WADS 2011

Shang YangStony Brook Univ.08/17/2011

1.CONVEX 2.CLOSED

3.STABS EVERY

ELEMENT

1.CONVEX

3.STABS EVERYONE

2.CLOSED

Esther M.Arkin

Christian Knauer

Claudia Dieckmann

Lena Schlipf

ShangYang

Joseph S.B.Mitchell

Valentin Polishchuk

Computing a convex transversal

1987: original problem proposed Arik Tamir (NYU CG Day 3/13)

* Comp. Vision, Graphics & Image Processing 49(2):152170

1990: 2d parallel segments solved Goodrich & Snoeyink*

Curve Reconstruction

Line Simplification

Motion Planning

Surface Reconstruction

Motivation

2d segments, squares, 3d ballsproved NP-hard from 3-SAT

Our contributions

?Disjoint segments,pseudodisks Polytime (DP)If vertices are from a given set

NP-Completeness Proof: Stabbing Arbitrary Segments

6

From 3-SAT

21

Additional Hardness Results

Finding a convex transversal for:

1. a set of unit-length segments in 2D2. a set of pairwise-disjoint segments in 3D3. a set of disjoint balls in 3D4. a set of disjoint unit balls in 3D Conjectured

is NP-Complete.

Polytime Algorithm: First Step

3-link chain:Bridge

Segment

Segment

chord

Chords Are NEVERINTERSECTEDBy Any InputSegments

Chords & Bridges

A Bridge Partitions the probleminto 2 halves

pq

t r

q

t

Bp

r

B

Stab(qp, tr, B)=1

Stab(pq, rt, B)=1

Succeed!!

DP Current State (pq, rt, qabt)p

q

tr

a

bb

ac

pq

t r

a

b

DP Recursion( abc Case 1)

Stab(pq, rt, qabt)

(c)

= Stab(pq, rt,qbt)

pq

t r

b

(c)

a

DP Recursion( abc Case 2)

Stab(pq, rt, qabt)

pq

t r

a

b

z

c

pq

t r

b

=Stab(pz, rt,zcbt)

z

ca

DP Recursion( abc Case 3)

Stab(pq, rt, qabt)

pq

t r

a

b

z

c

pq

t r

b

= Stab(pz, rt,zbt)

z

ca

DP Recursion( abc Case 4)

u

z

ca

q p

vrt

b z

c

Stab(pq, uz, qacz) & Stab(rt, vz, tbcz)

Base Case(1 & 2)

p

q

t

r

a

b

z

c

Stab(pq, rt

, qabt) = FA

LSE

p

q

t

r

a

b

Base Case(3)

Stab(pq, rt

, qabt) = TR

UE

q

t

Bp

r

BReview

Symmetry Detection

Stabbing with Regular PolygonPolynomial Time Algorithm

Optimization Versions of the Problem

• Maximize the number of objects stabbed by the convex transversal (DP)

• Minimize the length of the stabber:– TSP with Neighborhoods (TSPN)– Require convexity: Shortest convex stabber– Example: TSPN for lines

• Minimize movement of objects to make them have a convex stabber: optimal “Convexification”

35

Convexification

Fast 2-Approximation & PTAS

37

Let Q’ be the convex hull of Q, and let D be the maximum distance from a point, q, in Q to the boundary of Q’.

Convexification: 2-Approx

Q’

D

q

Convexification: 2-Approx

38

DOPT

Convexification: 2-Approx

39

D

OPT

Summary

• Settle the open problem in 2D:– Deciding existence of a convex transversal is NP-

complete, in general– If objects S are disjoint, or form set of

pseudodisks, then poly-time algorithm to decide, and to max # objects stabbed

• 3D: NP-complete, even for disjoint disks

40

Hard even for terrain stabbers!

Assumes candidate set P of corners of stabber is given.

3 Open Problems

Candidate Points NOT Given?

Allowing < k reflex vertices?

Fast 2d Unit Disk Case Solver?

Esther M.Arkin

Christian Knauer

Claudia Dieckmann

Lena Schlipf

Joseph S.B.Mitchell

Valentin Polishchuk

Thank you! Questions Please!

Shang Yang