Substructures in Geometric Arrangements and -nets Esther Ezra Duke University.

Substructures in Geometric Substructures in Geometric Arrangements Arrangements

and and -nets -nets

EstherEsther Ezra Ezra

Duke UniversityDuke University

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 planeAll portions of the plane that are covered by the triangles.

Union of simply-shaped bodiesUnion of simply-shaped bodies

The union boundary

The union has two holes

Motivation: Motion planningMotivation: Motion planning

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 spaceThe set of all legal placements of R, while translating R in d-space.

R does not intersect any of the obstacles in A

The workspace

collisionNo rotations

The configuration spaceThe configuration space

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

reference point

The free spaceThe free space

The free space is the complement of 1 i n Pi

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)

The Union as a substructure in The Union as a substructure in arrangementsarrangements

Union is a substructure of the arrangement

The union complexityThe 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!).

Previous results in 2D:Previous results in 2D:Fat objectsFat objects

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!

Each of the angles

O(n1+) , for any >0 . r

r’

r’/r ,and 1.

r diam(C) ,

D C, < 1 is a constant.

rC

D

Previous results in 3D:Previous results in 3D:Fat ObjectsFat Objects

Congruent cubesn 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

C

r diam(C) , D C, < 1 is a

constant.

D

Our results: New union bounds.Our results: New union bounds.Fat TetrahedraFat 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.

Our results: New union bounds.Our results: New union bounds.Infinite cylindersInfinite 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) .

Envelopes in Envelopes in dd-space-space

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) = min{F 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) .

The lower envelope is monotone.

O*(n2) for d=3 .

The sandwich regionThe 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 d=3, the complexity of the sandwich region is: O*(n2)

Main idea: Main idea: Reduce cylinders to envelopesReduce 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].

Small-size Small-size -nets for Axis--nets for Axis-Parallel Rectangles and Parallel Rectangles and

BoxesBoxes

Range SpacesRange Spaces

Range space (X, R) :

X – Ground set.

R – Ranges: Subsets of X .

|R| 2|X|

Abstract form: Hypergraphs.

X – vertices.

R – hyperedges.

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.

Geometric Range SpacesGeometric Range Spaces

X finite: Discrete model ,

X infinite: Continuous model

The hitting-set problemThe 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…

Art-gallery ApplicationArt-gallery Application

Useful application: art-galleryInput: Polygon P.

Goal: Find minimum-size set of guards (points in P) that “see” the entire P.

P

g

q

q’

g, q are visible. g, q’ are invisible

Visibility polygon

Pq

g

q’Vis(g)

Art-gallery ApplicationArt-gallery Application

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.

Pq

g

P

qg

Vis(g)

Vis(q)

Input:

C – set of clients .A - locations of antennas , 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.

Sensor networking applicationSensor networking application

Sensor networking applicationSensor networking application

Hitting-set instance:

X – antennas locations

R – (,1): unit disks with

client-centers. Observation:

covers iff is inside (,1)

Approximation for hitting setsApproximation for hitting setsFinding 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

-nets for range spaces-nets for range spacesGiven:

• A range space (X, R) , assume X is finite, |X| = n .

• A parameter 0 < < 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/ .

n

Bound does not depend on n.

Approximation for geometric hitting Approximation for geometric hitting setssets

The Bronimann-Goodrich technique / LP-relaxationIf (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!

An upper bound for the An upper bound for the - -net sizenet size

The -net theorem [Haussler-Welzl, 87]:

If the ranges are simply-shaped regions, then,

for any > 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.

Is the bound optimal?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 forgeometric range spaces. Ideally O(1/ ), but anything better than O(1/ log (1/ )) is `exciting‘ !

Achieved by points and intervals on the

real line.

Previous resultsPrevious 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

Our results [Aronov, Ezra, Sharir]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

Improved approximation factorsImproved approximation factorsfor geometric hitting setsfor 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 [0,1]d .

Main idea for axis parallel rectangles :Main idea for axis parallel rectangles :Use two-level samplingUse 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.

