842 lines
24 KiB
TeX
842 lines
24 KiB
TeX
\chapter{Sweep line algorithms}
|
|
|
|
\index{sweep line}
|
|
|
|
Many geometric problems can be solved using
|
|
\key{sweep line} algorithms.
|
|
The idea in such algorithms is to represent
|
|
the problem as a set of events that correspond
|
|
to points in the plane.
|
|
The events are processed in a sorted order
|
|
according to their x or y coordinate.
|
|
|
|
As an example, let us consider a problem
|
|
where there is a company that has $n$ employees,
|
|
and we know for each employee arrival and
|
|
leaving times on a certain day.
|
|
Our task is to calculate the maximum number of
|
|
employees that were in the office at the same time.
|
|
|
|
The problem can be solved by modelling the situation
|
|
so that each employee is assigned two events that
|
|
corresponds to the arrival and leaving times.
|
|
After sorting the events, we can go trough them
|
|
and keep track of the number of people in the office.
|
|
For example, the table
|
|
\begin{center}
|
|
\begin{tabular}{ccc}
|
|
person & arrival time & leaving time \\
|
|
\hline
|
|
Uolevi & 10 & 15 \\
|
|
Maija & 6 & 12 \\
|
|
Kaaleppi & 14 & 16 \\
|
|
Liisa & 5 & 13 \\
|
|
\end{tabular}
|
|
\end{center}
|
|
generates the following events:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.6]
|
|
\draw (0,0) rectangle (17,-6.5);
|
|
\path[draw,thick,-] (10,-1) -- (15,-1);
|
|
\path[draw,thick,-] (6,-2.5) -- (12,-2.5);
|
|
\path[draw,thick,-] (14,-4) -- (16,-4);
|
|
\path[draw,thick,-] (5,-5.5) -- (13,-5.5);
|
|
|
|
\draw[fill] (10,-1) circle [radius=0.05];
|
|
\draw[fill] (15,-1) circle [radius=0.05];
|
|
\draw[fill] (6,-2.5) circle [radius=0.05];
|
|
\draw[fill] (12,-2.5) circle [radius=0.05];
|
|
\draw[fill] (14,-4) circle [radius=0.05];
|
|
\draw[fill] (16,-4) circle [radius=0.05];
|
|
\draw[fill] (5,-5.5) circle [radius=0.05];
|
|
\draw[fill] (13,-5.5) circle [radius=0.05];
|
|
|
|
\node at (2,-1) {Uolevi};
|
|
\node at (2,-2.5) {Maija};
|
|
\node at (2,-4) {Kaaleppi};
|
|
\node at (2,-5.5) {Liisa};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
We go through the events from left to right
|
|
and maintain a counter.
|
|
Always when a person arrives, we increase
|
|
the value of the counter by one,
|
|
and when a person leaves,
|
|
we decrease the value by one.
|
|
The answer for the problem is the maximum
|
|
value of the counter during the algorithm.
|
|
|
|
In the example, the events are processed as follows:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.6]
|
|
\path[draw,thick,->] (0.5,0.5) -- (16.5,0.5);
|
|
\draw (0,0) rectangle (17,-6.5);
|
|
\path[draw,thick,-] (10,-1) -- (15,-1);
|
|
\path[draw,thick,-] (6,-2.5) -- (12,-2.5);
|
|
\path[draw,thick,-] (14,-4) -- (16,-4);
|
|
\path[draw,thick,-] (5,-5.5) -- (13,-5.5);
|
|
|
|
\draw[fill] (10,-1) circle [radius=0.05];
|
|
\draw[fill] (15,-1) circle [radius=0.05];
|
|
\draw[fill] (6,-2.5) circle [radius=0.05];
|
|
\draw[fill] (12,-2.5) circle [radius=0.05];
|
|
\draw[fill] (14,-4) circle [radius=0.05];
|
|
\draw[fill] (16,-4) circle [radius=0.05];
|
|
\draw[fill] (5,-5.5) circle [radius=0.05];
|
|
\draw[fill] (13,-5.5) circle [radius=0.05];
|
|
|
|
\node at (2,-1) {Uolevi};
|
|
\node at (2,-2.5) {Maija};
|
|
\node at (2,-4) {Kaaleppi};
|
|
\node at (2,-5.5) {Liisa};
|
|
|
|
\path[draw,dashed] (10,0)--(10,-6.5);
|
|
\path[draw,dashed] (15,0)--(15,-6.5);
|
|
\path[draw,dashed] (6,0)--(6,-6.5);
|
|
\path[draw,dashed] (12,0)--(12,-6.5);
|
|
\path[draw,dashed] (14,0)--(14,-6.5);
|
|
\path[draw,dashed] (16,0)--(16,-6.5);
|
|
\path[draw,dashed] (5,0)--(5,-6.5);
|
|
\path[draw,dashed] (13,0)--(13,-6.5);
|
|
|
|
\node at (10,-7) {$+$};
|
|
\node at (15,-7) {$-$};
|
|
\node at (6,-7) {$+$};
|
|
\node at (12,-7) {$-$};
|
|
\node at (14,-7) {$+$};
|
|
\node at (16,-7) {$-$};
|
|
\node at (5,-7) {$+$};
|
|
\node at (13,-7) {$-$};
|
|
|
|
\node at (10,-8) {$3$};
|
|
\node at (15,-8) {$1$};
|
|
\node at (6,-8) {$2$};
|
|
\node at (12,-8) {$2$};
|
|
\node at (14,-8) {$2$};
|
|
\node at (16,-8) {$0$};
|
|
\node at (5,-8) {$1$};
|
|
\node at (13,-8) {$1$};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
The symbols $+$ and $-$ indicate whether the
|
|
value of the counter increases of decreases,
|
|
and the value of the counter is shown below.
