Practical and parallel algorithms.

diff --git a/PLAN b/PLAN
index 41930d069e731a846e0d6faa45c951c313c4d0e4..81d21f3c83268e461eef7a64ef68bae08a8e5f08 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -60,13 +60,9 @@ Spanning trees:
 - Some algorithms (most notably Fredman-Tarjan) do not need flattening
 - use the notation for contraction by a set
 - mention bugs in Valeria's verification paper
 - Some algorithms (most notably Fredman-Tarjan) do not need flattening
 - use the notation for contraction by a set
 - mention bugs in Valeria's verification paper

-- more references on decision trees
-- Lemma: deletion of a non-MST edge does not alter the MST
-- mention that there are only a few algorithms based on the Red rule
+* Lemma: deletion of a non-MST edge does not alter the MST

 
 Related:
 
 Related:

-- practical considerations: katriel:cycle, moret:practice (mention pairing heaps)
-- parallel algorithms: p243-cole (see also remarks in Karger and pettie:minirand), pettie:parallel

 - K best trees
 - degree-restricted cases and arborescences
 - bounded expansion classes?
 - K best trees
 - degree-restricted cases and arborescences
 - bounded expansion classes?


@@ -86,10 +82,10 @@ Notation:
 
 - sort the table
 - G has to be connected, so m=O(n)
 
 - sort the table
 - G has to be connected, so m=O(n)

-- impedance mismatch in terminology: contraction of G along e vs. contraction of e.
+* impedance mismatch in terminology: contraction of G along e vs. contraction of e.

 - use \delta(X) notation
 - unify use of n(G) vs. n
 - use \delta(X) notation
 - unify use of n(G) vs. n

-- change the notation for contractions -- use double slash instead of the dot?
+* change the notation for contractions -- use double slash instead of the dot?

 - introduce \widehat\O early
 
 Typography:
 - introduce \widehat\O early
 
 Typography:

diff --git a/appl.tex b/appl.tex
index 82c2a1a9918d0e6e3c28456bb3c9a74edfdfac39..1b3b3da86058f8a3487de91bdb58eaf81419e364 100644 (file)
--- a/appl.tex
+++ b/appl.tex
@@ -144,4 +144,52 @@ There is also a~$\widetilde\O(n\cdot \poly(1/\varepsilon))$ time approximation
 algorithm for the MST weight in arbitrary metric spaces by Czumaj and Sohler \cite{czumaj:metric}.
 (This is still sub-linear since the corresponding graph has roughly $n^2$ edges.)
 
 algorithm for the MST weight in arbitrary metric spaces by Czumaj and Sohler \cite{czumaj:metric}.
 (This is still sub-linear since the corresponding graph has roughly $n^2$ edges.)
 

+%--------------------------------------------------------------------------------
+
+\section{Practical algorithms}
+
+So far, we were studying the theoretical aspects of the MST algorithms.
+How should we find the MST on a~real computer?
+
+Moret and Shapiro \cite{moret:expmst} have conducted an~extensive experimental
+study of performance of implementations of various MST algorithms on different
+computers. They have tested the algorithms on several sets of graphs, varying
+in the number of vertices (up to millions) and edge density (from constant to
+$n^{1/2}$. In almost all tests, the quickest algorithm was an~ordinary Prim's
+implemented with pairing heaps \cite{fredman:pairingheap}. The pairing heaps
+are known to perform surprisingly well in practice, but they still elude attempts
+at complete theoretical analysis. So far the best results are those of Pettie
+\cite{pettie:pairing}, who has proven that deletion of the minimum takes $\O(\log n)$
+time and all other operations $\O(2^{2\scriptscriptstyle\sqrt{\log\log n}})$; both bounds are amortized.
+
+The Moret's study however completely misses variants of the Bor\o{u}vka's algorithm
+and many of those have very promising characteristics, especially for planar graphs
+and minor-closed classes.
+
+Also, Katriel et al.~\cite{katriel:cycle} have proposed a~new algorithm based on~the
+Red rule. It is a~randomized algorithm which uses a~simplified form of the idea of edge
+filtering from the algorithm of Karger, Klein and Tarjan (see Section \ref{randmst}).
+The expected time complexity is slightly worse: $\O(n\log n + m)$. However, for dense
+graphs it performs very well in practice, even in comparison with the Moret's results.
+
+\paran{Parallel algorithms}
+Most of the early parallel algorithms for the MST are variants of the Bor\o{u}vka's algorithm.
+The operations on the individual trees are independent of each other, so they can run in parallel.
+There are $\O(\log n)$ steps and each of them can be executed in $\O(\log n)$ parallel time using
+standard PRAM techniques (see \cite{jaja:parallel} for the description of the model).
+
+Several better algorithms have been constructed, the best of which run in time $\O(\log n)$.
+Chong, Han and Lam \cite{chong:parallel} have recently discovered an~algorithm that achieves
+this time complexity even on the EREW PRAM --- the weakest of the parallel RAM's which does
+not allow concurrent reading nor writing to the same memory cell by multiple processors.
+In this model, the $\O(\log n)$ bound is clearly the best possible.
+
+As in the sequential models, the basic question still remains open: Is it
+possible to compute the MST in parallel on EREW PRAM, spending only linear
+work? This would of course imply existence of a~linear-time sequential
+algorithm, so none of the known parallel algorithms achieve that. Algorithms
+with linear work can be obtained by utilizing randomness, as shown for example
+by Pettie and Ramachandran \cite{pettie:minirand}, but so far only at the
+expense of higher time complexity.
+

