Commit e39ffd46 authored by Martin Mareš's avatar Martin Mareš
Browse files

Graphs: Do not use empty labels in figures

parent eeba6f56
......@@ -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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