|
|
The maximum value of the counter is 3
|
|
between Uolevi's arrival and Maija's leaving.
|
|
|
|
The running time of the solution is $O(n \log n)$,
|
|
because sorting the events takes $O(n \log n)$ time
|
|
and the rest of the algorithm takes $O(n)$ time.
|
|
|
|
\section{Intersection points}
|
|
|
|
\index{intersection point}
|
|
|
|
Given a set of $n$ line segments, each of them being either
|
|
horizontal or vertical, the problem is to efficiently
|
|
calculate the total number of intersection points.
|
|
For example, when the line segments are
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.5]
|
|
\path[draw,thick,-] (0,2) -- (5,2);
|
|
\path[draw,thick,-] (1,4) -- (6,4);
|
|
\path[draw,thick,-] (6,3) -- (10,3);
|
|
\path[draw,thick,-] (2,1) -- (2,6);
|
|
\path[draw,thick,-] (8,2) -- (8,5);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
there are three intersection points:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.5]
|
|
\path[draw,thick,-] (0,2) -- (5,2);
|
|
\path[draw,thick,-] (1,4) -- (6,4);
|
|
\path[draw,thick,-] (6,3) -- (10,3);
|
|
\path[draw,thick,-] (2,1) -- (2,6);
|
|
\path[draw,thick,-] (8,2) -- (8,5);
|
|
|
|
\draw[fill] (2,2) circle [radius=0.15];
|
|
\draw[fill] (2,4) circle [radius=0.15];
|
|
\draw[fill] (8,3) circle [radius=0.15];
|
|
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
It is easy to solve the problem in $O(n^2)$ time,
|
|
because we can go through all possible pairs of segments
|
|
and check if they intersect.
|
|
However, we can solve the problem more efficiently
|
|
in $O(n \log n)$ time using a sweep line algorithm.
|
|
|
|
The idea is to generate two types of events:
|
|
\begin{enumerate}[noitemsep]
|
|
\item[(1)] horizontal segment begins
|
|
\item[(2)] horizontal segment ends
|
|
\item[(3)] vertical segment
|
|
\end{enumerate}
|
|
|
|
The following events correspond to the example:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.6]
|
|
\path[draw,dashed] (0,2) -- (5,2);
|
|
\path[draw,dashed] (1,4) -- (6,4);
|
|
\path[draw,dashed] (6,3) -- (10,3);
|
|
\path[draw,dashed] (2,1) -- (2,6);
|
|
\path[draw,dashed] (8,2) -- (8,5);
|
|
|
|
\node at (0,2) {$1$};
|
|
\node at (5,2) {$2$};
|
|
\node at (1,4) {$1$};
|
|
\node at (6,4) {$2$};
|
|
\node at (6,3) {$1$};
|
|
\node at (10,3) {$2$};
|
|
|
|
\node at (2,3.5) {$3$};
|
|
\node at (8,3.5) {$3$};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
We go through the events from left to right
|
|
and use a data structure that maintains a set of
|
|
y coordinates where there is an active horizontal segment.
|
|
At event 1, we add the y coordinate of the segment
|
|
to the set, and at event 2, we remove the
|
|
y coordinate from the set.
|
|
|
|
Intersection points are calculated at event 3.
