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Title: Substructures in Geometric Arrangements and ? -nets Esther Ezra

1
Substructures in Geometric Arrangements and ?
-nets Esther Ezra
Duke University
2
Union of simply-shaped bodies

S S1, , Sn a collection of n simply-shaped
bodies in d-space.
The union of S consists of all region of ?d that
are covered by at least one element of S .
Example
Union of triangles in the plane
All portions of the plane that are
covered by the triangles.

The union has two holes
The union boundary
3
Motivation Motion planning
The workspace

Input
Robot R, a set A A1, , An of n disjoint
obstacles.
The robot and the obstacles are d-dimensional
bodies.
The robot moves around the obstacles.
Goal Construct the free space
The set of all legal placements
of R, while translating R in d-space.

collision
No rotations
R does not intersect any of the obstacles in A
4
The configuration space
reference point

The robot R is mapped to a point.
Each obstacle Ai is mapped to the set
Pi x ? ? d R(x) ? Ai ? ?
A point p in Pi corresponds to
an illegal placement of R
and vice versa.

The forbidden placements of R
The expanded obstacle
5
The free space

The free space is the complement of ?1? i? n Pi

6
The Union as a substructure in arrangements

The arrangement A(S) is the subdivision
of space induced by S .
The combinatorial complexity of A(S)
The maximal number of
vertices/edges/faces of A(S) ?(nd)

Union is a substructure of the arrangement
7
The union complexity

The problem
What is the maximal number of
vertices/edges/faces that form
the boundary of the union of the bodies in S ?
Trivial bound O(nd) (tight!).

8
Previous results in 2DFat objects
Each of the angles ? ?

n ?-fat triangles.
Number of holes in the union O(n) .
Union complexity O(n loglog n) . Matousek et
al. 1994
Fat curved objects (simple-shaped)
n convex ?-fat objects.
Union complexity O(n) Efrat, Sharir. 2000.
n ?-curved objects.
Union complexity O(n) Efrat, Katz. 1999.
Union complexity is one order of magnitude
smaller than the arrangement complexity!

r/r ? ? , and ? ?1.
O(n1?) , for any ? gt0 .
r
r
r ? ?? diam(C) , D ? C, ? lt 1 is a constant.
C
r
D
9
Previous results in 3DFat Objects

Congruent cubes
n arbitrarily aligned congruent cubes.
Union complexity O(n2) Pach, Safruti, Sharir
2003.
Simple curved objects
n congruent infinite cylinders.
Union complexity O(n2) Agarwal, Sharir 2000.
n ?-round objects.
Union complexity O(n2) Aronov et al. 2006.
Union complexity is one order of magnitude
smaller than the arrangement complexity!
Each of these bounds is nearly-optimal.

r ? ?? diam(C) , D ? C, ? lt 1 is a constant.
C
r
D
10
Our results New union bounds.Fat Tetrahedra

Ezra, Sharir 2007
n ?-fat tetrahedra in ?3, of arbitrary sizes
O(n2) .
n arbitrary side-length cubes O(n2) .
Bound is nearly optimal!
Settling a Conjecture of Pach, Safruti, Sharir ,
2003
In particular
Obtain a simpler proof for congruent cubes.

?
?
A cube can be decomposed into O(1) fat
tetrahedra.
11
Our results New union bounds.Infinite cylinders

Ezra 2008
n infinite cylinders in ?3 of arbitrary radii
O(n2) .
Bound is nearly optimal!
Settling a Conjecture of Agarwal, Sharir 2000.
In particular
Obtain a simpler proof for congruent infinite
cylinders.
It is crucial that the cylinders are infinite.
Otherwise, the union complexity is ?(n3) .

12
Envelopes in d-space
The lower envelope is monotone.

Input F F1, , Fn a collection of n
(d-1)-variate functions.
The lower envelope EF of F is the
pointwise minimum of these functions
EF(x) minF ? F F(x) , for x ? ?d-1 .
Sharir 1994
The complexity of the lower envelope
of n simple algebraic surfaces in
d-space is O(nd-1) .

