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Improve language

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Antti H S Laaksonen 2017-05-14 13:51:27 +03:00
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chapter30.tex
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@ -8,9 +8,9 @@ The idea in such algorithms is to represent
an instance of the problem as a set of events that correspond an instance of the problem as a set of events that correspond
to points in the plane. to points in the plane.
The events are processed in increasing order The events are processed in increasing order
according to their x or y coordinate. according to their x or y coordinates.
As an example, let us consider the following problem: As an example, consider the following problem:
There is a company that has $n$ employees, There is a company that has $n$ employees,
and we know for each employee their arrival and and we know for each employee their arrival and
leaving times on a certain day. leaving times on a certain day.
@ -19,8 +19,8 @@ employees that were in the office at the same time.
The problem can be solved by modeling 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. correspond to their arrival and leaving times.
After sorting the events, we can go through them After sorting the events, we go through them
and keep track of the number of people in the office. and keep track of the number of people in the office.
For example, the table For example, the table
\begin{center} \begin{center}
@ -122,7 +122,7 @@ The symbols $+$ and $-$ indicate whether the
value of the counter increases or decreases, value of the counter increases or decreases,
and the value of the counter is shown below. and the value of the counter is shown below.
The maximum value of the counter is 3 The maximum value of the counter is 3
between John's arrival time and Maria's leaving time. between John's arrival and Maria's leaving.
The running time of the algorithm is $O(n \log n)$, The running time of the algorithm is $O(n \log n)$,
because sorting the events takes $O(n \log n)$ time because sorting the events takes $O(n \log n)$ time
@ -213,9 +213,9 @@ $y_1$ and $y_2$, and add this number to the total
number of intersection points. number of intersection points.
To store y coordinates of horizontal segments, To store y coordinates of horizontal segments,
we can use a binary indexed or a segment tree, we can use a binary indexed or segment tree,
possibly with index compression. possibly with index compression.
Using such a structure, processing each event When such structures are used, processing each event
takes $O(\log n)$ time, so the total running takes $O(\log n)$ time, so the total running
time of the algorithm is $O(n \log n)$. time of the algorithm is $O(n \log n)$.
@ -295,7 +295,7 @@ Thus, it suffices to only consider points
that are located in those ranges, that are located in those ranges,
which makes the algorithm efficient. which makes the algorithm efficient.
For example, in the following picture the For example, in the following picture, the
region marked with dashed lines contains region marked with dashed lines contains
the points that can be within a distance of $d$ the points that can be within a distance of $d$
from the active point: from the active point:
@ -332,7 +332,7 @@ from the active point:
\end{center} \end{center}
The efficiency of the algorithm is based on the fact The efficiency of the algorithm is based on the fact
that such a region always contains that the region always contains
only $O(1)$ points. only $O(1)$ points.
We can go through those points in $O(\log n)$ time We can go through those points in $O(\log n)$ time
by maintaining a set of points whose x coordinate by maintaining a set of points whose x coordinate
@ -401,8 +401,9 @@ the convex hull is as follows:
an easy way 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 first locates the leftmost
in two parts: and rightmost points, and then
constructs the convex hull in two parts:
first the upper hull and then the lower hull. first the upper hull and then the lower hull.
Both parts are similar, so we can focus on Both parts are similar, so we can focus on
constructing the upper hull. constructing the upper hull.
@ -414,7 +415,7 @@ add each point to the hull.
Always after adding a point to the hull, Always after adding a point to the hull,
we make sure that the last line segment we make sure that the last line segment
in the hull does not turn left. in the hull does not turn left.
As long as this does not hold, we repeatedly remove the As long as it turns left, we repeatedly remove the
second last point from the hull. second last point from the hull.
The following pictures show how The following pictures show how