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datovky
ds2-notes
Commits
7358e5e9
Commit
7358e5e9
authored
Sep 06, 2021
by
Filip Stedronsky
Browse files
Succinct: progress
parent
328470af
Changes
3
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fs-succinct/Makefile
View file @
7358e5e9
TOP
=
..
PICS
=
sole sole_boxes sole_hilevel mixer
PICS
=
sole sole_boxes sole_hilevel mixer
composition
include
../Makerules
...
...
fs-succinct/composition.asy
0 → 100644
View file @
7358e5e9
//import ads;
//import flowchart;
//draw(roundrectangle("f", (0,0)));
//draw(roundrectangle("g", (1,-1)));
//draw(roundrectangle("h", (-1,-2)));
object f1 = draw("$f_1$", roundbox, (0,0), xmargin=0.5, ymargin=0.5);
object f2 = draw("$f_2$", roundbox, (1cm,-1cm), xmargin=0.5, ymargin=0.5);
object f3 = draw("$f_3$", roundbox, (-1cm,-1.5cm), xmargin=0.5, ymargin=0.5);
// XXX this does not work when setting unitsize
draw(point(f1, SE) -- point(f2, NW), Arrow);
draw(point(f1, SW) -- point(f3, NE), Arrow);
draw(point(f2, W) -- point(f3, E), Arrow);
draw(roundbox(bbox(), xmargin=0.35cm));
draw(point(f2, S) -- (xpart(point(f2, S)), -2.5cm), Arrow);
draw(point(f3, S) -- (xpart(point(f3, S)), -2.5cm), Arrow);
draw((xpart(point(f1, N)), 1cm) -- point(f1, N), Arrow);
label("$f$", (xpart(min(currentpicture)), ypart(max(currentpicture))) + (0.25cm, -0.25cm));
fs-succinct/succinct.tex
View file @
7358e5e9
...
...
@@ -284,6 +284,23 @@ want to set $C$ so that it satisfies the inequality $C \cdot Y \le 2^M$. Basical
we are asking the question how much information can we fit in
$
m
$
in addition to
the whole of
$
y
$
. Clearly we want
$
C
$
to be as high as possibly, thus we set
$
C :
=
\lfloor
2
^
M
/
Y
\rfloor
$
.
Now let us calculate the redundancy. First we shall note that we can compute redundancy
for
$
f
_
1
$
and
$
f
_
2
$
separately and add them up:
$$
\eqalign
{
r
(
f
)
&
=
M
+
\lceil\log
S
\rceil
-
\lceil\log
X
\rceil
-
\lceil\log
Y
\rceil
\cr
&
=
\left
(
M
-
\lceil\log
C
\rceil
-
\lceil\log
Y
\rceil\right
)
+
\left
(
\lceil\log
C
\rceil
+
\lceil\log
S
\rceil
-
\lceil\log
X
\rceil\right
)
\cr
&
=
r
(
f
_
2
)
+
r
(
f
_
1
)
}$$
}
This is just a telescopic sum. It works similarly for more complex mapping compositions:
as long as each intermediate result is used only once as an input to another mapping, you
can just sum the redundancies of all the mappings involved.
For example, if you have a mapping composition as in fig.
\figref
{
composition
}
,
you can easily see
$
r
(
f
)
=
r
(
f
_
1
)
+
r
(
f
_
2
)
+
r
(
f
_
3
)
$
.
\figure
[composition]
{
composition.pdf
}{}{
Mapping composition
}
First, we shall estimate
$
r
(
f
_
2
)
$
:
$$
\eqalign
{
r
(
f
_
2
)
&
=
M
-
\log
(
Y
\cdot
C
)=
M
-
\log
(
\overbrace
{
Y
\cdot
\lfloor
2
^
M
/
Y
\rfloor
}^{
\ge
2
^
M
-
Y
}
)
\cr
r
(
f
_
2
)
&
\le
M
-
\log
(
2
^
M
-
Y
)=
\log
{
2
^
M
\over
2
^
M
-
Y
}
=
\log
{
1
\over
1
-
{
Y
\over
2
^
M
}}}$$
\endchapter
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