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explain big-O notation better

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Motiejus Jakštys 2021-04-26 11:54:00 +03:00
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@ -205,7 +205,7 @@ thus convenient to analyze for both small and large scale generalization.
\includegraphics[width=.2\textwidth]{salvis-250k} \includegraphics[width=.2\textwidth]{salvis-250k}
\caption{Example scaled 1:250000.} \caption{Example scaled 1:250000.}
\end{subfigure} \end{subfigure}
\caption{Down-scaled original river (1:50000 and 1:250000).} \caption{Down-scaled original river.}
\label{fig:salvis-50-250} \label{fig:salvis-50-250}
\end{figure} \end{figure}
@ -339,12 +339,12 @@ This section defines vocabulary and terms as defined in the rest of the paper.
\item[Vertex] is a point on a plane, can be expressed by a pair of $(x,y)$ \item[Vertex] is a point on a plane, can be expressed by a pair of $(x,y)$
coordinates. coordinates.
\item[Line Segment (or Segment)] joins two vertices by a straight line. A \item[Line Segment] or \textsc{segment} joins two vertices by a straight
segment can be expressed by two coordinate pairs: $(x_1, y_1)$ and line. A segment can be expressed by two coordinate pairs: $(x_1, y_1)$
$(x_2, y_2)$. Line Segment and Segment are used interchangeably and $(x_2, y_2)$. Line Segment and Segment are used interchangeably
throughout the paper. 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. the real world. For example, a river or a coastline.
Geometrically, A line is a series of connected line segments, or, Geometrically, A line is a series of connected line segments, or,
@ -354,21 +354,29 @@ This section defines vocabulary and terms as defined in the rest of the paper.
\item[Bend] is a subset of a line that humans perceive as a curve. The \item[Bend] is a subset of a line that humans perceive as a curve. The
geometric definition is complex and is discussed in geometric definition is complex and is discussed in
section~\onpage{sec:definition-of-a-bend}. section~\ref{sec:definition-of-a-bend}.
\item[Baseline] is a line between bend's first and last vertex. \item[Baseline] is a line between bend's first and last vertex.
\item[Sum of inner angles] TBD. \item[Sum of inner angles] TBD.
\item[Algorithmic Complexity] also called \textsc{big o notation}, is a \item[Algorithmic Complexity] also called \textsc{big o notation}, is a
relative measure to explain how long will the algorithm run depending relative measure to explain how long will the algorithm runs depending
on it's input. For example, given $n$ objects and time complexity of on it's input. It is widely used in computing science when discussing
$O(n)$, the time it takes to execute the algorithm is proportional to the efficiency of a given algorithm.
$n$. Conversely, if complexity is $O(n^2)$, then the time it takes to
execute the algorithm is quadratic. $O$ notation was first suggested by For example, given $n$ objects and time complexity of $O(log(n))$, the
time it takes to execute the algorithm is logarithmic to $n$.
Conversely, if complexity is $O(n^2)$, then the time it takes to
execute the algorithm is quadratic depending on the input. Importantly,
if the input size doubles, the time it takes to run the algorithm
quadruples.
$O$ notation was first suggested by
Bachmann\cite{bachmann1894analytische} and Bachmann\cite{bachmann1894analytische} and
Laundau\cite{landau2000handbuch} in late XIX'th century, and adopted Landau\cite{landau2000handbuch} in late XIX'th century, and clarified
for computer science by Donald Knuth\cite{knuth1976big} in 1970s. and popularized for computing science by Donald
Knuth\cite{knuth1976big} in the 1970s.
\end{description} \end{description}