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19
bib.bib
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@ -150,3 +150,22 @@
year={1974},
publisher={Elsevier}
}
@article{knuth1976big,
title={Big omicron and big omega and big theta},
author={Knuth, Donald E},
journal={ACM Sigact News},
volume={8},
number={2},
pages={18--24},
year={1976},
publisher={ACM New York, NY, USA}
}
@book{bachmann1894analytische,
title={Die analytische zahlentheorie},
author={Bachmann, Paul},
volume={2},
year={1894},
publisher={Teubner}
}

135
mj-msc.tex
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@ -183,7 +183,7 @@ thus convenient to analyze for both small and large scale generalization.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{salvis-25k}
\caption{Example rivers for visual tests (1:25000)}
\caption{Example rivers for visual tests (1:25000).}
\label{fig:salvis-25}
\end{figure}
@ -197,15 +197,15 @@ river for a smaller scale is worthy.
\centering
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{salvis-50k}
\caption{Example scaled 1:50000}
\caption{Example scaled 1:50000.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.49\textwidth}
\centering
\includegraphics[width=.2\textwidth]{salvis-250k}
\caption{Example scaled 1:250000}
\caption{Example scaled 1:250000.}
\end{subfigure}
\caption{Down-scaled original river (1:50000 and 1:250000)}
\caption{Down-scaled original river (1:50000 and 1:250000).}
\label{fig:salvis-50-250}
\end{figure}
@ -276,6 +276,7 @@ many cases, corner cases are discussed and clarified.
Assume Euclidean geometry throughout this document, unless noted otherwise.
\subsection{Vocabulary and terminology}
\label{sec:vocab}
This section defines vocabulary and terms as defined in the rest of the paper.
@ -289,7 +290,7 @@ This section defines vocabulary and terms as defined in the rest of the paper.
$(x_2, y_2)$. Line Segment and Segment are used interchangeably
throughout the paper.
\item[Line] (or \textsc{linestring}) represents a single linear feature in
\item[Line], or \textsc{linestring}, represents a single linear feature in
the real world. For example, a river or a coastline.
Geometrically, A line is a series of connected line segments, or,
@ -305,6 +306,16 @@ This section defines vocabulary and terms as defined in the rest of the paper.
\item[Sum of inner angles] TBD.
\item[Algorithmic Complexity] also called \textsc{big o notation}, is a
relative measure to explain how long will the algorithm run depending
on it's input. For example, given $n$ objects and time complexity of
$O(n)$, the time it takes to execute the algorithm is proportional to
$n$. Conversely, if complexity is $O(n^2)$, then the time it takes to
execute the algorithm is quadratic. $O$ notation was first suggested by
Bachmann\cite{bachmann1894analytische} in late XIX'th century, and
adopted for computer science by Donald Knuth\cite{knuth1976big} in
1970s.
\end{description}
\subsection{Radians and Degrees}
@ -341,7 +352,7 @@ figure~\ref{fig:test-figures}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{test-figures}
\caption{Line geometries for automated test cases}
\caption{Line geometries for automated test cases.}
\label{fig:test-figures}
\end{figure}
@ -349,6 +360,25 @@ The full test suite can be executed with a single command, and completes in a
few seconds. Having an easily accessible test suite boosts confidence that no
unexpected bugs have snug in while modifying the algorithm.
\subsection{Reproducing generalizations in this paper}
\label{sec:reproducing-the-paper}
It is widely believed that the ability to reproduce the results of a published
study is important to the scientific community. In practice, however, it is
often hard to impossible: research methodologies, as well as algorithms
themselves, are explained in prose, which, due to the nature of the non-machine
language, lends itself to inexact interpretations.
This article, besides explaining the algorithm in prose, \emph{includes} the
program of the algorithm in a way that can be executed on reader's workstation.
On top of it, all the illustrations in this paper are generated using that
algorithm, from a predefined list of test geometries (test geometries were
explained in section~\ref{sec:automated-tests}).
