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explain big-O notation better

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Motiejus Jakštys 2021-05-19 22:57:48 +03:00 committed by Motiejus Jakštys
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mj-msc.tex
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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}
\caption{Example scaled 1:250000.}
\end{subfigure}
\caption{Down-scaled original river (1:50000 and 1:250000).}
\caption{Down-scaled original river.}
\label{fig:salvis-50-250}
\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 $(x,y)$
coordinates.
\item[Line Segment (or Segment)] joins two vertices by a straight line. A
segment can be expressed by two coordinate pairs: $(x_1, y_1)$ and
$(x_2, y_2)$. Line Segment and Segment are used interchangeably
\item[Line Segment] or \textsc{segment} joins two vertices by a straight
line. A segment can be expressed by two coordinate pairs: $(x_1, y_1)$
and $(x_2, y_2)$. 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,
@ -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
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[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
relative measure to explain how long will the algorithm runs depending
on it's input. It is widely used in computing science when discussing
the efficiency of a given algorithm.
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
Laundau\cite{landau2000handbuch} in late XIX'th century, and adopted
for computer science by Donald Knuth\cite{knuth1976big} in 1970s.
Landau\cite{landau2000handbuch} in late XIX'th century, and clarified
and popularized for computing science by Donald
Knuth\cite{knuth1976big} in the 1970s.
\end{description}