O(n2) for d3 .
13
The sandwich region

Agarwal et al. 1996, koltun sharir 2003
The complexity of the sandwich region enclosed
between
the lower envelope of n simple algebraic surfaces
in
d-space and the upper envelope of another such
collection
is O(nd-1) , for d ? 4.
For d3, the complexity
of the sandwich region
is O(n2)

14
Main idea Reduce cylinders to envelopes

Decompose space into vertical prism cells ?.
Partition the boundary of the cylindersinto
canonical strips.
Show that in each cell ? most of the union
vertices v appear on the sandwich region
enclosed between two envelopes of the strips.

Apply the bound O(n2) of Agarwal et al. 1996.
15
Small-size ? -nets for Axis-Parallel Rectangles
and Boxes
16
Range Spaces

Range space (X, R)
X Ground set.
R Ranges Subsets of X .
R ? 2X
Abstract form Hypergraphs.
X vertices.
R hyperedges.

17
Geometric Range Spaces

specification X ?d , R set of simply-shaped
regions in ?d .
X Points on the real line.
R Intervals.
X Points on the plane.
R halfplanes.
X Points on the plane.
R Disks.

X finite Discrete model , X infinite
Continuous model
18
The hitting-set problem

A hitting set for (X, R) is a subset H ? X, s.t.,
for any Q ? R , Q ? H ? ? .
Goal Find smallest hitting set.
A fundamental problem with various applications

19
Art-gallery Application

Useful application art-gallery
Input Polygon P.
Goal Find minimum-size set of
guards (points in P) that see the entire P.

Visibility polygon
P
q
P
q
g
g, q are visible. g, q are invisible
g
q
Vis(g)
q
20
Art-gallery Application
P

The actual goal is to cover P with a
minimum number of visibility polygons!
This is the so-called set-cover problem.
Observation q lies inside Vis(g) iff
g lies inside Vis(q) !
Hitting-set instance
X points in a polygonal region P.
R all visibility polygons in P.

q
g
Vis(g)
Vis(q)
P
q
g
21
Sensor networking application

Input
C set of clients c.
A - locations of antennas a, each of which with
a unit sensing radius.
Each antenna serves the clients within
its sensing distance
Goal Find minimum-size set of antennas that
serve all clients.
The actual goal is to find a smallest set of
disks
(centered in A) that covers C.

c
a
22
Sensor networking application

Hitting-set instance
X antennas locations
R D(c,1) unit disks with
client-centers.

c
a
Observation a covers c iff a is inside D(c,1)
23
Approximation for hitting sets

Finding a hitting set of smallest size is
NP-hard,
even for geometric range spaces!
Abstract range spaces
Greedy algorithm. Approximation factor O(1 log
X)
Best known approximation achieved in polynomial
time!
Geometric range spaces
Achieve improved approximation factor!
Approximation factor O(1 log OPT) ,
OPT size of the smallest hitting set.
Sometimes, the approximation factor is even
smaller!
This is achieved via ?-nets

24
? -nets for range spaces

Given
A range space (X, R) , assume X is finite, X
n .
A parameter 0 lt ? lt 1 ,
An ? -net for (X, R) is a subset N ? X that hits
every
range Q ? R, with Q ? X ? ? n .
N is a hitting set for all the heavy'' ranges.
Example
Points and intervals on the real line N 1/? .

Bound does not depend on n.
? n
25
Approximation for geometric hitting sets

The Bronimann-Goodrich technique / LP-relaxation
If (X, R) admits an ? -net of size f(1/? ) ,
then there exists a polynomial-time approximation
algorithm that reports a hitting set of size
O(f(OPT)) .
Idea
Assign weights on X s.t each range Q ? R becomes
heavy .
Construct an ? -net for the weighed range space.
Each range is hit by the ? -net.
Small-size ? -nets imply small approximation
factors!

26
An upper bound for the ? -net size

The ? -net theorem Haussler-Welzl, 87
If the ranges are simply-shaped regions, then,
for any ? gt 0, a random sample of size
O(1/? log (1/? ))
is an ? -net, with constant probability.
Remark
In fact, it is sufficient to assume that the
number
of ranges is only polynomial in n.

Bound does not depend on n.
27
Is the bound optimal?

