From f8a6a966591ffa5508fb97eaad76b37712654e15 Mon Sep 17 00:00:00 2001
From: Filip Stedronsky <p@regnarg.cz>
Date: Sun, 29 Aug 2021 19:27:09 +0200
Subject: [PATCH] Succinct: SOLE: proof
---
fs-succinct/succinct.tex | 11 ++++++++++-
1 file changed, 10 insertions(+), 1 deletion(-)
diff --git a/fs-succinct/succinct.tex b/fs-succinct/succinct.tex
index 80a20d4..405b798 100644
--- a/fs-succinct/succinct.tex
+++ b/fs-succinct/succinct.tex
@@ -187,11 +187,20 @@ 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 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.
+For the first pass, we want:
+$$\eqalign{
+(B+1)^2 &\le (B-3i)(B+3i+3)\cr
+B^2 + 2B + 1 &\le B^2 + 3B - 9i^2 - 9i\cr
+B &\ge 9i^2 + 9i + 1\cr
+}$$
+We know $B \ge 4n^2$ and $i \le {n+1\over 2}$. By plugging $i = {n+1\over 2}$
+and doing some algebraic manipulation, we can verify that the inequality holds.
+
+For the second pass, this is trivial, as $(B+i)(B-i) = B^2 - i^2 \le B^2$.
\section{Succinct representation of arbitrary-alphabet strings}
--
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