Commit f8a6a966 authored by Filip Stedronsky's avatar Filip Stedronsky
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Succinct: SOLE: proof

parent c52f9aac
......@@ -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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