845 lines
24 KiB
TeX
845 lines
24 KiB
TeX
\chapter{Sweep line algorithms}
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\index{sweep line}
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Many geometric problems can be solved using
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\key{sweep line} algorithms.
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The idea in such algorithms is to represent
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the problem as a set of events that correspond
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to points in the plane.
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The events are processed in increasing order
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according to their x or y coordinate.
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As an example, let us consider a problem
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where there is a company that has $n$ employees,
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and we know for each employee their arrival and
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leaving times on a certain day.
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Our task is to calculate the maximum number of
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employees that were in the office at the same time.
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The problem can be solved by modeling the situation
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so that each employee is assigned two events that
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corresponds to their arrival and leaving times.
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After sorting the events, we can go through them
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and keep track of the number of people in the office.
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For example, the table
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\begin{center}
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\begin{tabular}{ccc}
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person & arrival time & leaving time \\
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\hline
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John & 10 & 15 \\
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Maria & 6 & 12 \\
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Peter & 14 & 16 \\
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Lisa & 5 & 13 \\
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\end{tabular}
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\end{center}
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corresponds to the following events:
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\begin{center}
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\begin{tikzpicture}[scale=0.6]
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\draw (0,0) rectangle (17,-6.5);
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\path[draw,thick,-] (10,-1) -- (15,-1);
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\path[draw,thick,-] (6,-2.5) -- (12,-2.5);
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\path[draw,thick,-] (14,-4) -- (16,-4);
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\path[draw,thick,-] (5,-5.5) -- (13,-5.5);
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\draw[fill] (10,-1) circle [radius=0.05];
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\draw[fill] (15,-1) circle [radius=0.05];
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\draw[fill] (6,-2.5) circle [radius=0.05];
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\draw[fill] (12,-2.5) circle [radius=0.05];
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\draw[fill] (14,-4) circle [radius=0.05];
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\draw[fill] (16,-4) circle [radius=0.05];
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\draw[fill] (5,-5.5) circle [radius=0.05];
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\draw[fill] (13,-5.5) circle [radius=0.05];
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\node at (2,-1) {John};
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\node at (2,-2.5) {Maria};
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\node at (2,-4) {Peter};
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\node at (2,-5.5) {Lisa};
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\end{tikzpicture}
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\end{center}
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We go through the events from left to right
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and maintain a counter.
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Always when a person arrives, we increase
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the value of the counter by one,
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and when a person leaves,
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we decrease the value of the counter by one.
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The answer to the problem is the maximum
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value of the counter during the algorithm.
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In the example, the events are processed as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.6]
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\path[draw,thick,->] (0.5,0.5) -- (16.5,0.5);
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\draw (0,0) rectangle (17,-6.5);
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\path[draw,thick,-] (10,-1) -- (15,-1);
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\path[draw,thick,-] (6,-2.5) -- (12,-2.5);
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\path[draw,thick,-] (14,-4) -- (16,-4);
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\path[draw,thick,-] (5,-5.5) -- (13,-5.5);
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\draw[fill] (10,-1) circle [radius=0.05];
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\draw[fill] (15,-1) circle [radius=0.05];
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\draw[fill] (6,-2.5) circle [radius=0.05];
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\draw[fill] (12,-2.5) circle [radius=0.05];
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\draw[fill] (14,-4) circle [radius=0.05];
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\draw[fill] (16,-4) circle [radius=0.05];
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\draw[fill] (5,-5.5) circle [radius=0.05];
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\draw[fill] (13,-5.5) circle [radius=0.05];
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\node at (2,-1) {John};
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\node at (2,-2.5) {Maria};
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\node at (2,-4) {Peter};
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\node at (2,-5.5) {Lisa};
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\path[draw,dashed] (10,0)--(10,-6.5);
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\path[draw,dashed] (15,0)--(15,-6.5);
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\path[draw,dashed] (6,0)--(6,-6.5);
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\path[draw,dashed] (12,0)--(12,-6.5);
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\path[draw,dashed] (14,0)--(14,-6.5);
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\path[draw,dashed] (16,0)--(16,-6.5);
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\path[draw,dashed] (5,0)--(5,-6.5);
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\path[draw,dashed] (13,0)--(13,-6.5);
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\node at (10,-7) {$+$};
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\node at (15,-7) {$-$};
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\node at (6,-7) {$+$};
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\node at (12,-7) {$-$};
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\node at (14,-7) {$+$};
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\node at (16,-7) {$-$};
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\node at (5,-7) {$+$};
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\node at (13,-7) {$-$};
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\node at (10,-8) {$3$};
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\node at (15,-8) {$1$};
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\node at (6,-8) {$2$};
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\node at (12,-8) {$2$};
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\node at (14,-8) {$2$};
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\node at (16,-8) {$0$};
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\node at (5,-8) {$1$};
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\node at (13,-8) {$1$};
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\end{tikzpicture}
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\end{center}
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The symbols $+$ and $-$ indicate whether the
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value of the counter increases or decreases,
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and the value of the counter is shown below.
