diff --git a/om-graphs/graphs.tex b/om-graphs/graphs.tex index 140a21cc03b108b2ac98b57df6fb1d6ec85ee528..7334c602abf14e2286eb101ea4dc97c701c7d6d4 100644 --- a/om-graphs/graphs.tex +++ b/om-graphs/graphs.tex @@ -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.}