Packing Necklaces into a Box

1. 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 MitchellApplied Math and Statistics, Stony Brook University Anne PääkköComputer Science, University of Helsinki

2. Motivation • Birthday party ! • Lovesnecklaces

3. 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

5. 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

6. Formally… < L GIVEN: • Polygonal domain (rectangle) • obstacles • left side: “source” • right side: “sink” FIND: • Max number of necklaces polygonal source-sink path • vertices separated < L • centers of obstacle-free pairwise-disjoint unit disks

7. Our Solution: Bottommost Necklacesthrough Disk Packing

8. L+4/3 L+4/3 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 < 1stretch L + 4/3 > L

9. Maximal packing of 1/3-disks Animation A set of feasible necklaces albeit bead radius 1/3 < 1stretch L + 4/3 > L How many (compared to OPT)?

10. 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

11. OPT • Every bead contains 1/3-disk from packing • Exist ≥ OPTnecklaceswith 1/3-beadsand stretchL+4/3 < L + 4/3 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 L+4/3 we find

12. Implementation Hexagonal packing

13. Output

14. Output

15. Output

16. 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 L+4/3 we find

18. 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,…

19. 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

20. Continuous Flows Flow vector field σ: P → R2 Given: Polygonal domain P with holes source and sink S and T div σ = 0 inside P σ• n = 0 on ∂P\{S,T} |σ| ≤ 1 – capacity V = |s Sσ • n ds| = |s Tσ • n ds| MaxFlow Find σthat maximizes V Cut: Partition P S in one part, T in the other Capacity: Length of bd between parts counted within P (not within holes)

21. Discrete Network 2D Domain 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 • 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

22. Disjoint Paths in Graphs Related to Network Flows s t s t s t

23. Continuous MaxFlow/MinCut Theorem MaxFlow = MinCut = SPT-B path in critical graph [Strang’83, Mitchell’90]

24. Continuous Menger’s Theorem Max # of disjoint thick paths = MinCut’ = SPT-B in thresholded critical graph lij = bdij / airlane widthc [Arkin,Mitchcell,P’08]

25. Well Separated Paths Max # of disjoint thick paths = MinCut’ = SPT-B in thresholded critical graph lij = bdij / airlane widthc + 1 [Kröller,Mitchell,P]

26. MinCut Over Time

27. Disjoint Paths in Graphs Related to Network Flows s t s t s t

28. MinCost Flow Given: Polygonal domain P sources S and sinks T Flow vector field σ div σ = 0 inside P σ• n = 0 on ∂P\{S,T} |σ| ≤ 1 – capacity V = |s Sσ • n ds| = |sTσ • n ds| Min-Cost Flow Given V Find σthat minimizes cost Cost |ls| – length of streamline through s in S cost = sS|ls|ds

29. Continuous Flow Decomposition Theorem Flow = U of paths (e.g., streamlines) Continuous Flow Decomposition Theorem: Min-Cost Flow = U of shortest thick paths [Mitchell,P’07] linear #

30. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 1 1 1 s t 1 1 1 1 1 1 s t s t 1 1 1 Thick Paths Thick Pats –Right Model?

31. Real Flight Paths Chan, Refai and DeLaura AIAA Aviation Technology, Integration and Operations Conference, Belfast, 2007

32. Real Flight Paths Chan, Refai and DeLaura AIAA Aviation Technology, Integration and Operations Conference, Belfast, 2007

33. Real Flight Paths Chan, Refai and DeLaura AIAA Aviation Technology, Integration and Operations Conference, Belfast, 2007

34. How Do Pilots Treat Obstacles?

36. Beads = Triangles ??? • Time stretching maneuvers Temporary blockage

37. Templates Schoemig, Armbruster, Boyle, Haraldsdottir, Scharl IEEE/AIAA Digital Avionics Systems Conference, 2006

38. 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

40. More Requests Lower bound on stretch between beads No beads on top of sector boundaries Reachable region

41. Java Applet www.cs.helsinki.fi/group/compgeom/necklace/

42. More holding pattern larger stretch little stretch ground delay destination • Bottommost paths – long • mincost maxflow through the grid • Theory: NP-hard?

44. Map Labeling • Lines • routes • rivers • borders • Important distinction: • rivers are given • adding beads to given threads

45. Multicommodity Flows(Red/Bluepaths)

46. 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 L+4/3