From fb122d64110fccde02ed843467c91fcc586c96db Mon Sep 17 00:00:00 2001 From: Martin Mares <mj@ucw.cz> Date: Fri, 21 Feb 2020 18:01:03 +0100 Subject: [PATCH] Geometric: Typos --- 07-geom/geom.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/07-geom/geom.tex b/07-geom/geom.tex index 5e02aa2..7efda99 100644 --- a/07-geom/geom.tex +++ b/07-geom/geom.tex @@ -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. 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 -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 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). For every band, we construct a~$y$-tree containing all points in the band 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 $\O(n\log n)$ space: $\O(n)$ for the $x$-tree, $\O(n\log n)$ for all $y$-trees. -- GitLab