|
|
When there is a vertical segment between points
|
|
$y_1$ and $y_2$, we count the number of active
|
|
horizontal segments whose y coordinate is between
|
|
$y_1$ and $y_2$, and at this number to the total
|
|
number of intersection points.
|
|
|
|
An appropriate data structure for storing
|
|
y coordinates of horizontal segments is either
|
|
a binary-indexed tree or a segment tree,
|
|
possibly with index compression.
|
|
Using such structures, processing each event
|
|
takes $O(\log n)$ time, so the total running
|
|
time of the algorithm is $O(n \log n)$.
|
|
|
|
\section{Nearest points}
|
|
|
|
\index{nearest points}
|
|
|
|
Given a set of $n$ points, our next problem is
|
|
to find two points whose distance is minimum.
|
|
For example, if the points are
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
|
|
|
|
\draw (1,2) circle [radius=0.1];
|
|
\draw (3,1) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5.5,1.5) circle [radius=0.1];
|
|
\draw (6,2.5) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (9,1.5) circle [radius=0.1];
|
|
\draw (10,2) circle [radius=0.1];
|
|
\draw (1.5,3.5) circle [radius=0.1];
|
|
\draw (1.5,1) circle [radius=0.1];
|
|
\draw (2.5,3) circle [radius=0.1];
|
|
\draw (4.5,1.5) circle [radius=0.1];
|
|
\draw (5.25,0.5) circle [radius=0.1];
|
|
\draw (6.5,2) circle [radius=0.1];
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\begin{samepage}
|
|
we should find the following points:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
|
|
|
|
\draw (1,2) circle [radius=0.1];
|
|
\draw (3,1) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5.5,1.5) circle [radius=0.1];
|
|
\draw[fill] (6,2.5) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (9,1.5) circle [radius=0.1];
|
|
\draw (10,2) circle [radius=0.1];
|
|
\draw (1.5,3.5) circle [radius=0.1];
|
|
\draw (1.5,1) circle [radius=0.1];
|
|
\draw (2.5,3) circle [radius=0.1];
|
|
\draw (4.5,1.5) circle [radius=0.1];
|
|
\draw (5.25,0.5) circle [radius=0.1];
|
|
\draw[fill] (6.5,2) circle [radius=0.1];
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{samepage}
|
|
|
|
This problem can be also solved in $O(n \log n)$ time
|
|
using a sweep line algorithm.
|
|
We go through the points from left to right
|
|
and maintain a value $d$: the minimum distance
|
|
between two points so far.
|
|
At each point, we find the nearest point to the left.
|
|
If the distance is less than $d$, it is the
|
|
new minimum distance and we update
|
|
the value of $d$.
|
|
|
|
If the current point is $(x,y)$
|
|
and there is a point to the left
|
|
within a distance of less than $d$,
|
|
the x coordinate of such a point must
|
|
be between $[x-d,x]$ and the y coordinate
|
|
must be between $[y-d,y+d]$.
|
|
Thus, it suffices to only consider points
|
|
that are located in those ranges,
|
|
which makes the algorithm efficient.
|
|
|
|
For example, in the following picture the
|
|
region marked with dashed lines contains
|
|
the points that can be within a distance of $d$
|
|
from the active point:
|
|
\\
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
|
|
|
|
\draw (1,2) circle [radius=0.1];
|
|
\draw (3,1) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5.5,1.5) circle [radius=0.1];
|
|
\draw (6,2.5) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (9,1.5) circle [radius=0.1];
|
|
\draw (10,2) circle [radius=0.1];
|
|
\draw (1.5,3.5) circle [radius=0.1];
|
|
\draw (1.5,1) circle [radius=0.1];
|
|
\draw (2.5,3) circle [radius=0.1];
|
|
\draw (4.5,1.5) circle [radius=0.1];
|
|
\draw (5.25,0.5) circle [radius=0.1];
|
|
\draw[fill] (6.5,2) circle [radius=0.1];
|
|
|
|
\draw[dashed] (6.5,0.75)--(6.5,3.25);
|
|
\draw[dashed] (5.25,0.75)--(5.25,3.25);
|
|
\draw[dashed] (5.25,0.75)--(6.5,0.75);
|
|
\draw[dashed] (5.25,3.25)--(6.5,3.25);
|
|
|
|
\draw [decoration={brace}, decorate, line width=0.3mm] (5.25,3.5) -- (6.5,3.5);
|
|
\node at (5.875,4) {$d$};
|
|
\draw [decoration={brace}, decorate, line width=0.3mm] (6.75,3.25) -- (6.75,2);
|
|
\node at (7.25,2.625) {$d$};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
The efficiency of the algorithm is based on the fact
|
|
that the region limited by $d$ always contains
|
|
only $O(1)$ points.
