fixed typos, added another todo

streaming/streaming.tex

@@ -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