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datovky
ds2-notes
Commits
fb122d64
Commit
fb122d64
authored
5 years ago
by
Martin Mareš
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Geometric: Typos
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17300317
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07-geom/geom.tex
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fb122d64
...
@@ -153,7 +153,7 @@ with median $x$~coordinate in the root. We split the points to the two half-plan
...
@@ -153,7 +153,7 @@ with median $x$~coordinate in the root. We split the points to the two half-plan
and recurse on each half-plane, which constructs the left and right subtree.
and recurse on each half-plane, which constructs the left and right subtree.
During the recursion, we alternate coordinates. As the number of points in the
During the recursion, we alternate coordinates. As the number of points in the
current subproblem decreases by a~factor of two in every recursive call, we obtain
current subproblem decreases by a~factor of two in every recursive call, we obtain
a~tree
a~tree of height
$
\lceil\log
n
\rceil
$
.
a~tree of height
$
\lceil\log
n
\rceil
$
.
Time complexity of building can be analyzed using the recursion tree: since we can find a~median
Time complexity of building can be analyzed using the recursion tree: since we can find a~median
of~
$
m
$
items in time
$
\O
(
m
)
$
, we spend
$
\O
(
n
)
$
time per tree level. We have
$
\O
(
\log
n
)
$
of~
$
m
$
items in time
$
\O
(
m
)
$
, we spend
$
\O
(
n
)
$
time per tree level. We have
$
\O
(
\log
n
)
$
...
@@ -220,7 +220,7 @@ a~\em{band} in~$\R^2$ (an~open rectangle which is vertically unbounded).
...
@@ -220,7 +220,7 @@ a~\em{band} in~$\R^2$ (an~open rectangle which is vertically unbounded).
For every band, we construct a~
$
y
$
-tree containing all points in the band
For every band, we construct a~
$
y
$
-tree containing all points in the band
ordered by the
$
y
$
~coordinate.
ordered by the
$
y
$
~coordinate.
If the
$
x
$
-tree is balanced, every node lies in
$
\O
(
log n
)
$
subtrees.
If the
$
x
$
-tree is balanced, every node lies in
$
\O
(
\
log
n
)
$
subtrees.
So every point lies in
$
\O
(
\log
n
)
$
bands and the whole structure takes
So every point lies in
$
\O
(
\log
n
)
$
bands and the whole structure takes
$
\O
(
n
\log
n
)
$
space:
$
\O
(
n
)
$
for the
$
x
$
-tree,
$
\O
(
n
\log
n
)
$
for all
$
\O
(
n
\log
n
)
$
space:
$
\O
(
n
)
$
for the
$
x
$
-tree,
$
\O
(
n
\log
n
)
$
for all
$
y
$
-trees.
$
y
$
-trees.
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