Succinct: SOLE: fixes

fs-succinct/succinct.tex

@@ -172,17 +172,27 @@ increases throughout the encoding passes:
 \tightlist{o}
 \: If the original length was a multiple of $b$, we must add one block to complete padding.
 \: We always add one block with EOF character.
-\: Before the first pass, we may need to add an extra padding block to make number of blocks even.
+\: Before the first pass, we may need to add an extra padding block to make number of blocks even
+   (not shown in fig. \figref{sole}).
 \: Before the second pass, we always add an extra padding block to make number of blocks odd.
 \endlist
-In total, we add at most 4 blocks. Thus $r(n) \le 4b$. That is a constant
-and thus we have a succinct scheme.
 
-Also not that this representation is locally decodable and modifiable -- each input
+In total, we add at most 4 blocks. Thus $r(n) \le 4b$.
+
+For the scheme to work, we need to set $b \ge 2\log n + 2$ (so
+$B \ge 4n^2$). This gives us redundancy $r(n) = \O(\log n)$. Thus
+we have a succinct scheme. Also, one block fits into $\O(1)$ words
+on a RAM, so we can do constant-time arithmetic on the blocks.
+
+Note that this representation is locally decodable and modifiable -- each input
 block affects at most 4 output blocks.
 
 
-Now we need to check that all the alphabet translations are valid.
+Now we must check that all the alphabet transformations are valid, i.e., the
+output universe of each transformation is always at least as big as the input
+universe.
+
+
 
 \section{Succinct representation of arbitrary-alphabet strings}