Instructions how to re-generate all the visualizations are found in
appendix~\ref{sec:code-regenerate}. The visualization code serves as a good
example reference for anyone willing to start using the algorithm.
\section{Description of the implementation}
Like alluded in section~\ref{sec:introduction}, {\WM} paper skims over
@ -411,7 +441,7 @@ but with bends colored as polygons: each color is a distinctive bend.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig8-definition-of-a-bend}
\caption{Originally figure 8: detected bends are highlighted}
\caption{Originally figure 8: detected bends are highlighted.}
\label{fig:fig8-definition-of-a-bend}
\end{figure}
@ -431,14 +461,14 @@ when a single vertex is moved outwards the end of the bend.
\centering
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{fig5-gentle-inflection-before}
\caption{Before applying the inflection rule}
\caption{Before applying the inflection rule.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{fig5-gentle-inflection-after}
\caption{After applying the inflection rule}
\caption{After applying the inflection rule.}
\end{subfigure}
\caption{Originally figure 5: gentle inflections at the ends of the bend}
\caption{Originally figure 5: gentle inflections at the ends of the bend.}
\label{fig:fig5-gentle-inflection}
\end{figure}
@ -458,15 +488,15 @@ vertices to the next bend.
\centering
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{inflection-1-gentle-inflection-before}
\caption{Before applying the inflection rule}
\caption{Before applying the inflection rule.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{inflection-1-gentle-inflection-after}
\caption{After applying the inflection rule}
\caption{After applying the inflection rule.}
\end{subfigure}
\caption{Gentle inflection at the end of the bend when multiple vertices
are moved}
are moved.}
\label{fig:inflection-1-gentle-inflection}
\end{figure}
@ -494,58 +524,65 @@ The "quite small angle" was arbitrarily chosen to $\smallAngle$.
\subsection{Self-line Crossing When Cutting a Bend}
When bend's baseline crosses another bend, it is called self-crossing.
Self-crossing is undesirable in the upcoming operators, thus should be removed
following the rules of the article.
Self-crossing is undesirable for the upcoming bend manipulation operators, thus
should be removed. There are a few rules on when and how they should be removed
--- this section explains them in higher detail, discusses their time
complexity and applied optimizations. Figure~\ref{fig:fig6-selfcrossing} is
copied from the original article.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{fig6-selfcrossing-before}
\caption{Bend's baseline (dotted) is crossing a neighboring bend}
\caption{Bend's baseline (dotted) is crossing a neighboring bend.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{fig6-selfcrossing-after}
\caption{Self-crossing removed following the algorithm}
\caption{Self-crossing removed.}
\end{subfigure}
\caption{Originally figure 6: simple case of self-line crossing}
\caption{Originally figure 6: simple case of self-line crossing.}
\label{fig:fig6-selfcrossing}
\end{figure}
The original description does not go into detail which bends may self-cross, and which <TBD>
The self-line-crossing may happen not by the neighboring bend, but by any other
bend in the line. For example, the baseline of the bend may cross different
bends in between, as depicted in figure~\ref{fig:selfcrossing-1-non-neighbor}.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{selfcrossing-1-before}
\caption{Bend's baseline (dotted) is crossing a non-neighboring bend}
\caption{Bend's baseline (dotted) is crossing a non-neighboring bend.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.49\textwidth}
\includegraphics[width=\textwidth]{selfcrossing-1-after}
\caption{Self-crossing removed following the algorithm}
\caption{Self-crossing removed.}
\end{subfigure}
\caption{Self-crossing with non-neighboring bend}
\caption{Self-crossing with non-neighboring bend.}
\label{fig:selfcrossing-1-non-neighbor}
\end{figure}
Naively implemented, checking every bend with every bend is costs $O(n^2)$. In
other words, the time it takes to run the algorithm grows quadratically with
the with the number of vertices.
Looking at the {\WM} paper alone, it may seem like self-crossing may happen
only with the neighboring bend. This would mean an efficient $O(n)$
implementation\footnote{where $n$ is the number of bends in a line. See
explanation of \textsc{algorithmic complexity} in section~\ref{sec:vocab}.}.