 \endpart
 \endpart

diff --git a/biblio.bib b/biblio.bib
index 3e75568065f9fbd34d5642534001832b093c43da..9e569206053bf899d33eb3ed3e4cd7fedb42ac4f 100644 (file)
--- a/biblio.bib
+++ b/biblio.bib
@@ -1495,7 +1495,7 @@
   pages = {247--255},
 }
 
   pages = {247--255},
 }
 

-@article{375847,
+@article{ chong:parallel,

   author = {Ka Wong Chong and Yijie Han and Tak Wah Lam},
   title = {Concurrent threads and optimal parallel minimum spanning trees algorithm},
   journal = {Journal of the ACM},
   author = {Ka Wong Chong and Yijie Han and Tak Wah Lam},
   title = {Concurrent threads and optimal parallel minimum spanning trees algorithm},
   journal = {Journal of the ACM},


@@ -1508,3 +1508,42 @@
   publisher = {ACM},
   address = {New York, NY, USA},
 }
   publisher = {ACM},
   address = {New York, NY, USA},
 }

+
+@article{ moret:expmst,
+  title={{An Empirical Assessment of Algorithms for Constructing a Minimum Spanning Tree}},
+  author={Moret, B. M. E. and Shapiro, H. D.},
+  journal={Computational Support for Discrete Mathematics: DIMACS Workshop, March 12-14, 1992},
+  year={1994},
+  publisher={American Mathematical Society}
+}
+
+@article{ fredman:pairingheap,
+  title={{The pairing heap: A new form of self-adjusting heap}},
+  author={Fredman, M.L. and Sedgewick, R. and Sleator, D.D. and Tarjan, R.E.},
+  journal={Algorithmica},
+  volume={1},
+  number={1},
+  pages={111--129},
+  year={1986},
+  publisher={Springer}
+}
+
+@article{ pettie:pairing,
+  author = {Seth Pettie},
+  title = {Towards a Final Analysis of Pairing Heaps},
+  journal = {focs},
+  volume = {00},
+  year = {2005},
+  isbn = {0-7695-2468-0},
+  pages = {174-183},
+  doi = {http://doi.ieeecomputersociety.org/10.1109/SFCS.2005.75},
+  publisher = {IEEE Computer Society},
+  address = {Los Alamitos, CA, USA},
+}
+
+@book{ jaja:parallel,
+  title={{An introduction to parallel algorithms}},
+  author={J{\'a}J{\'a}, J.},
+  year={1992},
+  publisher={Addison Wesley Longman Publishing Co., Inc. Redwood City, CA, USA}
+}

diff --git a/macros.tex b/macros.tex
index 5877abacfe422071815ee42efd7d0583e583438c..3a734d87712883b650b6476015443e7c251e0577 100644 (file)
--- a/macros.tex
+++ b/macros.tex
@@ -384,7 +384,7 @@
 \def\problemn{\problem\labelx}
 \def\remn{\rem\labelx}
 
 \def\problemn{\problem\labelx}
 \def\remn{\rem\labelx}
 

-\def\paran#1{\para {\sl #1.\/}\enspace}
+\def\paran#1{\para {\sl #1.\/}\enspace\kern 0pt}

 
 \def\proof{\noindent {\sl Proof.}\enspace}
 \def\proofsketch{\noindent {\sl Proof sketch.}\enspace}
 
 \def\proof{\noindent {\sl Proof.}\enspace}
 \def\proofsketch{\noindent {\sl Proof sketch.}\enspace}