Solution Introduction to Computational Geometry Spring Homework Due Date BSB Professor

Solution Introduction to Computational Geometry Spring Homework Due

to Computational Geometry Spring Homework Due Date BSB Professor E mail Phone Suneeta

Solution Introduction to Computational Geometry Spring Homework

Due Date BSB Professor E mail Phone Suneeta

Solution Introduction to Computational Geometry

Solution Introduction

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computational geometry................/............................1
Introduction to Computational Geometry
50:198:473/56:198:573 (Spring 2016)
Homework:
4
Due Date:
4/13/2016
Ofce:
321 BSB
ProFessor:
Suneeta Ramaswami
E-mail:
[email protected]
URL:
http://crab.rutgers.edu/~rsuneeta
Phone:
(856)-225-6439
Homework Assignment 4
Give written (and legible) answers to the following questions. Please give succinct and mathe-
matically precise answers. Whenever you are asked to give an algorithm for a problem, you should
present the following: the algorithm, a justiFcation of its correctness, and a derivation of its run-
ning time. Write clear and convincing pseudo-code for your algorithms. All work must be done
independently.
I. (30 points)
We studied a randomized incremental algorithm for constructing the trapezoidal
decomposition. Recall that the geometric structure of the Fnal trapezoidal decomposition is
always the same, regardless of the order in which the segments are inserted. However, there
are instances of probabilistic constructions in which the Fnal geometric structure depends on
the particular order in which the objects are inserted. In this question, you will study one
such structure called the Binary Space Partition (BSP).
You are given a set of
n
non intersecting line segments in the plane, and you build a subdivision
recursively as follows: Initially, the subdivision contains no segments and only one face, the
entire space. When the Frst segment is inserted, this face is partitioned into two faces by
a splitting line that contains this segment. In general, as each segment is added, every face
of the existing subdivision that intersects this segment is split by the line containing the
segment. An example is shown in ±igure 1. In (a), the order of insertion is
s
1
,s
2
,s
3
,s
4
,s
5
.
In (b), the order of insertion is
s
4
,s
5
,s
3
,s
1
,s
2
. Observe that di²erent orders of insertions of
the segments result in di²erent subdivisions.
When a splitting line is added, each
future
segment which intersects this splitting line is
said to be
cut
by the line. (Therefore, if we say that a segment
v
is cut by a splitting line
containing
u
, then
v
must have been inserted after
u
.) In ±igure 1(a), segment
s
3
is cut by
the line extending segment
s
1
, and segments
s
3
and
s
5
are cut by the line extending
s
2
. Note
that segment
s
1
is not considered to be cut by
s
2
(because
s
2
was inserted after
s
1
).
1. Given
n
segments in general position, prove that the number of faces of the Fnal BSP
subdivision is equal to
n
+ 1 plus the total number of cuts (verify this in the Fgure
above).
Hint:
Use induction.
2. Give an example of a set of
n
nonintersecting line segments such that for one insertion
order, the resulting BSP has size
O
(
n
) and for another insertion order, the BSP has size
?(
n
2
).
3. Given two line segments
u
and
v
, deFne
index
(
u,v
) as follows: If the line containing
u
does not intersect
v
, then
index
(
u,v
) =
?
. Otherwise, deFne the index to be the number
of line segments that intersect this line and lie between
v
and the closest endpoint
of
u
. (In the Fgure below,
index
(
u,v
) = 3. In ±igure 1(a),
index
(
s
2
,s
5
) = 1, and
index
(
s
1
,s
3
) = 0.) Prove that if the segments are inserted in random order, then the
probability that
u
cuts
v
is at most
1
index
(
u,v
)+2
.

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