However, as one can see in figure~\ref{fig:selfcrossing-1-non-neighbor}, it may
not be the case: any other bend in the line may be crossing it.
It is possible to optimize this step and skip checking some of the bends. Only
bends whose sum of inner angles is $\pi$ can ever self-cross. If the value is
less than $\pi$, it cannot cross other bends. That way, only a fraction of
bends need to be checked.
If one translates the requirements to code in a straightforward way, it would
be quite computationally expensive: naively implemented, complexity of checking
every bend with every bend is $O(n^2)$. In other words, the time it takes to
run the algorithm grows quadratically with the with the number of vertices.
It is possible to optimize this step and skip checking most of the bends. Only
bends whose sum of inner angles is larger than $\pi$ can ever self-cross. If
the value is less than $\pi$, it cannot cross other bends. That way, only a
fraction of bends need to be checked. The worst-case complexity is still
$O(n^2)$, when all bends' inner angles are larger than $\pi$, but, assuming no
more than $20\%$ of the bends' inner angles are larger than $\pi$, the time it
takes to run this piece of the algorithm drops by $80\%$.
\subsection{Attributes of a Single Bend}
\textsc{Compactness Index} is "the ratio of the area of the polygon over the
circle whose circumference length is the same as the length of the
circumference of the polygon" \cite{wang1998line}. Given a bend, its
@ -555,9 +592,9 @@ compactness index is calculated as follows:
\item Construct a polygon by joining first and last vertices of the bend.
\item Calculate area of the polygon $P$.
\item Calculate area of the polygon.
\item Calculate perimeter of the polygon $u$. The same value is the
\item Calculate perimeter $u$ of the polygon. The same value is the
circumference of the circle.
\item Given circle's perimeter $u$, circle's area $A$ is:
@ -599,8 +636,8 @@ Two conditions must be true to claim that a bend is isolated:
\begin{enumerate}
\item \textsc{average curvature} of neighboring bends, should be larger
than the "candidate" bend's curvature; this implementation arbitrarily
chose $\isolationThreshold$.
than the "candidate" bend's curvature. The article did not offer a
value, this implementation arbitrarily chose $\isolationThreshold$.
\item Bends on both sides of the "candidate" should be longer than a
certain value. This implementation does not (yet) define such a
@ -617,7 +654,7 @@ To find out whether two bends are similar, they are compared by 3 components:
\item Baseline length
\end{enumerate}
These 3 components represent a point in the 3-dimensional space, and Euclidean
Components 1, 2 and 3 represent a point in a 3-dimensional space, and Euclidean
distance $d$ between those is calculated to differentiate between bends $p$ and
$q$:
@ -627,7 +664,7 @@ $q$:
(baseline_p-baseline_q)^2}
\]
The more similar the bends are, the smaller the distance $d$.
The smaller the distance $d$, the more similar the bends are.
\subsection{Elimination Operator}
@ -651,19 +688,17 @@ The more similar the bends are, the smaller the distance $d$.
\section{Code listings}
\subsection{Reproducing the generalizations in this paper}
\subsection{Re-generating this paper}
\label{sec:code-regenerate}
We strongly believe in the ability to reproduce the results is critical for any
scientific work. To make it possible for this paper, all source files and
accompanying scripts have been attached to the PDF. To re-generate this
document and its accompanying graphics, run this script (assuming name of
this document is {\tt mj-msc-full.pdf}):
Like explained in section~\ref{sec:reproducing-the-paper}, illustrations in
this paper are generated from a small list of sample geometries. To observe
the source geometries or regenerate this paper, run this script (assuming
name of this document is {\tt mj-msc-full.pdf}):
\inputcode{bash}{extract-and-generate}
This was tested on Linux Debian 11 with upstream packages only.
\subsection{Algorithm code listings}
%\subsection{Algorithm code listings}
%\inputcode{postgresql}{wm.sql}
\end{appendices}