Commit 9882fc7d authored by Parth Mittal's avatar Parth Mittal
Browse files

fixed typos, added another todo

parent 16df9964
......@@ -106,7 +106,7 @@ can be easily combined.
\:\em{Init}:
$C[1\ldots t][1\ldots k] \= 0$, where $k \= \lceil 2 / \varepsilon \rceil$
and $t \= \lceil \log(1 / \delta) \rceil$.
\:: Choose $t$ independent hash functions $h_1, \ldots h_t : [n] \to [k]$, each
\:: Choose $t$ independent hash functions $h_1, \ldots , h_t : [n] \to [k]$, each
from a 2-independent family.
\:\em{Process}($x$):
\::For $i \in [t]$: $C[i][h_i(x)] \= C[i][h_i(x)] + 1$.
......@@ -114,7 +114,7 @@ can be easily combined.
\endalgo
Note that the algorithm needs $\O(tk \log m)$ bits to store the table $C$, and
$\O(t \log n)$ bits to store the hash functions $h_1, \ldots h_t$, and hence
$\O(t \log n)$ bits to store the hash functions $h_1, \ldots , h_t$, and hence
uses $\O(1/\varepsilon \cdot \log (1 / \delta) \cdot \log m
+ \log (1 / \delta)\cdot \log n)$ bits. It remains to show that it computes
a good estimate.
......@@ -298,7 +298,7 @@ Recall that $\E[Y_r] = d / 2^r$, so the terms in the first sum can be bounded
using Chebyshev's inequality. The second sum is equal to the probability of
the event $[t \geq s]$, that is, the event $Y_{s - 1} \geq c / \varepsilon^2$
(since $z$ is only increased when $B$ becomes larger than this threshold).
We will simply use Markov's inequality to bound this event.
We will use Markov's inequality to bound the probability of this event.
Putting it all together, we have:
$$\eqalign{
......@@ -327,7 +327,12 @@ The counter $z$ requires only $\O(\log \log n)$ bits, and $B$ has
$\O(1 / \varepsilon^2)$ entries, each of which needs $\O( \log n )$ bits.
Finally, the hash function $h$ needs $\O(\log n)$ bits, so the total space
used is dominated by $B$, and the algorithm uses $\O(\log n / \varepsilon^2)$
space.
space. As before, if we use the median trick, the space used increases to
$\O(\log\delta \cdot \log n / \varepsilon^2)$.
(TODO: include the version of this algorithm where we save space by storing
$(g(a), {\tt tz}(h(a)))$ instead of $(a, {\tt tz}(h(a)))$ in $B$ for some
hash function $g$ as an exercise?)
\endchapter
......
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