Theorem Komlos, Pach, Woeginger 92
The bound is tight!
The construction
Artificial on abstract hypergraphs
(non-geometric!).
No lower bound better than ?(1/? ) is known in
geometry.
What is the actual bound? O(1/? ) ?
Goal Obtain smaller bounds for
geometric range spaces. Ideally O(1/? ),
but anything better than O(1/? log (1/? )) is
exciting !

Achieved by points and intervals on the real line.
28
Previous results

Points and halfspaces in 2D, 3D.
O(1/? ) Matousek 92,
Pyrga, Ray 08, Har-Peled et al. 08
Points and disks, or pseudo-disks in 2D O(1/? )
Matousek, Seidel, Welzl 90, Pyrga, Ray 08.

Pseudo-disks
29
Our results Aronov, Ezra, Sharir

Points and axis-parallel rectangles in the plane.
? -net size is O(1/? log log (1/? )) .
Points and axis-parallel boxes in 3-space.
? -net size is O(1/? log log (1/? )) .
Points and ?-fat triangles in the plane.
? -net size is O(1/? log log (1/? )) .
Points uniformly distributed over the unit-cube,
and axis-parallel boxes in d-space.
? -net size is O(1/? log log (1/? )) .

Each of the angles ? ?
30
Improved approximation factorsfor geometric
hitting sets

Ranges previous
bound new bound
Axis-parallel rectangles log OPT
log log OPT
Axis-parallel 3-boxes log OPT
log log OPT
?-fat triangles log OPT
log log OPT
Axis-parallel d-boxes log OPT
log log OPT

Uniformly distributed points in 0,1d .
31
Main idea for axis parallel rectangles Use
two-level sampling

Primary sampling step
Obtain an initial sample S of 1/? points of X.
Each rectangle Q with Q ? S ? ,
contains at most O(? n log (1/? )) points of X.
Second sampling step (repair step)
In each heavy rectangle Q ? R ,
with Q ? S ? (bad), sample additional
points to guarantee that Q is stabbed by the net.

S
According to the ? -net theorem
Q
bad contains at least ? n points
32
Main idea

repair step
On average, each heavy rectangle Q
must satisfy Q ? S ? ? .
The number of bad rectangles is small.
It is sufficient to consider a set M of
representative rectangles, instead of R.
M is defined over the points of S.
M f(1/? ) (does not depend on n).

S
and so does points sampled at the repair step.
M
Q
33
The ? -net construction

Input X - a set of n points.
Parameters r 1/? .
Primary sample
Produce a random sample S ? X, s.t.,
each point is chosen with probability r/n .
S is part of the ? -net. ES r.
Generate the set M of all maximal S-empty
rectangles,
with respect to S.

The representative rectangles
34
The set of maximal S-empty rectangles
Each rectangle M ? M is defined by ? 4 points of
S, and M ? S ? . M is a representative set for
R For each input heavy rectangle Q, with Q ? S
? , expand Q until each of its sides touches
a point of S or continues to ? . Apply
repair-step on M.
S
Otherwise, done!
M
Q
35
The repair step

Consider a heavy rectangle M, with M ? X t ?
n/r ,
1 ? t ? log r .
Second sampling step
Construct 1/ t -net NM inside M ,
by sampling O(t log t ) points in M.
According to the ? -net theorem, each input
(empty) rectangle Q ? R, Q ? M,
with Q ? n/r , must be stabbed by NM !

According to the ? -net theorem
The excess of M
M
Q
The universe size is now t ? n/r
36
The final ? -net

Output
The union of S and ?M ? M NM .
What is the expected size of the ? -net ?
ES E ?t?1 t log t Mt
Exponential Decay Lemma
Chazelle, Friedman 90, Agarwal Matousek,
Schwarzkopf. 98
E M?t O( 2-t E M ) ,
The number of bad rectangles decreases
exponentially!

Mt rectangles in M with excess t
Expected number of maximal empty rectangles
37
An improved ? -net
No improvement yet

Theorem E M O(r log r)
The expected ? -net size is O(1/? log (1/? )) .
Observation
Choose a slightly larger primary sample
S O(r) S O(r log log r)
and repair only rectangles M with excess t ? c?
log log r .
Using the Exponential Decay Lemma E M?t
o(r).
The number of bad rectangles is only sublinear in
r !

