From: Martin Mares <mj@ucw.cz>
Date: Tue, 4 Mar 2008 15:57:04 +0000 (+0100)
Subject: Graphs with sorted weights.
X-Git-Tag: printed~198
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=c78f73fe35f6e307261d50acd2ab722832deabfe;p=saga.git

Graphs with sorted weights.
---

diff --git a/PLAN b/PLAN
index f0c63b4..6383397 100644
--- a/PLAN
+++ b/PLAN
@@ -50,7 +50,6 @@ Spanning trees:
 - \cite{pettie:onlineverify} online lower bound
 - mention Steiner trees
 - mention matroids
-- sorted weights
 - mention disconnected graphs
 - Euclidean MST
 - Some algorithms (most notably Fredman-Tarjan) do not need flattening
@@ -67,7 +66,6 @@ Models:
 - add more context from thorup:aczero, also mention FP operations
 - refs on Cartesian trees
 - update notation.tex
-- iteration of Q-Heaps
 
 Ranking:
 
diff --git a/adv.tex b/adv.tex
index f50821d..730e955 100644
--- a/adv.tex
+++ b/adv.tex
@@ -522,11 +522,24 @@ MST of a~graph with integer edge weights can be found in time $\O(m)$ on the Wor
 We will combine the Iterated Jarn\'\i{}k's algorithm with the Q-heaps from section \ref{qheaps}.
 We modify the first pass of the algorithm to choose $t=\log n$ and use the Q-heap tree instead
 of the Fibonacci heap. From Theorem \ref{qh} and Remark \ref{qhtreerem} we know that the
-operations on the Q-heap tree run in constant time, so the whole first phase takes time~$\O(m)$.
+operations on the Q-heap tree run in constant time, so the modified first phase takes time~$\O(m)$.
 Following the analysis of the original algorithm in the proof of Theorem \ref{itjarthm} we obtain
 $t_2\ge 2^{t_1} = 2^{\log n} = n$, 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
diff --git a/mst.tex b/mst.tex
index de4ffa9..866bcc1 100644
--- a/mst.tex
+++ b/mst.tex
@@ -463,7 +463,7 @@ structure would be sufficient, as long as it has $\O(\log n)$ amortized complexi
 per operation. For example, we can label vertices with identifiers of the
 corresponding components and always recolor the smaller of the two components.
 
-\thm
+\thm\id{kruskal}%
 Kruskal's algorithm finds the MST of a given graph in time $\O(m\log n)$
 or $\O(m\timesalpha(n))$ if the edges are already sorted by their weights.