Commit b662898c by Filip Stedronsky

### Succinct: mixers intro

parent bd3aaf80
 TOP=.. 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; mixer(0,0,r); 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: \tightlist{o} \: $S = \O(\sqrt{X})$ \endlist } \endchapter
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