# HG changeset patch
# User Markus Kaiser <markus.kaiser@in.tum.de>
# Date 1401739243 -7200
# Node ID c7e07d48caeea8bb2e95a879c40c565178199387
# Parent  b0f1f2800dce7c2e1e486172e0f6483f51ae54c0
fix tree metrics inequality; give nodes a name in bisection argument

diff -r b0f1f2800dce -r c7e07d48caee minimum_bisection/presentation.pdf
Binary file minimum_bisection/presentation.pdf has changed
diff -r b0f1f2800dce -r c7e07d48caee minimum_bisection/presentation.tex
--- a/minimum_bisection/presentation.tex	Mon Jun 02 12:17:38 2014 +0200
+++ b/minimum_bisection/presentation.tex	Mon Jun 02 22:00:43 2014 +0200
@@ -590,7 +590,7 @@
             \addtolength{\jot}{.5em}
             \begin{alignat}{2}
                 d_\ell(u, v) &\leq M_{uv} & \forall u, v \in V\\
-                \sum_{u, v \in V} c_{uv} \ell_{uv} &\leq \Oh(\log n) \sum_{u, v \in V} c_{uv} M_{uv}
+                \sum_{u, v \in V} c_{uv} M_{uv} &\leq \Oh(\log n) \sum_{u, v \in V} c_{uv} d_\ell(u, v)
             \end{alignat}
         \endgroup
     \end{theorem}
@@ -911,7 +911,7 @@
     \frametitle{Tree Bisections}
 
     \begin{lemma}[]
-        For any tree $T$ and any $S \subseteq V$ we have
+        For any spanning tree $T$ and any $S \subseteq V$ we have
         \begin{align}
             c(\delta(S)) \leq c_T(\delta(S))
         \end{align}
@@ -932,6 +932,10 @@
         \path (el)
             ++ (-30:2.3) node {\structure{$S$}};
 
+        \draw
+            (b) node[below=0.05] {\structure{u}}
+            (j) node[below=0.05] {\structure{v}};
+
         \begin{pgfonlayer}{marked}
             \foreach \source/\dest in {b/j,f/c,e/c,e/g}
             \draw [tree edge, tumblue, dash pattern=on 7pt off 7pt] (\source.center) -- (\dest.center);