Recall r 1/?
M ? X ? n/r
38
Quadratic Lower bound construction
A staircase construction Each point in the upper
staircase is matched with each point in the lower
staircase.
?(r2) empty rectangles.
We can prune away most of these rectangles and
remain only with O(r log r) rectangles .
39
An O(s log s) bound for M

Key observation
Consider a vertical line l,
and all points to its left.
Claim
The number of maximal
S-empty rectangles, anchored
at l is only linear.
Next step Use a tree decomposition
built on top of X in order to obtain the
O(r log r) bound.

l
Q
Q
?v
l1
l3
l2
l2
l3
l3
l3
40

Efficient sensor placement in a polygonal domain

41
Problem statement
P

Input A polygon P of unit area.
Goal Find minimum-size set of sensors
(points in P) that monitor P.
? size of the smallest set of sensors.
A polynomial-time approximation algorithm
Unknown!

g
Vis(g)
Under the continuous model.
42
Our result Agarwal, Ezra, Ganjugunte

For any 0 ? ? ? 1
Obtain a polynomial time approximation
algorithm to monitor (1- ? ) of the polygon!
The approximation factor O(log (? / ? ))
Use a landmark-based approach
Randomly sample points inside P,
and cover only them. If the sample is
sufficiently large, it is guaranteed, with high
probability,
that (1- ? ) of the polygon is covered!

P
g
43
An ? -net application

Define the range space (X, R)
X P
R P \ ?i1, ? Vis(xi) x1, , x? ? P
Observation An ? -net N for (X, R) hits all
ranges of R with
area ? ? , and misses all ranges of area ? ? .
If we manage to cover N with ? visibility
polygons,
then their area ? 1 - ? .

Complement of the union of each collection of ?
visibility polygons
Thus the complement of this area is 1 - ? !
The ? - net is the landmark set.
44
Approximation algorithms for geometric hitting
sets
45
hitting sets for (X,R)

Input Range space (X, R) , m X , n R .
? size of the smallest hitting set.
previous algorithms
Greedy O(nm ? )
Bronimann-Goodrich O(m? n?2 )
Running time can be improved to O((n m) ? )
for both algorithms
Axis-parallel rectangles.
Planar regions with near-linear union complexity.

46
Our result Agarwal, Ezra, Sharir

Obtain a O(log n) approximation in near-linear
time ,
when the union complexity of R is near-linear.
Applied in both discrete and continuous models.
Specifically
Union complexity of R is O(n ?(n)) .
Obtain an approximation factor of O(?(?) log n)
in (randomized expected) time O(m n) .

? (?) is a slowly growing function.
The running time is O(n) for the continuous model
47
Our result Axis-parallel rectangles

Discrete model
Approximation factor O(log ? log n)
Running time O(m n)
Continuous model (axis-parallel boxed in
d-space)
Approximation factor O(logd-1?)
Running time O(n log n)
Fast implementation of the Bronimann-Goodrich
algorithm
Approximation factor O(log ?) in any dimension d.
Running time O(m n ?d1)

Union complexity quadratic
48
Open problems

Improve our upper bound O(1/? log log (1/? ))
for points and axis-parallel rectangles.Conserva
tive goal Obtain a weak ? -net of size o(1/?
log log (1/? )) .
Extend our bound to points and axis-parallel
boxes in d ? 4.Best known upper bound O(1/?
log (1/? )) .
Dual range spaces for rectangles and points.Best
known upper bound O(1/? log (1/? )) .Can
improve to O(1/? log log (1/? )) ?

The points of the ? -net are not necessarily
chosen from X .
p
49
Thank you
50
Sensor networking Extensions

Sensor networking within a polygonal domain
An art-gallery problem can be interpreted as a
sensor-networking problem within a polygonal
domain and infinite sensing radius.
Each guard is a sensor.

P
g
Vis(g)
51
Sensor networking Extensions

Unit sensing radius
Each range is the intersection of a visibility
polygon and a unit disk
(centered at the sensor).
Hitting-set instance
X points in a polygonal region P.
R intersections of each visibility polygon
with a source in P and a unit disk.

