Given a sequence $x \in \{ 1,2,3...,\vert \Sigma \vert \}^*$ one wants to create a sketch of it say $s(x)$ of size $\frac{2c}{3}k (ln^2 k)$ bits. And that seems to be achieved as follows,
- pick at random 2k-wise independent hash functions,
$h : [n \vert \Sigma\vert] \rightarrow [(k/3)ln (k)$
$h^0 : [n \vert \Sigma\vert] \times [2ln(k)] \rightarrow [c ln^2 (k)]$
(where apparently $c = 72e$ helps!)
For each new element $\sigma_i \in x$ the following update is done to the sketch
Let $bucket = h((i-1)\vert \Sigma\vert + \sigma_i )$
For $t \in \{1,2,3..,2ln(k)\}$
$sub-bucket = h^0((i-1)\vert \Sigma\vert + \sigma_i,t )$
$d = (bucket-1)2cln^3k + (t-1)cln^2(k) + sub-bucket$
$s(x)_d = s(x)_d \oplus 1 $
Can someone kindly explain as to what exactly happened here? I am not seeing the intuition at all..