According to the -net theorem

“bad” = contains at least n points

S

Q

Main ideaMain idea

repair step:On average, each heavy rectangle Qmust satisfy Q S . The number of “bad” rectangles is small.

It is sufficient to consider a set of representative rectangles, instead of R. is defined over the points of S. || = f(1/ ) (does not depend on n).

and so does #points sampled at the repair step.

Q

M

S

The The -net construction-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. E{|S|} = r.

Generate the set of all maximal S-empty rectangles,with respect to S. The representative

rectangles

The set of maximal S-empty The set of maximal S-empty rectanglesrectangles

Each rectangle M is defined

by 4 points of S, and M S = .

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 .Q

M

Otherwise, done!

S

The repair stepThe 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

Q

MThe excess of M

The “universe” size is now t n/r

The final The final -net-net Output:

The union of S and M NM .

What is the expected size of the -net ?

E{|S|} + E{ t1 t log t |t | }

Exponential Decay Lemma:[Chazelle, Friedman 90], [Agarwal Matousek, Schwarzkopf. 98]

E{ |t | } = O( 2-t E{ || }) ,

The number of bad rectangles decreases exponentially!

t = # rectangles in with excess t

Expected number of maximal empty

rectangles

An improved An improved -net-net

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{ |t| } = o(r).

The number of bad rectangles is only sublinear in r !

No improvement yet…

|M X | n/r

Recall r = 1/

Quadratic Lower bound constructionQuadratic 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 .

An O(An O(ss log log ss) bound for ) bound for ||||

Key observation:Consider a vertical line , and all points to its left.

Claim:The number of maximal S-empty rectangles, anchored at is only linear.

Next step: Use a tree decomposition built on top of X in order to obtain the O(r log r) bound. v

Q

Q’

1 3 ‘22 ‘‘3‘33

Efficient sensor placement in a Efficient sensor placement in a polygonal domainpolygonal domain

Problem statementProblem statement

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!

P

g

Vis(g)

Under the continuous model.

Our result: Our result: [Agarwal, Ezra, Ganjugunte][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

An An -net application-net application

Define the range space (X, R) :

X = P

R = {P \ i=1,… 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.

Approximation algorithms for Approximation algorithms for geometric hitting setsgeometric hitting sets

hitting sets for (X,R)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 + n2 )

Running time can be improved to O*((n + m) )

for both algorithms:

• Axis-parallel rectangles.

• Planar regions with near-linear union complexity.

Our result [Agarwal, Ezra, Sharir]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

Our result: Axis-parallel rectanglesOur 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 + d+1)

Union complexity: quadratic

• Improve our upper bound O(1/ log log (1/ )) for points and axis-parallel rectangles.Conservative 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/ )) ?

Open problemsOpen problems

p

The points of the -net are not necessarily chosen from X .

Thank youThank you

Sensor networking: ExtensionsSensor 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)


Unit sensing radius

Each range is the intersection of a visibility

polygon and a unit disk

(centered at the sensor).


X – points in a polygonal region P.

R – intersections of each visibility polygon

with a source in P and a unit disk.

P

Vis(g)

g


Camera sensors

Each range is the intersection of a visibility

polygon and a wedge

(with an apex at the sensor).


X – points in a polygonal region P.

R – intersections of each visibility

polygon with a source in P and a wedge.

P

Vis(g)g

The The -net construction-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. E{|S|} = s.

Generate the set of all maximal S-empty rectangles,with respect to S. The representative

rectangles

An improved An improved -net-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{ |t | } = O( 2-t E{ || }) .

E{ |t| } = O(s log s / polylog s) = o(s) = o(r).

The number of bad rectangles is only sublinear in r !

No improvement yet…

|M X | n/r

Recall r = 1/

Bounding Bounding |||| The expected -net size is thus:

E{|S|} + E{ || } t1 t log t 2-t

Goal: Show that E{ || } 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/

Bounding the final Bounding the final -net size-net size

The expected size of the final -net is:

E{|S|} + E{ || } tc 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.

An O(An O(ss log log ss) bound for ) bound for ||||

Key observation:Consider a vertical line , and all points to its left.

