preliminaries a first attempt · a first attempt (cont.) ¥sort the edges of each slab by...

tisdag 6 april 2004 SMD156 Computational Geometry - Håkan Jonsson 1

SMD156Lecture 8

Point Location

SymbolicPerturbation

Randomized Algorithm

Trapezoidal Maps



Preliminaries

• A map is a planar subdivision !a DCEL".

• A point location query is to ask in which region!face" of a map a given !query" point lie.

• We strive to pre-process the map to enablehigh-speed point location queries.# Building a data structure suitable to search.


A first attempt

• Let P be a planar subdivision with n edges.

• Partition the plane into slabs by cutting alongvertical lines through vertices of P.

• Slabs# are partitioned by parts of edges, and

# contain no vertices.

• !You have met slabs before in this course, butwhere..?"# Exercise 2.1!


A first attempt (cont.)

• Sort the slabs by x-coordinate.# Then we can locate the slab that contains a query

point in O!log n" time.


A first attempt (cont.)

• Sort the edges of each slab by y-coordinate.# Then we can locate the face within a slab that contains a

query point in O!log n" time.• Two neighboring edges bound the same face.

• Bad news: Storage needed is O!n2".• Good news: A query takes just O!log n" time!

n/4


Trapezoidal maps• Non-crossing line segments are disjoint or share a

common end point.• A set of non-crossing line segments are in general

position if two distinct end points never lie on thesame vertical line.# Unrealistical

• Let S be a set of n non-crossing line segmentsenclosed in a rectangular box R.

• The trapezoidal map !or vertical/trapezoidaldecomposition" T!S" is obtained by drawing two verticalextensions from every end point in S - one upwardsand one downwards - until they hit another segmentor R.# The upper vertical extension and the lower vertical extension.


Example


Lemma 6.1

• $Each face in a trapezoidal map of a set S ingeneral position has one or two vertical sidesand exactly two non-vertical sides.$


Lemma 6.2

• $The trapezoidal map T!S" of a set S of n linesegments in general position contains at most6n+4 vertices and at most 3n+1 trapezoids.$# Trapezoidal maps are planar graphs.

# Euler%s formula.

# A trapezoidal map has linear size.


Trapezoids

• The top and bottom edgeof a trapezoid.

• Its point leftp !rightp analogously defined":

• top, bottom, leftp and rightp defines thetrapezoid.


Trapezoids (cont)

• Two trapezoids are adjacent if they meet alonga vertical edge.


A Randomized Incremental Algorithm

• Algorithm TrapezoidalMap!S"1. Determine a bounding box R that contains all of

the segments of S.

2. Initialize T, the trapezoidal map structure and D,the search structure:

• !T is the whole box R and D has only onenode corresponding to T."

3. Compute a random permutation s1, s2, ..., sn ofthe segments of S.

4. For each segment si !in this order", … .. .


The algorithm

. .. …• Find the set of trapezoids in T properly

intersected by si

• Remove them from T and replace them bythe new trapezoids that appear because of theinsertion of si

• Remove the leaves of D for the oldtrapezoids and create leaves for the new ones.Link them to the existing inner nodes byadding some new inner nodes.


About the algorithm

• This algorithm computes a trapezoidal mapand a search structure !a DAG" that leads tothe trapezoid containing a query point.

• Expected ... !Theorem 6.3"# …time to construct: O!n log n"

# …space to store: O!n"

# …time to query: O!log n"


tisdag 6 april 2004 SMD156 Computational Geometry - Håkan Jonsson 17 tisdag 6 april 2004 SMD156 Computational Geometry - Håkan Jonsson 18

tisdag 6 april 2004 SMD156 Computational Geometry - Håkan Jonsson 19 tisdag 6 april 2004 SMD156 Computational Geometry - Håkan Jonsson 20

Degenerate cases

• Two assumptions:1. General position !no distinct end points of line

segments share x-coordinate"

2. No query point on a vertical line through an endpoint of a line segment.

• One solution: Rotate &around origo, forinstance'.# Small rotations lead to numerical problems.

• Better: Symbolic $rotations$.


Symbolic perturbation

• A shear transformation:

• ! should be small and not change the x-orderof the input points.


Symbolic perturbation (cont)

• We store points as !x,y", not !x + !y, y".

• We need two operations to compute atrapezoidal map:# Determine on which side of the vertical line

through one given point p another given point qlies.

# Given a line segment with end points p1 and p2

determine if a given point q, on a vertical line thatintersects the line segment, lies above or below oron the line segment.



• Consider the first operation on the points !xp + !yp, yp"

and !xq + !yq, yq".# If xp(xq then the order between xp and xq gives the answer!

• This was how ! was chosen.

# If, on the other hand, xp=xq then the order between yp and yq

gives the answer.

• There is no need to know the value of !!

• Also the second operation can be performed withoutknowing the value of !!# NB! Slightly more involved; read pages 139- 140!

• So, we now got rid of the first assumption.



• In fact, also the second assumption !that noquery point may lie on a vertical line throughan end point of a line segment" can beremoved.

Theorem 6.5$All previous results are valid also without thetwo assumptions on the input we made.$

preliminaries a first attempt · a first attempt (cont.) ¥sort the edges of each slab by...

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