Packing Necklaces into a Box

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Packing Necklaces into a Box
Valentin Polishchuk Helsinki Institute for
Information Technology, University of Helsinki
and Helsinki University of Technology
Joint work with
Estie Arkin , Joe Mitchell Applied Math
and Statistics, Stony Brook University
Anne Pääkkö Computer Science,
University of Helsinki
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Motivation

• Birthday party !
• Loves necklaces

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• Presents on a table

Loving aunt
Beads
Thread
4

• Presents on a table

Loving aunt
Make lots of necklaces and put them on the table
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Necklaces should
Algorithmic question Lay down a maximum
number of necklaces

• Go all the way left to right
• Go around the presents
• Be pairwise-disjoint

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Formally

• GIVEN
• Polygonal domain
• (rectangle)
• obstacles
• left side source
• right side sink
• FIND
• Max number of necklaces
• polygonal source-sink path
• vertices separated
• centers of obstacle-free pairwise-disjoint unit
  disks

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Our Solution Bottommost Necklacesthrough Disk
Packing
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Maximal packing of 1/3-disks

• Bottommost disk at source bead
• Rightmost reachable with straight line segment
  bead
• Rightmost reachable bead
• Until sink is reached
• Pop disks touched by thread

A set of feasible necklaces albeit
bead radius 1/3 L
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Maximal packing of 1/3-disks

• Animation

A set of feasible necklaces albeit
bead radius 1/3 L
How many (compared to OPT)?
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Fact
Maximal packing of 1/3-disks

• Obstacle-free unit diskfully contains a 1/3-disk
  from maximal packing
• 2/3-disk
• Is there a center of a disk from packing?
• no place 1/3-disk
• yes inside the unit disk

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OPT

• Every bead contains 1/3-disk from packing
• Exist OPT necklaceswith 1/3-beadsand
  stretchL4/3

Bottommost packing uppermost path maxflow
alg No necklace is lost
If exist K necklaces with unit-disk beads and
stretch L
K necklaces with 1/3-disk beads and stretch
L4/3
we find
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Implementation
Hexagonal packing
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Output
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Output
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Output
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Sell Output to Little Girls Inc.?

• If exist K necklaces
• with unit-disk beads
• and stretch L

K necklaces with 1/3-disk beads and stretch
L4/3
we find
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(No Transcript)
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Who Else would be Interested?

• Air Traffic Management path planning
• Given
• Domain 2D airspace
• source and sink
• Obstacles hazardous weather systems
• Find
• Thick source-sink paths
• planes with protected airspace zones (disks)
• not intersecting obstacles

Max of Paths, Shortest Paths,
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Motivations

• VLSI wire thickness
• Robotics circular robot
• Sensor field
• Short paths
• Close to bd
• congestion
• Well-separated
• Medial axis
• Long
• Shortest paths
• given separation
• Air Traffic Management safety margins

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Continuous Flows

• Given Polygonal domain P
• with holes
• source and sink S and T
• div s 0 inside P
• s n 0 on ?P\S,T
• s 1 capacity
• V s S s n ds s T s n ds
• MaxFlow
• Find s that maximizes V

Flow vector field s P ? R2
Cut Partition P S in one part, T in the other
Capacity Length of bd between parts
counted within P (not within holes)
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Discrete Network 2D Domain

• Source and sink nodes
• Cut
• partition nodes
• capacity
• edges that cross
• Flow
• integers on arcs

• Source and sink edges
• Cut
• partition domain
• capacity
• length of the boundary
• Flow
• vector field

1
1
1
s
t
1
1
1
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Disjoint Paths in Graphs
Related to Network Flows
s
t
s
t
s
t
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Continuous MaxFlow/MinCut Theorem
Strang83, Mitchell90

• MaxFlow
• MinCut
• SP T-B path in critical graph

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Continuous Mengers Theorem
Arkin,Mitchcell,P08

• Max of disjoint thick paths
• MinCut
• SP T-B in thresholded critical graph
• lij bdij / airlane widthc

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Well Separated Paths
Kröller,Mitchell,P

• Max of disjoint thick paths
• MinCut
• SP T-B in thresholded critical graph
• lij bdij / airlane widthc 1

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MinCut Over Time
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Disjoint Paths in Graphs
Related to Network Flows
s
t
s
t
s
t
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MinCost Flow

• Given Polygonal domain P
• sources S and sinks T
• Flow
• vector field s
• div s 0 inside P
• s n 0 on ?P\S,T
• s 1 capacity
• V s S s n ds sT s n ds
• Min-Cost Flow
• Given V
• Find s that minimizes cost

Cost ls length of streamline through s in
S cost sS lsds
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Continuous Flow Decomposition Theorem

• Flow U of paths (e.g., streamlines)
• Continuous Flow
• Decomposition Theorem
• Min-Cost Flow U of shortest thick paths

Mitchell,P07
linear
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Thick Paths
Thick Pats Right Model?
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Real Flight Paths
Chan, Refai and DeLaura AIAA Aviation Technology,
Integration and Operations Conference, Belfast,
2007
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Real Flight Paths
Chan, Refai and DeLaura AIAA Aviation Technology,
Integration and Operations Conference, Belfast,
2007
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Real Flight Paths
Chan, Refai and DeLaura AIAA Aviation Technology,
Integration and Operations Conference, Belfast,
2007
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How Do Pilots Treat Obstacles?
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Beads Triangles
???

• Time stretching maneuvers

Temporary blockage
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Templates
Schoemig, Armbruster, Boyle, Haraldsdottir,
Scharl IEEE/AIAA Digital Avionics Systems
Conference, 2006
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Paths with Wiggle Room
Schoemig, Armbruster, Boyle, Haraldsdottir,
Scharl IEEE/AIAA Digital Avionics Systems
Conference, 2006 AIAA Modeling and Simulation
Technologies Conference and Exhibit 2006
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(No Transcript)
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More Requests
Lower bound on stretch between beads
No beads on top of sector boundaries
Reachable region
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Java Applet

• www.cs.helsinki.fi/group/compgeom/necklace/

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More

• Bottommost paths long
• mincost maxflow through the grid
• Theory NP-hard?

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(No Transcript)
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Map Labeling

• Lines
• routes
• rivers
• borders
• Important distinction
• rivers are given
• adding beads to given threads

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Multicommodity Flows (Red/Blue paths)
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What we Learnt
If exist K necklaces with unit-disk beads and
stretch L
E
E

• Angry Little Girls Inc.
• How to make aunts happy
• in the dual sense
• Implemented bottommost paths
• Necklaces in ATM

we find
K necklaces with 1/3-disk beads and stretch
L4/3
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(No Transcript)