|
|
We can go through those points in $O(\log n)$ time
|
|
by maintaining a set of points whose x coordinate
|
|
is between $[x-d,x]$ and that are sorted according
|
|
to the y coordinate.
|
|
|
|
The time complexity of the algorithm is $O(n \log n)$,
|
|
because it goes through $n$ points and
|
|
finds for each point the nearest point to the left
|
|
in $O(\log n)$ time.
|
|
|
|
\section{Convex hull}
|
|
|
|
The \key{convex hull} is the smallest convex polygon
|
|
that contains all points in a given set.
|
|
Convexity means that a line segment between
|
|
any two vertices of the polygon is completely
|
|
inside the polygon.
|
|
A good intuitive definition is that we surround
|
|
the points using a tight rope.
|
|
|
|
\begin{samepage}
|
|
For example, for the points
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{samepage}
|
|
the convex hull is as follows:
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\draw (0,0)--(4,-1)--(7,1)--(6,3)--(2,4)--(0,2)--(0,0);
|
|
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\index{Andrew's algorithm}
|
|
|
|
A good way to construct the convex hull
|
|
is \key{Andrew's algorithm}
|
|
that works in $O(n \log n)$ time.
|
|
The algorithm constructs the convex hull
|
|
in two steps:
|
|
first the upper hull and then the lower hull.
|
|
Both steps are similar, so we can focus on
|
|
constructing the upper hull.
|
|
|
|
We sort the points primarily according to
|
|
x coordinates and secondarily according to y coordinates.
|
|
After this, we go through the points and always
|
|
add the new point to the hull.
|
|
After adding a point we check using cross products
|
|
whether the tree last point in the hull turn left.
|
|
If this holds, we remove the middle point from the hull.
|
|
After this we keep checking again the three last points
|
|
and removing points, until the three last points
|
|
don't turn left.
|
|
|
|
The following pictures show how
|
|
Andrew's algorithm works:
|
|
\\
|
|
\begin{tabular}{ccccccc}
|
|
\\
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(1,1);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(1,1)--(2,2);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,2);
|
|
\end{tikzpicture}
|
|
\\
|
|
1 & & 2 & & 3 & & 4 \\
|
|
\end{tabular}
|
|
\\
|
|
\begin{tabular}{ccccccc}
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,2)--(2,4);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2)--(4,-1);
|
|
\end{tikzpicture}
|
|
\\
|
|
5 & & 6 & & 7 & & 8 \\
|
|
\end{tabular}
|
|
\\
|
|
\begin{tabular}{ccccccc}
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2)--(4,-1)--(4,0);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2)--(4,0);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2)--(4,0)--(4,3);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(3,2)--(4,3);
|
|
\end{tikzpicture}
|
|
\\
|
|
9 & & 10 & & 11 & & 12 \\
|
|
\end{tabular}
|
|
\\
|
|
\begin{tabular}{ccccccc}
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3)--(5,2);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3)--(5,2)--(6,1);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3)--(5,2)--(6,1)--(6,3);
|
|
\end{tikzpicture}
|
|
\\
|
|
13 & & 14 & & 15 & & 16 \\
|
|
\end{tabular}
|
|
\\
|
|
\begin{tabular}{ccccccc}
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3)--(5,2)--(6,3);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(4,3)--(6,3);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(6,3);
|
|
\end{tikzpicture}
|
|
& \hspace{0.1cm} &
|
|
\begin{tikzpicture}[scale=0.3]
|
|
\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
|
|
\draw (0,0) circle [radius=0.1];
|
|
\draw (4,-1) circle [radius=0.1];
|
|
\draw (7,1) circle [radius=0.1];
|
|
\draw (6,3) circle [radius=0.1];
|
|
\draw (2,4) circle [radius=0.1];
|
|
\draw (0,2) circle [radius=0.1];
|
|
|
|
\draw (1,1) circle [radius=0.1];
|
|
\draw (2,2) circle [radius=0.1];
|
|
\draw (3,2) circle [radius=0.1];
|
|
\draw (4,0) circle [radius=0.1];
|
|
\draw (4,3) circle [radius=0.1];
|
|
\draw (5,2) circle [radius=0.1];
|
|
\draw (6,1) circle [radius=0.1];
|
|
|
|
\draw (0,0)--(0,2)--(2,4)--(6,3)--(7,1);
|
|
\end{tikzpicture}
|
|
\\
|
|
17 & & 18 & & 19 & & 20
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|