References etc.

This commit is contained in:
Antti H S Laaksonen 2017-02-25 18:57:13 +02:00
parent 3b312e9caf
commit 34d8288b73
2 changed files with 33 additions and 8 deletions

View File

@ -22,7 +22,9 @@ problem and no efficient algorithm is known for solving the problem.
\index{Eulerian path}
An \key{Eulerian path} is a path
An \key{Eulerian path}\footnote{L. Euler (1707--1783) studied such paths in 1736
when he solved the famous Königsberg bridge problem.
This was the birth of graph theory.} is a path
that goes exactly once through each edge in the graph.
For example, the graph
\begin{center}
@ -222,7 +224,8 @@ from node 2 to node 5:
\index{Hierholzer's algorithm}
\key{Hierholzer's algorithm} is an efficient
\key{Hierholzer's algorithm}\footnote{The algorithm was published
in 1873 after Hierholzer's death \cite{hie73}.} is an efficient
method for constructing
an Eulerian circuit.
The algorithm consists of several rounds,
@ -395,7 +398,8 @@ so we have successfully constructed an Eulerian circuit.
\index{Hamiltonian path}
A \key{Hamiltonian path} is a path
A \key{Hamiltonian path}\footnote{
W. R. Hamilton (1805--1865) was an Irish mathematician.} is a path
that visits each node in the graph exactly once.
For example, the graph
\begin{center}
@ -481,12 +485,12 @@ Also stronger results have been achieved:
\begin{itemize}
\item
\index{Dirac's theorem}
\key{Dirac's theorem}:
\key{Dirac's theorem} \cite{dir52}:
If the degree of each node is at least $n/2$,
the graph contains a Hamiltonian path.
\item
\index{Ore's theorem}
\key{Ore's theorem}:
\key{Ore's theorem} \cite{ore60}:
If the sum of degrees of each non-adjacent pair of nodes
is at least $n$,
the graph contains a Hamiltonian path.
@ -525,7 +529,7 @@ It is possible to implement this solution in $O(2^n n^2)$ time.
\index{De Bruijn sequence}
A \key{De Bruijn sequence} is a string that contains
A \key{De Bruijn sequence}\footnote{N. G. de Bruijn (1918--2012) was a Dutch mathematician.} is a string that contains
every string of length $n$
exactly once as a substring, for a fixed
alphabet of $k$ characters.
@ -546,7 +550,7 @@ and each edge adds one character to the string.
The following graph corresponds to the above example:
\begin{center}
\begin{tikzpicture}
\begin{tikzpicture}[scale=0.8]
\node[draw, circle] (00) at (-3,0) {00};
\node[draw, circle] (11) at (3,0) {11};
\node[draw, circle] (01) at (0,2) {01};
@ -633,7 +637,8 @@ a complete tour will be found quickly.
\index{heuristic}
\index{Warnsdorff's rule}
\key{Warnsdorff's rule} is a simple and effective heuristic
\key{Warnsdorff's rule}\footnote{This heuristic was proposed
in Warnsdorff's book \cite{war23} in 1823.} is a simple and effective heuristic
for finding a knight's tour.
Using the rule, it is possible to efficiently construct a tour
even on a large board.

View File

@ -44,6 +44,11 @@
A note on two problems in connexion with graphs.
\emph{Numerische Mathematik}, 1(1):269--271, 1959.
\bibitem{dir52}
G. A. Dirac.
Some theorems on abstract graphs.
\emph{Proceedings of the London Mathematical Society}, 3(1):69--81, 1952.
\bibitem{edm65}
J. Edmonds.
Paths, trees, and flowers.
@ -117,6 +122,11 @@
A dynamic programming approach to sequencing problems.
\emph{Journal of the Society for Industrial and Applied Mathematics}, 10(1):196--210, 1962.
\bibitem{hie73}
C. Hierholzer and C. Wiener.
Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren.
\emph{Mathematische Annalen}, 6(1), 30--32, 1873.
\bibitem{hop71}
J. E. Hopcroft and J. D. Ullman.
A linear list merging algorithm.
@ -169,6 +179,11 @@
An $O(n \log n)$ algorithm for finding all repetitions in a string.
\emph{Journal of Algorithms}, 5(3):422--432, 1984.
\bibitem{ore60}
Ø. Ore.
Note on Hamilton circuits.
\emph{The American Mathematical Monthly}, 67(1):55, 1960.
\bibitem{pac13}
J. Pachocki and J. Radoszweski.
Where to use and how not to use polynomial string hashing.
@ -213,4 +228,9 @@
Dimer problem in statistical mechanics -- an exact result.
\emph{Philosophical Magazine}, 6(68):1061--1063, 1961.
\bibitem{war23}
H. C. von Warnsdorff.
\emph{Des Rösselsprungs einfachste und allgemeinste Lösung}.
Schmalkalden, 1823.
\end{thebibliography}