Point Location Strategies

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Point Location Strategies

• Idit Haran
• 13.4.2005

2
Outline

• Problem Description
• Other strategies
• Our strategy - Landmarks
• Benchmark
• Working with conic arrangementsand some number
  theory
• Future work
• Summary

3
Point Location Problem Description

• Given
• A map (which divides space of interest into
  regions)
• A query point q (specified by coordinates)
• Find the region of the map containing q

• Requirement
• (segments/conics)
• Fast query
• Reasonable preprocess time
• Good data structure (small space)

INPUT
OUTPUT
4
Other Strategies

• Strategies implemented in CGAL
• Naïve
• Walk along line
• Random Incremental Algorithm (Trapezoidal)
• Triangulation
• Other strategies
• Persistent search trees

5
Naïve Approach (CGAL)
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Walk Along a Line
7
Random Incremental Algorithm (Trapezoidal)
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Vertical Decomposition

• Assume a bounding box.
• Extend the vertical line from each vertex upward
  and downward, until it touches another segment.

9
Trapezoidal Maps

• Contains triangles and trapezoids.
• Each trapezoid or triangle is determined by two
  vertices and two segments.
• A refinement of the original subdivision.

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Trapezoidal Maps

• Theorem In a trapezoidal map of n segments,
  there are at most 6n4 vertices, and at most 3n1
  faces.
• Proof
• Vertices
• 2n original vertices
• 4n 2 extensions for each original vertex.
• 4 Vertices of the bounding box.
• Total 6n4
• Faces Similar

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Trapezoidal Map Data Structure

• For each Trapezoid store
• The vertices that define its right and left
  sides.
• The (up to two) neighboring trapezoids on right
  and left.
• (Optional) The neighboring trapezoids from above
  and below. This number might be linear, so we
  store only the leftmost of these.

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The DAG Search Structure
DAG Directed Acyclic Graph
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Using the DAG Search Structure
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Triangulation
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Persistent Search Trees (Sarnak-Tarjan)
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Other Strategies - Summary
More Algorithms on Delaunay Triangulations The
Delaunay Hierarchy, Jump Walk
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Point location using Landmarks
l
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Choosing the Landmarks

• Number of landmarks
• Distribution of the landmarks in the arrangement
• Geometric entities of the arrangement
• Vertices
• Edge Midpoints
• Points inside the faces Preprocess the
  arrangement into trapezoids and triangles
• Independent of the arrangement geometry

Grid
Random
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Quality of the landmarks Arrangement Distance
(AD)

• The Arrangement Distance (AD) between two points
  is the number pf faces in which the straight
  line segment that connect these points passes.
• Arrangement Distance ? Euclidean Distance

AD 0
AD 4
v
p
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Nearest Neighbor Search Structure

• Input
• Landmarks
• Query point q
• Question
• Find nearest landmark l to the query point q
• Answer
• Voronoi?
• - Again, point location problem !

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Kd-trees

• The kd-tree is a powerful data structure that is
  based on recursively subdividing a set of points
  with alternating axis-aligned hyper-planes.
• The classical kd-tree uses O(dn lgn)
  preprocessing time, O(dn) space and answers
  queries in time logarithmic in n, but exponential
  in d

l1
l3
l2
l4
l5
l7
l6
l8
l9
l10
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Kd-trees - Construction
l1
l9
l3
l2
l5
l6
l3
l2
l4
l5
l7
l6
l10
l8
l7
l8
l9
l10
l4
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Kd-trees - Query
l1
4
6
l9
7
l3
l2
l5
l6
8
l3
l2
5
l4
l5
l7
l6
9
10
3
l10
l8
l7
l8
l9
l10
2
1
l4
11
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Using Kd-trees with the landmarks

• Quick
• Approximate nearest neighbor
• Operations on points only
• Round all points to double
• Inexact computation.
• Final result is always exact !

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Landmarks point location Query
Find Nearest Landmark
Walk
Decide on Startup Face
Is query point in face ?
Cross to Next Face
No
Yes
Query Point Located
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Landmarks point location Query

• Decide on startup face
• From vertex
• Check all incident face to the vertex, and find
  the face in qs direction.
• From Edge
• Choose between the two incident face to the edge.
  
• From Face
• This is the startup face.

