Maximum Thick Paths in Static and Dynamic Environments

1. Maximum Thick Paths in Static and Dynamic Environments Valentin Polishchuk joint work with Esther Arkin and Joseph Mitchell

2. VLSI: Multiple Paths

3. Robotics: Moving Objects

4. ATM: Moving Obstacles

5. Motivation • Domain – 2D airspace • Obstacles – hazardous weather systems • Objects – planes with PAZs • disks Goal: • Determine max throughput (“capacity”)

6. Formulation Given • Polygonal domain • outer polygon • moving holes • known trajectories • speed ≤ 1 • Source/sink edges • Entry and exit time intervals Find • maximum number of trajectories for unit disks • speed ≤ 1 • disjoint from each other and obstacles • enter/exit during entry/exit time intervals Source Sink OPT

7. NP-Hard to Approximate • Does there exist one thin (radius-0) paths? • NP-hard [Canny and Reif’87] • Does there exist α•OPT radius-bpaths? • NP-hard for any α,b Need disks move faster

8. Just Moving Faster is not Enough • Do there exist OPT paths each of length ≤ L amidst static obstacles? • NP-hard [MP’07] Entry/exit intervals + Speed bound = Bound on paths length • Does there exist OPT paths for disks with max speed v ? • NP-hard for any v < ∞ Need to compromise on radius + speed

9. Our Result If exist OPT trajectories unit disks speed ≤ 1 We find OPT trajectories radius Ω(1) speed O(1) Pseudopolynomial time

10. Lifting to (x,y,t)

11. Tubes

12. Obstacles

13. Time Slicing

14. Maximal Packings of Disks R = 1/3 – Dt / 2

15. time t + Δt t Cylinder Around a Path 1 - Dt

16. Δt time 1- Dt 1- 3/2 Dt t + Δt = 3R Speed ≤2/ Dt t 1 - Dt

17. Radius-R Slope ≤ 2/Dt Tubes

18. Radius-R Slope ≤ 2/Dt Tubes

19. S T Motion Graph (Time-Expanded Network) Speed ≤2/ Dt # of S-T paths ≥OPT

20. time t + Δt t Intersect Between Time Slices

21. S Motion Graph T ≤ 1 is used by paths # of S-T paths respecting intersections ≥OPT

22. MaxPaths with Forbidden Pairs ≤ 1 ≤ 1 ≤ 1 INAPPROXIMABLE S T B Jim Orlin: A C B A s t C = MaxPaths with Forbidden Pairs Max independent set

23. S MaxPaths with Forbidden Pairs T

24. time t + Δt t Deep Penetration

25. Deep Penetration: Top View Deep Penetration: Only in Pairs

26. time t + Δt t A Deeply Penetrating Pair

27. time t + Δt t Non-Deep Penetration

28. S T Motion Graph # of S-T paths respecting intersections ≥OPT

29. Modified Motion Graph Deeply penetrating pair node of capacity 1 # of S-T paths in the Modified Motion Graph ≥ # of S-T paths respecting intersections in Motion Graph≥OPT

30. Max # of S-T paths in the Modified Motion Graph ≥ OPT A Deeply Penetrating Pair Non-Deeply Penetrating node of capacity 1 Done!

31. S The Algorithm T node of capacity 1 Find max # of disjoint S-T paths in the Modified Motion Graph

32. If there exist OPT paths for unit discsmoving with speed ≤ 1 we find, for any Δt < 1/2 ≥ OPT paths for discs of radius (1/6 – 1/4 •Δt)2moving with speed ≤ 10 / Δt

33. Static Obstacles: Strongly Polynomial Time (a representation of the paths) Continuous Flow Decomposition Theorem[MP’07] Continuous Menger’s Theorem: Max # of disjoint thick paths = SPT-B in thresholded critical graph lij = bdij / path widthc Open: (Strongly) polynomial-time approximation algorithm for dynamic obstacles

34. See our algorithm animated at the video session tomorrow

40. Motivation • Domain – 2D airspace • Obstacles – hazardous weather system • Objects – planes with PAZs • disks Goal: • Determine max throughput (“capacity”)