From bd3aaf809fbca9a4df32d7e968a1a14341bb319f Mon Sep 17 00:00:00 2001
From: Filip Stedronsky <p@regnarg.cz>
Date: Mon, 30 Aug 2021 18:40:48 +0200
Subject: [PATCH] Succinct: fixup
---
fs-succinct/succinct.tex | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/fs-succinct/succinct.tex b/fs-succinct/succinct.tex
index 0f772d9..457cbc7 100644
--- a/fs-succinct/succinct.tex
+++ b/fs-succinct/succinct.tex
@@ -165,9 +165,7 @@ odd-numbered blocks have smaller alphabets than even-numbered ones. The second
pass runs phase shifted by one block and converts the variable-alphabet blocks
into blocks with alphabet $[B]$.
-What is the redundancy of this scheme?
-
-Now we can finally analyze redundancy. Let us count how the number of blocks
+What is the redundancy of this scheme? Let us count how the number of blocks
increases throughout the encoding passes:
\tightlist{o}
\: If the original length was a multiple of $b$, we must add one block to complete padding.
@@ -215,9 +213,9 @@ from the alphabet $(B+1)^2$, output the alphabet $B^2$ and the part of the
information that did not fit into the output is passed as a
``carry''\foot{Sometimes the alternative term {\it spill} is used instead.} to
the next encoding box (similarly to how carrying works when doing addition).
-See fig. \figref{sole_boxes}.
+See fig. \figref{sole_boxes}. We will also call these boxes {\it mixers}.
-\figure[sole_boxes]{sole_boxes.pdf}{}{SOLE interpreted as a chain of encoding boxes}
+\figure[sole_boxes]{sole_boxes.pdf}{}{SOLE interpreted as a chain of mixers}
The start and end of the encoding are irregular, but we will ignore that for now.
An important property of these boxes is that outgoing carry does not depend on incoming
@@ -225,7 +223,7 @@ carry (unlike in addition). This allows for local decoding and modification. Oth
a single input change could affect the whole output. Now we can describe this scheme
in a more abstract, high-level way (fig. \figref{sole_hilevel}).
-\figure[sole_hilevel]{sole_hilevel.pdf}{}{SOLE high-level block diagram}
+\figure[sole_hilevel]{sole_hilevel.pdf}{}{SOLE high-level mixer diagram}
In our case, the input alphabet size is always $(B+1)^2$, the output alphabet size
is $B^2$ and the carry alphabet sizes form the sequence $B+3i$. Given that the output
@@ -233,4 +231,6 @@ alphabet is smaller than the input alphabet, it makes sense that the carry alpha
has to increase in size to accomodate the accumulating information that did not fit
into the output. The final carry is then used to output some extra blocks at the end.
+\subsection{Generalizing the mixer concept}
+
\endchapter
--
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