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The maximum value of the counter is 3
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between John's arrival time and Maria's leaving time.
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The running time of the algorithm is $O(n \log n)$,
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because sorting the events takes $O(n \log n)$ time
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and the rest of the algorithm takes $O(n)$ time.
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\section{Intersection points}
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\index{intersection point}
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Given a set of $n$ line segments, each of them being either
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horizontal or vertical, consider the problem of
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counting the total number of intersection points.
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For example, when the line segments are
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\begin{center}
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\begin{tikzpicture}[scale=0.5]
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\path[draw,thick,-] (0,2) -- (5,2);
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\path[draw,thick,-] (1,4) -- (6,4);
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\path[draw,thick,-] (6,3) -- (10,3);
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\path[draw,thick,-] (2,1) -- (2,6);
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\path[draw,thick,-] (8,2) -- (8,5);
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\end{tikzpicture}
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\end{center}
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there are three intersection points:
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\begin{center}
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\begin{tikzpicture}[scale=0.5]
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\path[draw,thick,-] (0,2) -- (5,2);
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\path[draw,thick,-] (1,4) -- (6,4);
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\path[draw,thick,-] (6,3) -- (10,3);
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\path[draw,thick,-] (2,1) -- (2,6);
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\path[draw,thick,-] (8,2) -- (8,5);
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\draw[fill] (2,2) circle [radius=0.15];
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\draw[fill] (2,4) circle [radius=0.15];
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\draw[fill] (8,3) circle [radius=0.15];
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\end{tikzpicture}
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\end{center}
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It is easy to solve the problem in $O(n^2)$ time,
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because we can go through all possible pairs of segments
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and check if they intersect.
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However, we can solve the problem more efficiently
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in $O(n \log n)$ time using a sweep line algorithm.
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The idea is to generate three types of events:
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\begin{enumerate}[noitemsep]
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\item[(1)] horizontal segment begins
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\item[(2)] horizontal segment ends
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\item[(3)] vertical segment
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\end{enumerate}
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The following events correspond to the example:
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\begin{center}
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\begin{tikzpicture}[scale=0.6]
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\path[draw,dashed] (0,2) -- (5,2);
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\path[draw,dashed] (1,4) -- (6,4);
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\path[draw,dashed] (6,3) -- (10,3);
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\path[draw,dashed] (2,1) -- (2,6);
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\path[draw,dashed] (8,2) -- (8,5);
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\node at (0,2) {$1$};
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\node at (5,2) {$2$};
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\node at (1,4) {$1$};
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\node at (6,4) {$2$};
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\node at (6,3) {$1$};
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\node at (10,3) {$2$};
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\node at (2,3.5) {$3$};
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\node at (8,3.5) {$3$};
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\end{tikzpicture}
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\end{center}
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We go through the events from left to right
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and use a data structure that maintains a set of
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y coordinates where there is an active horizontal segment.
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At event 1, we add the y coordinate of the segment
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to the set, and at event 2, we remove the
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y coordinate from the set.
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Intersection points are calculated at event 3.
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When there is a vertical segment between points
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$y_1$ and $y_2$, we count the number of active
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horizontal segments whose y coordinate is between
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$y_1$ and $y_2$, and add this number to the total
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number of intersection points.
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An appropriate data structure for storing
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y coordinates of horizontal segments is either
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a binary indexed tree or a segment tree,
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possibly with index compression.