Claim:The number of maximal S-empty rectangles,anchored at is only linear.

Handling a query rectangle Q:One of the halves Q’of Q contains at least n/(2r) points. Q’ is anchored at .Expand Q’ on “heavier” side of .

Q

Q’

Tree decompositionTree decomposition

• Build balanced binary tree on X, sorted by x-coordinate

• Stop expansion of when nodes have n/r points.

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) . 1 3 ‘22 ‘‘3‘33

v

Each node is a vertical strip

Query rectangle QQuery rectangle Q

For an input rectangle Q with n/r points:

Find the first (highest) node of whose bounding line meets Q.

Expand Q within the “heavier”

strip v bounded by .

The maximal S-empty anchored

rectangles comprise the representative

set for R.

v

Q

Q’

1 3 ‘22 ‘‘3‘33

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/ )) .

ExtensionsExtensions

Number of maximal empty anchored orthants

is only linear!

No need to decompose space.

Dual (Geometric) Range SpacesDual (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

The set-cover problemThe 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).

-nets for dual range spaces-nets for dual range spaces

Useful for the set cover problem.

-net for (R, X*) is a subset N R that

covers all points at depth |R| .

An -net is a set cover for all the deep points.

Example:

Intervals and points on the real line: |N| = 1/ .

depth(p) = #ranges that cover p X.

n

Range space (R, X*), s.t., for each S R, |S| = m, the union S has (a small) complexity o(m log m) : o(1/ log (1/ )) .[Clarkson, Varadarajan 07]

Specifically:The complexity of the union is O(m (m)) -net size is O(1/ (1/ )).

Previous resultsPrevious results

() is a slowly growing function.

More about the Clarkson-More about the Clarkson-VaradarajanVaradarajan techniquetechnique

Example: disks (or pseudo-disks) and points

Input: A set S of m disks.

Union complexity: O(m) .

[kedem et al. 86]

-net size is O(1/ ) .

Example: fat triangles and points

Input: A set S of m -fat triangles.

Union complexity: O(m loglog m) .

[Matousek et al. 1994]

-net size is O(1/ log log (1/ )) .

Each of the angles

Our results: DualOur results: Dual

The complexity of the union is O(m (m))

-net size is O(1/ log (1/ )) .

Fat triangles:

Union complexity: O(m loglog m)

-net size is O(1/ log log log (1/ )) .Locally -fat objects:

Union complexity: O(m polylog m)

-net size is O(1/ log log (1/ )) .

And several other improved factors.

O D

area(D O) area(D)

0 < 1

Use an initial sample s r and then the Exponential

Decay Lemma.

Is the bound optimal?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 forgeometric range spaces. Ideally O(1/ ), but anything better than O(1/ log (1/ )) is `exciting‘ !

Achieved by points and intervals on the

real line.

Bounding the Bounding the -net size-net size

Exponential Decay Lemma:[Chazelle, Friedman 90], [Agarwal Matousek, Schwarzkopf. 98]

E{ |t| } = O( 2-t E{ | '| }) ,where:• S' is a smaller random sample, each point chosen withprobability s/(t n) .• t - all maximal S-empty rectangles M with tM t .• ' - all maximal S'-empty rectangles.

Bounding the final Bounding the final -net size-net size

A very useful tool:

Exponential Decay Lemma:

[Chazelle, Friedman 90], [Agarwal Matousek, Schwarzkopf. 98]

E{ |t| } = O( 2-t E{ || }) ,

where t is all maximal S-empty rectangles M with tM t .

The number of heavy rectangles decreases exponentially!

A nearly-linear bound for A nearly-linear bound for ||||

Fix a node v of and its strip v :

Xv = S v , Sv = S v

Lemma:

The number of maximal Sv-empty

anchored rectangles in v is O(Sv) .

At a fixed level i of , overall numberis O(s) .

Overall: O(s log r) .

v

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Substructures in Geometric Arrangements and -nets Esther Ezra Duke University.

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Transcript of Substructures in Geometric Arrangements and -nets Esther Ezra Duke University.