P
g
Vis(g)
52
Sensor networking Extensions

Camera sensors
Each range is the intersection of a visibility
polygon and a wedge
(with an apex at the sensor).
Hitting-set instance
X points in a polygonal region P.
R intersections of each visibility
polygon with a source in P and a wedge.

P
g
Vis(g)
53
The ? -net construction

Input X - a set of n points.
Parameters r 1/? , s ? r (s is slightly
larger than r).
Primary sample
Produce a random sample S ? X, s.t.,
each point is chosen with probability s/n .
S is part of the ? -net. ES s.
Generate the set M of all maximal S-empty
rectangles,
with respect to S.

The representative rectangles
54
An improved ? -net

Theorem E M O(s log s)
The expected ? -net size is O(1/? log (1/? )) .
Idea
Choose s O(r log log r) for the primary sample,
and repair only rectangles M with excess t ? c?
log log r .
Observation Using the Exponential Decay Lemma
E M?t O( 2-t E M ) .
E M?t O(s log s / polylog s) o(s)
o(r).
The number of bad rectangles is only sublinear in
r !

No improvement yet
Recall r 1/?
M ? X ? n/r
55
Bounding M

The expected ? -net size is thus
ES E M ? ?t?1 t log t ? 2-t
Goal Show that E M is o(s log s) .
Upper bound O(s2) .
Each rectangle is determined by
its two opposite corners.
problem
The bound O(s2) is bad for the analysis,
and yields an ? -net of size O(1/? 2 ) !

Recall s ? 1/?
56
Bounding the final ? -net size

The expected size of the final ? -net is
ES E M ? ?t?c loglog r t log t ? 2-t
O(r log log r) o(r) O(r log log r)
O(1/? log log (1/? )) .
Note
Any bound on E M of the form O(s polylog s)
yields an ? -net of that size.

Extensions Axis-parallel 3-boxes. ?-fat
triangles.
57
An O(s log s) bound for M

Key observation
Consider a vertical line l,
and all points to its left.
Claim
The number of maximal S-empty rectangles,
anchored at l is only linear.
Handling a query rectangle Q
One of the halves Qof Q contains
at least n/(2r) points. Q is anchored at l.
Expand Q on heavier side of l .

l
l
Q
Q
58
Tree decomposition
Each node is a vertical strip

Build balanced binary tree T on X, sorted by
x-coordinate
Stop expansion of T when nodes have ?n/r points.
T has O(log r) O(log s) levels.
At each level
maximal S-empty anchored rectangles O(s)
Overall (over all levels) O(s log s) .

?v
l1
l3
l2
l2
l3
l3
l3
59
Query rectangle Q

For an input rectangle Q with ? n/r points
Find the first (highest) node of T
whose bounding line l meets Q.
Expand Q within the heavier
strip ?v bounded by l .
The maximal S-empty anchored
rectangles comprise the representative
set for R.

Q
Q
?v
l1
l3
l2
l2
l3
l3
l3
60
Extensions

Axis-parallel boxes in 3-space
E M O(s log3 s)
?-fat triangles in the plane
E M O(s log2 s)
Axis-parallel boxes in d-space,
with points uniformly distributed over the
unit-cube
E M O(s logd-1 s)
The expected size of the ? -net is O(1/? log log
(1/? )) .

Number of maximal empty anchored orthants is only
linear!
No need to decompose space.
61
Dual (Geometric) Range Spaces

Flip roles of X and R, and obtain (R, X) .
R set of regions in ?d ,
X Rp p ? X, Rp r r ? R , r contains
p .
R Intervals.
X Subsets of intervals
containing a common point in ?1 .
R Disks.
X Subsets of Disks
containing a common point in ?2

p
p
62
The set-cover problem

Primal A hitting set for (X, R) is a subset H ?
X, s.t., for any Q ? R , Q ? H ? ? .
Dual A set cover for (X, R) is a subset S ? R,
s.t., any x ? X is covered by S .
A set cover for (X, R) is a hitting set for (R,
X)
Finding a set cover of smallest size is NP-hard!
(even for geometric range spaces).
Achieve improved approximation factors via ?-nets
(using the Bronimann-Goodrich technique /
LP-relaxation).

63
? -nets for dual range spaces