shang yang stony brook univ. 08/17/2011 1.convex 2.closed 3.stabs every element
TRANSCRIPT
![Page 1: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/1.jpg)
WADS 2011
Shang Yang
Stony Brook Univ.08/17/2011
1.CONVEX 2.CLOSED
3.STABS EVERY
ELEMENT
![Page 2: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/2.jpg)
1.CONVEX
3.STABS EVERYONE
2.CLOSED
Esther M.Arkin
Christian Knauer
Claudia Dieckmann
Lena Schlipf
ShangYang
Joseph S.B.Mitchell
Valentin Polishchuk
![Page 3: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/3.jpg)
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*
![Page 4: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/4.jpg)
Curve Reconstruction
Line Simplification
Motion Planning
Surface Reconstruction
Motivation
![Page 5: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/5.jpg)
2d segments, squares, 3d ballsproved NP-hard from 3-SAT
Our contributions
?Disjoint segments,pseudodisks Polytime (DP)If vertices are from a given set
![Page 6: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/6.jpg)
NP-Completeness Proof: Stabbing Arbitrary Segments
6
From 3-SAT
![Page 7: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/7.jpg)
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.
![Page 8: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/8.jpg)
Polytime Algorithm: First Step
3-link chain:Bridge
Segment
Segment
chord
![Page 9: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/9.jpg)
Chords Are NEVERINTERSECTEDBy Any InputSegments
![Page 10: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/10.jpg)
Chords & Bridges
![Page 11: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/11.jpg)
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!!
![Page 12: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/12.jpg)
DP Current State (pq, rt, qabt)
pq
tr
a
bb
ac
![Page 13: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/13.jpg)
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
![Page 14: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/14.jpg)
DP Recursion( abc Case 2)
Stab(pq, rt, qabt)
p
q
t r
a
b
z
c
pq
t r
b
=Stab(pz, rt,zcbt)
z
ca
![Page 15: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/15.jpg)
DP Recursion( abc Case 3)
Stab(pq, rt, qabt)
p
q
t r
a
b
z
c
pq
t r
b
= Stab(pz, rt,zbt)
z
ca
![Page 16: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/16.jpg)
DP Recursion( abc Case 4)
u
z
ca
q p
vrt
b z
c
Stab(pq, uz, qacz) & Stab(rt, vz, tbcz)
![Page 17: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/17.jpg)
Base Case(1 & 2)
p
q
t
r
a
b
z
c
Stab(p
q, rt,
qabt)
= FALS
E
![Page 18: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/18.jpg)
p
q
t
r
a
b
Base Case(3)
Stab(p
q, rt,
qabt)
= TRUE
![Page 19: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/19.jpg)
q
t
Bp
r
BReview
![Page 20: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/20.jpg)
Symmetry Detection
Stabbing with Regular PolygonPolynomial Time Algorithm
![Page 21: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/21.jpg)
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
![Page 22: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/22.jpg)
Convexification
Fast 2-Approximation & PTAS
![Page 23: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/23.jpg)
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
![Page 24: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/24.jpg)
Convexification: 2-Approx
38
DOPT
![Page 25: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/25.jpg)
Convexification: 2-Approx
39
D
OPT
![Page 26: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/26.jpg)
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.
![Page 27: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/27.jpg)
3 Open Problems
Candidate Points NOT Given?
Allowing < k reflex vertices?
Fast 2d Unit Disk Case Solver?
![Page 28: Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT](https://reader037.vdocument.in/reader037/viewer/2022110103/56649f335503460f94c507c4/html5/thumbnails/28.jpg)
Esther M.Arkin
Christian Knauer
Claudia Dieckmann
Lena Schlipf
Joseph S.B.Mitchell
Valentin Polishchuk
Thank you! Questions Please!
Shang Yang