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.
-- 
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