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Explain Flajolet Martin Algorithm with example.

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**Flajolet-Martin algorith**m 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:**

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.

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.

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.

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.

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.

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.

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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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?

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