From: Martin Mares <mj@ucw.cz>
Date: Sat, 19 Apr 2008 15:39:40 +0000 (+0200)
Subject: Boruvka steps.
X-Git-Tag: printed~75
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=3ca44dcf441e97d58f98e719cc837aeb27b69fab;p=saga.git

Boruvka steps.
---

diff --git a/PLAN b/PLAN
index f4a4f51..723d98b 100644
--- a/PLAN
+++ b/PLAN
@@ -61,7 +61,6 @@ Spanning trees:
 - rename theorem on Minimality by order
 - introduce Cut rule and Cycle rule earlier
 - Lemma: deletion of a non-MST edge does not alter the MST
-- use the name "Boruvka step"
 
 Related:
 - practical considerations: katriel:cycle, moret:practice (mention pairing heaps)
diff --git a/adv.tex b/adv.tex
index 3f1ae00..b5ca232 100644
--- a/adv.tex
+++ b/adv.tex
@@ -373,15 +373,13 @@ show a~faster integer algorithm soon.)
 As we already noted, the improved Jarn\'\i{}k's algorithm runs in linear time
 for sufficiently dense graphs. In some cases, it is useful to combine it with
 another MST algorithm, which identifies a~part of the MST edges and contracts
-the graph to increase its density. For example, we can perform several
-iterations of the Contractive Bor\o{u}vka's algorithm and find the rest of the
-MST by the Active Edge Jarn\'\i{}k's algorithm.
+the graph to increase its density. For example, we can perform several Bor\v{u}vka
+steps  and find the rest of the MST by the Active Edge Jarn\'\i{}k's algorithm.
 
 \algn{Mixed Bor\o{u}vka-Jarn\'\i{}k}
 \algo
 \algin A~graph~$G$ with an edge comparison oracle.
-\:Run $\log\log n$ iterations of the Contractive Bor\o{u}vka's algorithm (\ref{contbor}),
-  getting a~MST~$T_1$.
+\:Run $\log\log n$ Bor\o{u}vka steps (\ref{contbor}), getting a~MST~$T_1$.
 \:Run the Active Edge Jarn\'\i{}k's algorithm (\ref{jarniktwo}) on the resulting
   graph, getting a~MST~$T_2$.
 \:Combine $T_1$ and~$T_2$ to~$T$ as in the Contraction lemma (\ref{contlemma}).
@@ -641,7 +639,7 @@ on~$T$. The algorithm runs in linear time, for example because trees are planar
 (Theorem \ref{planarbor}). We therefore spend $\O(n)$ comparisons in it.
 
 As~$T'$ has~$n$ leaves and it is a~complete branching tree, it has at most~$n$ internal vertices,
-so~$n(T')\le 2n$ as promised. Since the number of passes of the Bor\o{u}vka's
+so~$n(T')\le 2n$ as promised. Since the number of iterations of the Bor\o{u}vka's
 algorithm is $\O(\log n)$, the depth of the Bor\o{u}vka tree must be logarithmic as well.
 
 For each query path $T[x,y]$ we find the lowest common ancestor of~$x$ and~$y$
diff --git a/mst.tex b/mst.tex
index 06a1af8..871d92d 100644
--- a/mst.tex
+++ b/mst.tex
@@ -584,7 +584,7 @@ at the beginning of the $i$-th iteration by $G_i$ (starting with $G_0=G$) and th
 of vertices and edges of this graph by $n_i$ and $m_i$ respectively.
 
 \lemma\id{contiter}%
-The $i$-th iteration of the algorithm (also called the Bor\o{u}vka step) can be carried
+The $i$-th iteration of the algorithm (also called the \df{Bor\o{u}vka step}) can be carried
 out in time~$\O(m_i)$.
 
 \proof
diff --git a/opt.tex b/opt.tex
index fc80d9e..98464e9 100644
--- a/opt.tex
+++ b/opt.tex
@@ -1011,7 +1011,7 @@ resulting graph.
 \:$G_A \= (G / \bigcup_i C_i) \setminus R^\C$. \cmt{the contracted graph}
 \:$F_A \= \msf(G_A)$ calculated by the Iterated Jarn\'\i{}k's algorithm (\ref{itjar}).
 \:$G_B \= \bigcup_i F_i \cup F_A \cup R^\C$. \cmt{combine subtrees with corrupted edges}
-\:Run two iterations of the Contractive Bor\o{u}vka's algorithm (\ref{contbor}) on~$G_B$,
+\:Run two Bor\o{u}vka steps (iterations of the Contractive Bor\o{u}vka's algorithm, \ref{contbor}) on~$G_B$,
   getting a~contracted graph~$G_C$ and a~set~$F_B$ of MST edges.
 \:$F_C \= \mst(G_C)$ obtained by a~recursive call to this algorithm.
 \:Return $F_B \cup F_C$.