From: Martin Mares <mj@ucw.cz>
Date: Mon, 14 Apr 2008 13:27:14 +0000 (+0200)
Subject: Intro to dynamic MSF.
X-Git-Tag: printed~98
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=9c25ac49377d8a6e1f5d09b1bae61c640fe08e17;p=saga.git

Intro to dynamic MSF.
---

diff --git a/biblio.bib b/biblio.bib
index fe0e1df..b4e14b1 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -1354,4 +1354,25 @@
   year={2000}
 }
 
-
+@techreport{ henzinger:twoec,
+  title = {Fully dynamic 2-edge-connectivity algorithm in polylogarithmic time per operation},
+  author = {Monika Rauch Henzinger and Valerie King},
+  type = {Technical note},
+  institution = {Digital Equipment Corp., Systems Research Ctr.},
+  address = {130 Lytton Rd., Palo Alto, CA, 94301, USA},
+  number = {1997-004},
+  month = {12 Jun},
+  year = {1997},
+  url = {http://gatekeeper.dec.com/pub/DEC/SRC/technical-notes/abstracts/src-tn-1997-004.html}
+}
+
+@book{ overmars:ds,
+  author = {Mark H. Overmars},
+  title = {Design of Dynamic Data Structures},
+  year = {1987},
+  isbn = {038712330X},
+  publisher = {Springer-Verlag New York, Inc.},
+  address = {Secaucus, NJ, USA},
+  series={{Lecture Notes in Computer Science}},
+  volume={156}
+}
diff --git a/dyn.tex b/dyn.tex
index 8ed00ea..4f4f7ae 100644
--- a/dyn.tex
+++ b/dyn.tex
@@ -460,4 +460,32 @@ we apply the trick from Example \ref{accel} and store~$F_0$ in a~ET-tree with $a
 This does not hurt the complexity of insertions and deletions, but allows for faster queries.
 \qed
 
+%--------------------------------------------------------------------------------
+
+\section{Dynamic MSF}
+
+Most of the early algorithms for dynamic connectivity also imply $\O(n^\varepsilon)$
+algorithms for dynamic maintenance of the MSF. Henzinger and King \cite{henzinger:twoec,henzinger:randdyn}
+have generalized their randomized connectivity algorithm to maintain the MSF in $\O(\log^5 n)$ time per
+operation, or $\O(k\log^3 n)$ if only~$k$ different values of edge weights are allowed. They have solved
+the decremental version of the problem first (which starts with a~given graph and only edge deletions
+are allowed) and then presented a~general reduction from the fully dynamic MSF to its decremental version.
+Holm, de Lichtenberg and Thorup \cite{holm:polylog} have followed the same path: they have modified
+their dynamic connectivity algorithm to solve the decremental MSF in $\O(\log^2 n)$ and obtained
+the fully dynamic MSF working in $\O(\log^4 n)$ per operation.
+
+We will present a~new algorithm which will reach the same time complexity in
+a~much easier way, and which also behaves better when the set of possible
+weights small (of size at most $\poly(n)$). The algorithm is based on
+a~different idea: We will first observe that the dynamic connectivity algorithm
+described in the previous section can be used for maintaining the MSF in time
+$\O(\log^2 n)$ if only two values of edge weights are allowed. This will serve
+as a~building block in an~$\O(\log k\cdot\log^2 n)$ time algorithm for
+$k$~different edge weights. Finally, partial rebuilding of this structure
+will bring the $\O(\log^4 n)$ bound for the MSF with unrestricted weights.
+
+
+
+
+
 \endpart