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Antti H S Laaksonen 2017-02-18 21:15:20 +02:00
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luku30.tex
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@ -17,7 +17,7 @@ leaving times on a certain day.
Our task is to calculate the maximum number of Our task is to calculate the maximum number of
employees that were in the office at the same time. employees that were in the office at the same time.
The problem can be solved by modelling the situation The problem can be solved by modeling the situation
so that each employee is assigned two events that so that each employee is assigned two events that
corresponds to their arrival and leaving times. corresponds to their arrival and leaving times.
After sorting the events, we can go through them After sorting the events, we can go through them
@ -167,7 +167,7 @@ and check if they intersect.
However, we can solve the problem more efficiently However, we can solve the problem more efficiently
in $O(n \log n)$ time using a sweep line algorithm. in $O(n \log n)$ time using a sweep line algorithm.
The idea is to generate two types of events: The idea is to generate three types of events:
\begin{enumerate}[noitemsep] \begin{enumerate}[noitemsep]
\item[(1)] horizontal segment begins \item[(1)] horizontal segment begins
\item[(2)] horizontal segment ends \item[(2)] horizontal segment ends
@ -209,7 +209,7 @@ horizontal segments whose y coordinate is between
$y_1$ and $y_2$, and add this number to the total $y_1$ and $y_2$, and add this number to the total
number of intersection points. number of intersection points.
An appropriate data structure for An appropriate data structure for storing
y coordinates of horizontal segments is either y coordinates of horizontal segments is either
a binary indexed tree or a segment tree, a binary indexed tree or a segment tree,
possibly with index compression. possibly with index compression.
@ -269,7 +269,7 @@ we should find the following points:
\end{samepage} \end{samepage}
This is another example of a problem This is another example of a problem
that can be also solved in $O(n \log n)$ time that can be solved in $O(n \log n)$ time
using a sweep line algorithm. using a sweep line algorithm.
We go through the points from left to right We go through the points from left to right
and maintain a value $d$: the minimum distance and maintain a value $d$: the minimum distance
@ -345,8 +345,6 @@ that contains all points of a given set.
Convexity means that a line segment between Convexity means that a line segment between
any two vertices of the polygon is completely any two vertices of the polygon is completely
inside the polygon. inside the polygon.
An intuitive definition for a convex hull
is that it surrounds the given points using a tight rope.
\begin{samepage} \begin{samepage}
For example, for the points For example, for the points
@ -393,8 +391,8 @@ the convex hull is as follows:
\index{Andrew's algorithm} \index{Andrew's algorithm}
\key{Andrew's algorithm} is an easy algorithm \key{Andrew's algorithm} provides
that can be used to an easy way to
construct the convex hull for a set of points construct the convex hull for a set of points
in $O(n \log n)$ time. in $O(n \log n)$ time.
The algorithm constructs the convex hull The algorithm constructs the convex hull
@ -410,7 +408,7 @@ add the new point to the hull.
After adding a point we check using cross products After adding a point we check using cross products
whether the tree last point in the hull turn left. whether the tree last point in the hull turn left.
If this holds, we remove the middle point from the hull. If this holds, we remove the middle point from the hull.
After this we keep checking again the three last points After this we keep checking the three last points
and removing points, until the three last points and removing points, until the three last points
do not turn left. do not turn left.