# Distinct Elements Problem

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07-Feb-2016Category

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### Transcript of Distinct Elements Problem

Ariel Rosenfeld

Input: a stream of m integers i1, i2, ..., im. (over 1,,n)Output: the number of distinct elements in the stream.

Example count the distinct number of IP addresses you encounter.

Bit vector of size n (mark 1 when encountered)Keeping all m integers and naively answer.Sort and count

O(min{n,mlogm})

a determinitic exact algorithm is impossible using o(n) bits.A deterministic approximation algorithm for this problem providing a (1 1/1000)-approximation using o(n) bits is impossible.

Pick random hash function h : [n] [0, 1]Calculate z = ministream h(i)Output 1/z 1

Same ints gets same hash value.

We will show that the output is a good approximation.

This is idealized for 2 reasons:1.We dont have perfect precision.2. We need n bits at least to remember the randomness associated with every i.

Lets ignore it for now

S = {j1,jt} (unique elements in the stream)

h(j1), ..., h(jt) = X1, ..., Xt are independent variables from Unif[0, 1]

Z = min{Xi}

P=10101F(x)11

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(HW)

We get a bounded variance.

q increases -> better approximation Chebyshev

We want a function that doesn't need n bits or more to represent.

So we will use k-wise independent hash functions (H) each can be represented using a small number of bits (log|H|).In lecture.

An example - Set q > k a prime power, and dene Hpoly,k to be the set of all degree (k 1) polynomials in Fq[x]. Hpoly,k is a k-wise independent family.

Size: qkNeeds: k log q bits.

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