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Landmarks point location Query

• Is query point in face?
• Count the number of edges on fs boundary that
  are above q.
• Odd q is inside f found !
• Even need to cross to the next face.

q
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Landmarks point location Query

• Cross to next face
• Create a segment s from the landmark l to the
  query point q
• find the edge e in f that intersects s
• flip e to get a new face f.
• check if q is in f

f
e
f
s
q
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Benchmark

• Categories Query Preprocess time
• Algorithms Naïve, Walk, Triangle, RIC, 2
  variants of landmarks algorithm.
• Arrangements random and degenerate. similar
  results.

Random
Onebig
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Benchmark Query Arrangements of random line
segments
Time in milliseconds
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Benchmark Preprocess Arrangements of random
line segments
Time in seconds
Naïve and Walk algorithms do not require any
specific preprocessing besides constructing the
arrangement
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Landmarks Algorithm Analysis
Algorithm performance for varying number of
random landmarks
Results on an arrangement of 500 original
segments, include 56,000 edges and 28,000
vertices
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Landmarks Algorithm Analysis
Algorithm performance vs. the type of landmarks
used
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Landmarks Algorithm Analysis
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Conics

• Differences from segments
• Bench results
• Problems invoked, and their solutions
• Cross to next face
• The chord method
• Number type theory

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Conics differences from segments

• More complex curves (every operator on the curves
  takes longer)
• Working with irrational number types
• Basically the same methods work (except triangle)

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Benchmark Query Arrangements of random conic
arcs
Time in seconds
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Conics - Cross to Next Face

• Find exact intersection point a complex
  operation on conics !
• Instead use a simple method to decide what edge
  to flip.
• The idea check vertical order on the left and
  right boundaries of the common x-range.

e
l
l
e
s
e
s
q
q
q
s
l
x-range
x-range
x-range
(a) (b)
(c)
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Whats the problem with the vertices in an
arrangement of conic arcs ?
v
p
Problem 1vertex may be complex irrational
number (algebraic with degree 4)
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The Chord Method

• Given
• Unit circle equation x2 y2 1
• The point (-1,0) is on the circle
• Find
• Rational point v close to v
• Solution
• Every line passing through this point isy
  t(x1) where t is the slope of the line.
• solve x2 (t(x1))2 1

x2y21
v
yt(x1)
(-1,0)
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One Rational Point

• Apparently, not all conics with rational (or even
  integer) coefficients have rational points on
  them
• For example, the circle x2 y2 3 does
  not have any rational points on it.

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Does a conic have rational solution?

• Change the equation into homogenic coordinates
  
• A r x2 s y2 t xy u x v y w 0
  ? (x?x/z,
  y?y/z)B r x2 s y2 t xy u xz v yz
  wz2 0
• (Rational solution to A ? Integer solution to
  B)
• Theory 1 if there is a solution for every
  congruence modulo pk for every prime p and every
  k, then there is a solution.
• Theory 2 Check only p2 and ps that divide
  the determinant of the equation coefficients. It
  is also sufficient to check only for k lt 3 the
  largest exponent in the determinant
  factorization.

3 Variables
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How to find a rational solution (if there is
one)

• Theoremif a conic equation has an integer
  solution,
• then the size of the solution ? (3F),
• where F is the sum of the absolute value of the
  coefficients.

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What to do in practice?

• Change the equation into homogenic coordinates
• Go over all small pk and check if there is a
  solution modulo pk.
• One NO no solution.
• All Yes
• search for pairs (x,y) so that xlt3F, ylt3F
• check whether z is also an integer.
• YES we found a solution.
• All NO there is no solution.
• Complexity (3F)2

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If there are no rational points on the conic?

• Euclids algorithm to approximate real number x
  into rational number r

x a0 1/x1, x1gt1, a0 - integer x1 a1
1/x2, x2gt1, a1 - interger Finally, r a0
1/(a1 (1/ (a2 .) ) ) pn/qn.
If x is irrational, and the algorithm ends after
n stages, then x - pn/qn lt 1/qn2
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How to find ??

• ? the approximation value.should stay close
  to the vertex.
• Find the minimum distance between v and other
  vertices. or
• Find the minimum distance to all edges/vertices
  in the adjacent faces to the vertex.

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Landmarks future work

• Rationalization of the vertices
• More types of landmarks
• Combination of different landmarks types
• Memory usage of the algorithms

Halton sequence Hammersley Points
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Summary

• Point location using landmarks
• Fast query
• Efficient preprocess, using inexact computation
• Low memory
• Simple to understand and implement
• Many ways to improvements different search
  structure, other choice of points