Commit b662898c authored by Filip Stedronsky's avatar Filip Stedronsky
Browse files

Succinct: mixers intro

parent bd3aaf80
PICS=sole sole_boxes sole_hilevel
PICS=sole sole_boxes sole_hilevel mixer
include ../Makerules
import succinct_common;
real r = 1.5;
real dist=2;
draw((-dist,0)--(-r,0), e_arrow);
draw((r,0)--(dist,0), e_arrow);
draw((0,-r)--(0,-dist), e_arrow);
draw((0,dist)--(0,r), e_arrow);
label((0, dist), "\vbox{\hbox{$x\in[X]$}\hbox{\eightrm (input)}}", N);
label((-dist, 0), "\vbox{\hbox{$y\in[Y]$}\hbox{\eightrm (carry in)}}", W);
label((dist, 0), "\vbox{\hbox{$s\in[S]$}\hbox{\eightrm (carry out)}}", E);
label((0, -dist), "\vbox{\hbox{$m\in[2^M]$}\hbox{\eightrm (output)}}", S);
label((0, 0), "$t\in [T]$");
......@@ -141,7 +141,7 @@ the alphabet [88] (by the simple transformation of $8x + y$). We can then split
that character again into two in a different way. For example into two characters
from alphabets [9] and [10]. This can be accomplished by simple division with
remainder: if the original character is $z\in [88]$, we transform in into
$\lfloor z / 10\rfloor$ and $(z \;{\rm mod}\; 10)$. For example, if we start
$\lfloor z / 10\rfloor$ and $(z \bmod 10)$. For example, if we start
with the characters 6 and 5, they first get combined to form $6\cdot 8 + 5 = 53$
and then split into 5 and 3.
......@@ -233,4 +233,33 @@ into the output. The final carry is then used to output some extra blocks at the
\subsection{Generalizing the mixer concept}
\figure[mixer]{mixer.pdf}{}{General structure of a mixer}
At a high level, a mixer can be thought of as a mapping $f: [X]\times[Y] \rightarrow [2^M]\times[S]$
with the property that when $(m,s) = f(x,y)$, $s$ depends only on $x$.
Internally, the a mixer is
always implemented as a composition of two mappings, $f_1$ that transforms $x \rightarrow (t,s)$ and $f2$
that transforms $(y,t) \rightarrow m$. See fig. \figref{mixer}. Both $f_1$ and $f_2$ must be injective
so that the encoding is reversible.
The mappings $f_1$ and $f_2$ themselves are trivial alphabet translations similar to what we
used in the SOLE encoding. You can for example use $f_1(x) = (\lceil x/S \rceil, x \bmod S)$
and $f_2(y,t) = t\cdot Y + y$.
Thus implementing the mixer is simple as long as the parameters allow its existence. A mixer
with parameters $X$, $Y$, $S$, $M$ can exist if and only if there exists $T$ such that
$S\cdot T \le X$ and $2^M \le T\cdot Y$ (once again, the alphabet translations need their
range to be as large as their domain in order to work).
\subsection{On the existence of certain kinds of mixers}
Now we would like to show that mixers with certain parameters do exist.
\lemma{For $X,Y \le 2^w$ there exists a mixer $f: [X]\times[Y] \rightarrow [2^M]\times[S]$ such that:
\: $S = \O(\sqrt{X})$
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