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Using such structures, processing each event
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takes $O(\log n)$ time, so the total running
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time of the algorithm is $O(n \log n)$.
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\section{Closest pair problem}
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\index{closest pair}
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Given a set of $n$ points, our next problem is
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to find two points whose distance is minimum.
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For example, if the points are
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\begin{center}
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\begin{tikzpicture}[scale=0.7]
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\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
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\draw (1,2) circle [radius=0.1];
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\draw (3,1) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5.5,1.5) circle [radius=0.1];
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\draw (6,2.5) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (9,1.5) circle [radius=0.1];
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\draw (10,2) circle [radius=0.1];
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\draw (1.5,3.5) circle [radius=0.1];
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\draw (1.5,1) circle [radius=0.1];
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\draw (2.5,3) circle [radius=0.1];
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\draw (4.5,1.5) circle [radius=0.1];
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\draw (5.25,0.5) circle [radius=0.1];
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\draw (6.5,2) circle [radius=0.1];
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\end{tikzpicture}
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\end{center}
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\begin{samepage}
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we should find the following points:
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\begin{center}
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\begin{tikzpicture}[scale=0.7]
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\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
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\draw (1,2) circle [radius=0.1];
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\draw (3,1) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5.5,1.5) circle [radius=0.1];
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\draw[fill] (6,2.5) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (9,1.5) circle [radius=0.1];
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\draw (10,2) circle [radius=0.1];
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\draw (1.5,3.5) circle [radius=0.1];
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\draw (1.5,1) circle [radius=0.1];
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\draw (2.5,3) circle [radius=0.1];
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\draw (4.5,1.5) circle [radius=0.1];
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\draw (5.25,0.5) circle [radius=0.1];
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\draw[fill] (6.5,2) circle [radius=0.1];
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\end{tikzpicture}
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\end{center}
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\end{samepage}
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This is another example of a problem
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that can be solved in $O(n \log n)$ time
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using a sweep line algorithm\footnote{Besides this approach,
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there is also an
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$O(n \log n)$ time divide-and-conquer algorithm \cite{sha75}
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that divides the points into two sets and recursively
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solves the problem for both sets.}.
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We go through the points from left to right
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and maintain a value $d$: the minimum distance
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between two points seen so far.
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At each point, we find the nearest point to the left.
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If the distance is less than $d$, it is the
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new minimum distance and we update
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the value of $d$.
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If the current point is $(x,y)$
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and there is a point to the left
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within a distance of less than $d$,
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the x coordinate of such a point must
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be between $[x-d,x]$ and the y coordinate
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must be between $[y-d,y+d]$.
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Thus, it suffices to only consider points
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that are located in those ranges,
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which makes the algorithm efficient.
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For example, in the following picture the
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region marked with dashed lines contains
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the points that can be within a distance of $d$
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from the active point:
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\begin{center}
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\begin{tikzpicture}[scale=0.7]
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\draw (0,0)--(12,0)--(12,4)--(0,4)--(0,0);
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\draw (1,2) circle [radius=0.1];
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\draw (3,1) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5.5,1.5) circle [radius=0.1];
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\draw (6,2.5) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (9,1.5) circle [radius=0.1];
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\draw (10,2) circle [radius=0.1];
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\draw (1.5,3.5) circle [radius=0.1];
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\draw (1.5,1) circle [radius=0.1];
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\draw (2.5,3) circle [radius=0.1];
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\draw (4.5,1.5) circle [radius=0.1];
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\draw (5.25,0.5) circle [radius=0.1];
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\draw[fill] (6.5,2) circle [radius=0.1];
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\draw[dashed] (6.5,0.75)--(6.5,3.25);
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\draw[dashed] (5.25,0.75)--(5.25,3.25);
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\draw[dashed] (5.25,0.75)--(6.5,0.75);
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\draw[dashed] (5.25,3.25)--(6.5,3.25);
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\draw [decoration={brace}, decorate, line width=0.3mm] (5.25,3.5) -- (6.5,3.5);
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\node at (5.875,4) {$d$};
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\draw [decoration={brace}, decorate, line width=0.3mm] (6.75,3.25) -- (6.75,2);
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\node at (7.25,2.625) {$d$};
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\end{tikzpicture}
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\end{center}
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The efficiency of the algorithm is based on the fact
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that such a region always contains
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only $O(1)$ points.
