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datovky
ds2-notes
Commits
e39ffd46
Commit
e39ffd46
authored
3 years ago
by
Martin Mareš
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Graphs: Do not use empty labels in figures
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om-graphs/graphs.tex
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e39ffd46
...
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@@ -40,7 +40,7 @@ $c_1,\dots c_n$ in its leaves (in this order) and inner nodes contain the minimu
children. Note that each node represents a subpath of~
$
F
$
with leaves being the single
vertices.
\figure
[]
{
range-tree.pdf
}{}{
An example of a range tree for path on eight vertices.
\figure
{
range-tree.pdf
}{}{
An example of a range tree for path on eight vertices.
Marked subtrees cover the subpath~
$
2
\to
6
$
.
}
\theorem
{
Static path representation via range tree can perform
\em
{
path query
}
,
...
...
@@ -73,7 +73,7 @@ This way, other operations can work as if there were no marks and path updates c
performed in~
$
\O
(
\log
n
)
$
time. Note that this lazy approach requires other operations to
always traverse the tree top-down in order to see correct values in the nodes.
\figure
[]
{
lazy-update.pdf
}{}{
Example of range tree traversal with marks. We wish to travel
\figure
{
lazy-update.pdf
}{}{
Example of range tree traversal with marks. We wish to travel
from
$
x
$
to
$
z
$
. The node~
$
x
$
is marked, with
$
\delta
=
+
4
$
, so we need to increase value
stored in~
$
x
$
by~4 and transfer mark to both children of~
$
x
$
. Then we can visit~
$
x
$
and
move along to~
$
y
$
. Node~
$
y
$
is also marked now, so we update~
$
y
$
and transfer mark to both
...
...
@@ -107,7 +107,7 @@ This gives us the decomposition of the tree into heavy paths that are connected
edges. The decomposition can be easily found using depth-first search in
linear time.
\figure
[]
{
heavy-light.pdf
}{}{
Example of heavy-light decomposition. Top part shows a tree
\figure
{
heavy-light.pdf
}{}{
Example of heavy-light decomposition. Top part shows a tree
with heavy paths marked by thick lines. Numbers in parenthesis show the value of
$
s
(
v
)
$
(ones are omitted). Bottom part shows the tree after compression of non-trivial heavy
paths.
}
...
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