Succinct: SOLE diagrams

fs-succinct/Makefile

 TOP=..
+PICS=sole sole_even
 
 include ../Makerules

fs-succinct/sole.asy

+import sole_common;
+blocks(0 ... concat(array(6,"B"), new string[] {"...", "B", "EOF"}));
+thruarrows(0,0,6);
+thruarrows(0,7,2);
+blocks(1 ... concat(array(6, "B+1"), array(1, "..."), array(2, "B+1")) );
+mixarrows(1,0,6);
+mixarrow(1, 7);
+block(1, 9, "0");
+blocks(2,  "B", "B+3", "B-3", "B+6", "B-6", "B+9", "...", "B-i","B+j", "B-j");
+thruarrow(2, 0);
+mixarrows(2, 1, 6);
+mixarrow(2, 8);
+thruarrow(1, 9);
+blocks(3 ... concat(array(6,"B"), array(1, "..."), array(3, "B")));
+
+passlabel(0, "Add EOF");
+passlabel(1, "Pass 1");
+passlabel(2, "Pass 2");

fs-succinct/sole_even.asy

+import sole_common;
+
+blocks(0, "...", "B", "B", "EOF");
+mixarrow(0, 1); 
+thruarrow(0,3);
+blocks(1, "...", "B+i", "B-i", "B+j");
+mixarrows(1, 0, 4);
+blocks(2, "...", "B", "B", "B");

fs-succinct/succinct.tex

@@ -156,6 +156,34 @@ possible input combinations).
 We will use this kind of alphabet re-encoding by pair heavily in the SOLE
 encoding. The best way to explain the exact scheme is with a diagram:
 
+\figure[sole]{sole.pdf}{}{SOLE alphabet re-encoding scheme}
+
+There are two re-encoding phases. The first transforms blocks with alphabet
+$[B+1]$ into blocks with variable alphabet sizes (of the form of alternating
+$[B+3k]$, $[B-3k]$). The second phase runs phase shifted by one block and converts
+the variable-alphabet blocks into blocks with alphabet $[B]$.
+
+What is the redundancy of this scheme? That depends on whether the original
+number of blocks (after padding) is even or odd. Figure \figref{sole} showed
+the case for an odd number. For an even number, the ending is a little bit different
+(fig. \figref{sole_even}).
+
+You can easily check that this is reversible: after decoding phase 2, the last block
+will always be either 0 (for the odd case) or EOF (for the even case).
+
+\figure[sole_even]{sole_even.pdf}{}{SOLE alphabet re-encoding scheme}
+
+Now we can finally analyze redundancy. 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.
+\: We always add one block with EOF character.
+\: In the first pass, number of blocks does not increase.
+\: In the second pass, we may need to add an extra padding block to make number of blocks odd.
+\endlist
+In total, we add at most 3 blocks. Thus $r(n) \le 3B$. That is a constant
+and thus we have a succinct scheme.
+
 
 \section{Succinct representation of arbitrary-alphabet strings}