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We can go through those points in $O(\log n)$ time
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by maintaining a set of points whose x coordinate
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is between $[x-d,x]$, in increasing order according
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to their y coordinates.
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The time complexity of the algorithm is $O(n \log n)$,
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because we go through $n$ points and
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find for each point the nearest point to the left
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in $O(\log n)$ time.
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\section{Convex hull problem}
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A \key{convex hull} is the smallest convex polygon
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that contains all points of a given set.
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Convexity means that a line segment between
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any two vertices of the polygon is completely
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inside the polygon.
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\begin{samepage}
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For example, for the points
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\begin{center}
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\begin{tikzpicture}[scale=0.7]
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\draw (0,0) circle [radius=0.1];
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\draw (4,-1) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (6,3) circle [radius=0.1];
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\draw (2,4) circle [radius=0.1];
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\draw (0,2) circle [radius=0.1];
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\draw (1,1) circle [radius=0.1];
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\draw (2,2) circle [radius=0.1];
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\draw (3,2) circle [radius=0.1];
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\draw (4,0) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5,2) circle [radius=0.1];
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\draw (6,1) circle [radius=0.1];
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\end{tikzpicture}
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\end{center}
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\end{samepage}
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the convex hull is as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.7]
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\draw (0,0)--(4,-1)--(7,1)--(6,3)--(2,4)--(0,2)--(0,0);
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\draw (0,0) circle [radius=0.1];
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\draw (4,-1) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (6,3) circle [radius=0.1];
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\draw (2,4) circle [radius=0.1];
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\draw (0,2) circle [radius=0.1];
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\draw (1,1) circle [radius=0.1];
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\draw (2,2) circle [radius=0.1];
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\draw (3,2) circle [radius=0.1];
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\draw (4,0) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5,2) circle [radius=0.1];
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\draw (6,1) circle [radius=0.1];
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\end{tikzpicture}
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\end{center}
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\index{Andrew's algorithm}
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\key{Andrew's algorithm} \cite{and79} provides
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an easy way to
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construct the convex hull for a set of points
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in $O(n \log n)$ time.
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The algorithm constructs the convex hull
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in two parts:
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first the upper hull and then the lower hull.
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Both parts are similar, so we can focus on
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constructing the upper hull.
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First, we sort the points primarily according to
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x coordinates and secondarily according to y coordinates.
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After this, we go through the points and
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add each point to the hull.
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Always after adding a point to the hull,
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we make sure that the last line segment
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in the hull does not turn left.
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As long as this holds, we repeatedly remove the
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second last point from the hull.
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The following pictures show how
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Andrew's algorithm works:
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\\
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\begin{tabular}{ccccccc}
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\\
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\begin{tikzpicture}[scale=0.3]
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\draw (-1,-2)--(8,-2)--(8,5)--(-1,5)--(-1,-2);
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\draw (0,0) circle [radius=0.1];
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\draw (4,-1) circle [radius=0.1];
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\draw (7,1) circle [radius=0.1];
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\draw (6,3) circle [radius=0.1];
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\draw (2,4) circle [radius=0.1];
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\draw (0,2) circle [radius=0.1];
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\draw (1,1) circle [radius=0.1];
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\draw (2,2) circle [radius=0.1];
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\draw (3,2) circle [radius=0.1];
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\draw (4,0) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5,2) circle [radius=0.1];
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\draw (6,1) circle [radius=0.1];
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|
|
\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];
|
|
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\draw (1,1) circle [radius=0.1];
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\draw (2,2) circle [radius=0.1];
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\draw (3,2) circle [radius=0.1];
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\draw (4,0) circle [radius=0.1];
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\draw (4,3) circle [radius=0.1];
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\draw (5,2) circle [radius=0.1];
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\draw (6,1) circle [radius=0.1];
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\draw (0,0)--(0,2)--(2,4)--(6,3)--(7,1);
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\end{tikzpicture}
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\\
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17 & & 18 & & 19 & & 20
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\end{tabular}
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