Added the chapter with examples.

diff --git a/adv.tex b/adv.tex
index 0eae4d235694800ab3eee2cf088019e896b07180..7ae951d2e6f8cfa884dfef45943bc9afded54808 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -516,7 +516,7 @@ construct a~heap of size $\Omega(\log^{(k)} n)$ with constant time per operation
 we can get a~linear-time algorithm for MST. This is the case when the weights are
 integers:
 
-\thmn{MST for graphs with integer weights, Fredman and Willard \cite{fw:transdich}}
+\thmn{MST for graphs with integer weights, Fredman and Willard \cite{fw:transdich}}\id{intmst}%
 MST of a~graph with integer edge weights can be found in time $\O(m)$ on the Word-RAM.
 
 \proof
@@ -528,19 +528,6 @@ Following the analysis of the original algorithm in the proof of Theorem \ref{it
 $t_2\ge 2^{t_1} = 2^{\log n} = n$, so the algorithm stops after the second phase.
 \qed
 
-\para
-We can also use this technique if the edge weights are not integers, but they
-are already sorted. We already knew that the Kruskal's algorithm runs in time
-$\O(m\alpha(n))$ in such cases (Theorem \ref{kruskal}), but we can do better:
-
-\corn{MST for graphs with sorted edges}
-For a~graph with edges already sorted by their weights, we can find
-the MST in time $\O(m)$ on the Word-RAM.
-
-\proof
-We renumber the weights to $1,\ldots,m$, which does not change the MST
-(Lemma \ref{mstiso}), and find the MST using the previous theorem.
-
 \rem
 Gabow et al.~\cite{gabow:mst} have shown how to speed up the Iterated Jarn\'\i{}k's algorithm to~$\O(m\log\beta(m,n))$.
 They split the adjacency lists of the vertices to small buckets, keep each bucket
@@ -554,5 +541,35 @@ which need careful handling, so we omit the description of this algorithm.
 
 %\section{Verification of minimality}
 
+%--------------------------------------------------------------------------------
+
+\section{Special classes of graphs}
+
+Finally, we will focus our attention on various special classes of graphs
+which frequently occur in practice.
+
+\examplen{Graphs with sorted edges}
+When the edges are already sorted by their weights, we can use the Kruskal's
+algorithm to find the MST in time $\O(m\timesalpha(n))$ (Theorem \ref{kruskal}).
+We however can do better: As the minimality of a~spanning tree depends only on the
+order of weights and not on the actual values (Theorem \ref{mstthm}), we can
+renumber the weights to $1, \ldots, m$ and find the MST using the Fredman-Willard
+algorithm for integer weights. According to Theorem \ref{intmst} it runs in
+time $\O(m)$ on the Word-RAM.
+
+\examplen{Graphs with non-unique edge weights}
+
+\examplen{Graphs with a~small number of distinct weights}
+
+\examplen{Graphs with floating-point weights}
+
+\examplen{Graphs with bounded degrees}
+
+\examplen{Euclidean MST}
+
+\examplen{Approximating the MST}
+\cite{chazelle:mstapprox},
+\cite{czumaj:euclidean},
+\cite{czumaj:metric}.
 
 \endpart

diff --git a/biblio.bib b/biblio.bib
index 304eb3cd27ad2dea9604638877eee3a7a3a17d7e..b638f1b1ec15a7a5117095898016e82f300253a2 100644 (file)
--- a/biblio.bib
+++ b/biblio.bib
@@ -892,3 +892,38 @@ inproceedings{ pettie:minirand,
   year={2003},
   publisher={ACM Press New York, NY, USA}
 }
+
+@article{ chazelle:mstapprox,
+  title={{Approximating the Minimum Spanning Tree Weight in Sublinear Time}},
+  author={Chazelle, B. and Rubinfeld, R. and Trevisan, L.},
+  journal={SIAM Journal on Computing},
+  volume={34},
+  pages={1370},
+  year={2005},
+  publisher={SIAM}
+}
+
+@inproceedings{ czumaj:metric,
+ author = {Artur Czumaj and Christian Sohler},
+ title = {Estimating the weight of metric minimum spanning trees in sublinear-time},
+ booktitle = {STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing},
+ year = {2004},
+ isbn = {1-58113-852-0},
+ pages = {175--183},
+ location = {Chicago, IL, USA},
+ doi = {http://doi.acm.org/10.1145/1007352.1007386},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+}
+
+@inproceedings{ czumaj:euclidean,
+ author = {Artur Czumaj and Funda Erg\"{u}n and Lance Fortnow and Avner Magen and Ilan Newman and Ronitt Rubinfeld and Christian Sohler},
+ title = {Sublinear-time approximation of Euclidean minimum spanning tree},
+ booktitle = {SODA '03: Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms},
+ year = {2003},
+ isbn = {0-89871-538-5},
+ pages = {813--822},
+ location = {Baltimore, Maryland},
+ publisher = {Society for Industrial and Applied Mathematics},
+ address = {Philadelphia, PA, USA},
+}