From: Martin Mares <mj@ucw.cz>
Date: Fri, 22 Feb 2008 17:13:07 +0000 (+0100)
Subject: Fix proof of the local contractive algorithm.
X-Git-Tag: printed~229
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;ds=sidebyside;h=f8fcc3700c80471ec8578cf4b77f47230947ab8d;p=saga.git

Fix proof of the local contractive algorithm.
---

diff --git a/PLAN b/PLAN
index 8039758..a397476 100644
--- a/PLAN
+++ b/PLAN
@@ -45,8 +45,6 @@ TODO:
 
 Spanning trees:
 
-- prove the density bound
-- fix proof of the local contraction alg
 - cite Eisner's tutorial \cite{eisner:tutorial}
 - \cite{pettie:onlineverify} online lower bound
 - mention Steiner trees
diff --git a/adv.tex b/adv.tex
index f4dc358..bf78611 100644
--- a/adv.tex
+++ b/adv.tex
@@ -194,10 +194,11 @@ of the previous algorithm (\ref{mstmcc}), we observe that all the $G_i$'s
 are minors of the graph~$G$ given as the input.
 
 For the choice $t=4\varrho$, the Lemma on low-degree vertices (\ref{lowdeg})
-guarantees that at least $n_i/2$ edges get selected in the $i$-th iteration.
-Hence at least a half of the vertices participates in contractions, so
-$n_i\le 3/4\cdot n_{i-1}$. Therefore $n_i\le n\cdot (3/4)^i$ and the algorithm terminates
-after $\O(\log n)$ iterations.
+guarantees that at the beginning of the $i$-th iteration, at least $n_i/2$ vertices
+have degree at most~$t$. Each selected edge removes one such vertex and
+possibly increases the degree of another, so at least $n_i/4$ edges get selected.
+Hence $n_i\le 3/4\cdot n_{i-1}$ and therefore $n_i\le n\cdot (3/4)^i$ and the
+algorithm terminates after $\O(\log n)$ iterations.
 
 Each selected edge belongs to $\mst(G)$, because it is the lightest edge of
 the trivial cut $\delta(v)$ (see the Blue Rule in \ref{rbma}).