in the reference section.

  7. The standard deviation of R is a constant: $ \sigma (R)=1.12$. (In other words, R can be off by about 1 for 1 - 0.68 = 32% of the observations, off by 2 for about 1 - 0.95 = 5% of the observations, off by 3 for 1 - 0.997 = 0.3% of the observations using the Empirical rule of statistics). This implies that our count can be off by a factor of 2 for 32% of the observations, off by a factory of 4 for 5% of the observations, off by a factor of 8 for 0.3% of the observations and so on.

Example:

$ S = { 1,3,2,1,2,3,4,3,1,2,3,1} $

$h(x) = (6x +1) \ mod \ 5$

Assume |b| = 5

x h(x) Rem Binary r(a)
1 7 2 00010 1
3 19 4 00100 2
2 13 3 00011 0
1 7 2 00010 1
2 13 3 00011 0
3 19 4 00100 2
4 25 0 00000 5
3 19 4 00100 2
1 7 2 00010 1
2 13 3 00011 0
3 19 4 00100 2
1 7 2 00010 1

R = max( r(a) ) = 5

So no. of distinct elements = $N = 2^R = 2^5 = 32$

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  • We may want to know how many different elements have appeared in the stream.

  • For example, we wish to know how many distinct users visited the website till now or in last 2 hours.

  • If no of distinct elements required to process many streams then keeping data in main memory is challenge.

  • FM algorithm gives an efficient way to count the distinct elements in a stream.

  • It is possible to estimate the no. of distinct elements by hashing the elements of the universal set to a bit string that is sufficiently long.

  • The length of the bit string must be sufficient that there are more possible results of the hash function than there are elements in the universal set.

  • Whenever we apply a hash function h to a stream element a, the bit string h(a) will end in some number of oS, possibly none.

  • Call this as tail length for a hash.

  • Let R be the maximum tail length of any a seen so far in the stream.

  • Then we shall use estimate $2^R$ for the number of distinct elements seen in the stream.

  • Consider a stream as:

S = {1, 2, 1, 3}

Let hash function be 2x + 2 mod 4

  • When we apply the hash function we get reminder represented in binary as follows:

000, 101, 000 considering bit string length as 3.

  • Maximum tail length R will be 3.

  • No of distinct elements will be $2^R = 2^3 = 8$

  • Here the estimates may be too large or too low depending on hash function.

  • We may apply multiple hash functions and combine the estimate to get near accurate values.

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