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11/14/2003 10:00AM

11/14/2003 10:00AM. Valentin Polishchuk Preliminary Examination Geometric Motion Planning Problems with Applications Adviser: Professor Joseph Mitchell. Research Directions. Motion Planning Problems Geometric Algorithms for Air Traffic Management. Motion Planning Problems.

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11/14/2003 10:00AM

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  1. 11/14/2003 10:00AM Valentin Polishchuk Preliminary Examination Geometric Motion Planning Problems with Applications Adviser: Professor Joseph Mitchell

  2. Research Directions • Motion Planning Problems • Geometric Algorithms for Air Traffic Management

  3. Motion Planning Problems

  4. Geometric domain Items in it Agent (Robot), which travels and moves the items Initial position Rules/Constraints What happens after a move What is forbidden Target configuration Sink A Puzzle

  5. Integral Orthohedral Version Rectilinear domain, possibly with holes Vertices have integral coordinates Domain is “pixeled” The agent occupies 1 pixel An item occupies 1 pixel The dual graph

  6. Feasibility • Is it possible at all to move from the initial configuration to the target? • Optimality • Cost associated with agent’s move • What is the min cost of rearranging the items?

  7. The Snow Blower Problem • Domain – driveway • Items – snow of depth 1 on every pixel • Agent – snowblower • Rules • the snow from a pixel entered (if any) is put onto an adjacent pixel (Left, Forward or Right) • if a boundary pixel is entered, the snow from it (if any) is thrown away • it’s forbidden to put the snow on the pixel already containing snow • Target configuration: depth 0 snow everywhere, clear driveway Related problems: Lawn-mowing Problem Milling Problem

  8. Optimization Cost: proportional to the number of pixels (with or without snow) visited NP-hard Reduction from Hamiltonian Cycle problem for cubic subgrid graphs ([PV], [Buro]) [PV] Christos H. Papadimitriou, Umesh V. Vazirani: On Two Geometric Problems Related to the Traveling Salesman Problem. J. Algorithms 5(2): 231-246 (1984) [Buro] M. Buro, “Simple Amazons Endgames and their Connection to Hamilton Circuits in Cubic Subgrid Graphs”, The Second International Conference on Computers and Games (CG2000), Hamamatsu Japan

  9. Reduction The region is clearly clearable (feasible instance) The graph is Hamiltonian iff the SBP can be solved in at most n moves. n is the number of nodes in the graph. Optimization problem is NP-hard even for regions with maximum pixel degree 3

  10. An Approximation • Assumption: no deg 1 nodes • Take any SB tour • Detour (cost 2) at degree 3 nodes • Get a Chinese Postman tour (visiting every edge) in the dual graph

  11. Analysis • Thus, CP ≤ SB + 2d3 • CP/SB ≤ 1 + 2d3/SB ≤ 3 since SB ≥ d3 • 3-APX for max pixel degree 3 regions SBP and TSP • SBP is TSP in the dual graph (if its maxdeg ≤ 3) • TSP APX – SBP APX • Christofides’ heuristic, 3/2-APX • Can we do better (at least in some graphs)?

  12. TSP in Grid Graphs • For general grid graphsCP ≤ TSP + 2d3 + 4d4 • CP/TSP ≤ 1 + (2d3 + 4d4)/TSP better guarantee than 3/2 when d3 + 2d4 < TSP/4 • sparse grid graphs: d3 + 2d4 < n/4 • sparse cubic subgrid graphs d4 = 0, d3 ≤ n/4 Open Question Can CP ever actually do better than Christofides’ algorithm?

  13. Comparison with Previous Work Dense (simple, solid) grid graphs rather then sparse were studied [Ntafos], [AFM] • 6/5-APX When is an APX necessary? When the problem is NP-hard TSP is hard if HC is hard • HC for dense graphs is poly [UL] • HC is hard for sparse grid graphs • modification of the graph from [PV] • [AFM] E.M. Arkin, S.P. Fekete, J. S. B. Mitchell: “Approximation Algorithms for Lawn Mowing and Milling”, CGTA 17(1-2), October 2000, pp. 25—50 • [Ntafos] S. Ntafos. Watchman routes under limited visibility. Comp. Geom. Theory and Appl, 1(3):149--170, 1992 • [UL] C. Umans and W. Lenhart. “Hamiltonian cycles in solid grid graphs”. In Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci., pages 496--507, 1997 • [PV] Christos H. Papadimitriou, Umesh V. Vazirani: On Two Geometric Problems Related to the Traveling Salesman Problem. J. Algorithms 5(2): 231-246 (1984)

  14. Sidewalks • Sidewalk contains no 2-by-2 square • Dual graph contains no 4-cycle – thin graph maxdeg = 3, but not thin Clearable Thin, but not maxdeg = 3 Not clearable

  15. Sidewalks (cont.) • HC for cubic subgrid graphs reduces to SBP, both hard • HC for thin grid graphs – hard? No. A by-product: Polynomial algorithm for HC search problem in thin grid graphs

  16. Feasibility • maxdeg = 3 regions • always feasible • Sidewalks • DFS: from a given entrance pixel O(n), n is the number of pixels in the domain [SBARG] • General case • DFS: Partial clearing doesn’t change feasibility. In O(n) moves every pixel is visited, in O(n) the snow from a pixel is thrown away; so it’s O(n2) A by-product – membership of the optimization problem in NP. So, SBP is NP-complete [SBARG] The Stony Brook Algorithms Reading Group, 2001

  17. Variations on SBP D – max snow depth through which the SB can move • D > 1 • if D > 1 then any region is clearable starting from any pixel on the outer boundary of the region • if D > 2 then any region is clearable starting from any pixel on the boundary of the region • if D > 2 then any region is clearable starting from any pixel on the boundary of the region and for any initial direction of entrance of the SB into the region • Snow Shovel Problem Scoop snow and walk through cleared region • Non-square tiling • Beehive Clearing Problem – any region is clearable • Non-convex tiling – ask Mauritz Escher

  18. Open Problems • Snow Plower Problem = SBP with fixed throw direction • Analyze/improve the complexity of the algorithm for HC finding in thin grid graphs • APX for TSP in thin grid graphs • APX

  19. The Box Mover Problem • Domain – warehouse • Items – boxes • Agent – warehouse-keeper (robot), pushing and/or pulling and/or lifting boxes • Rules • don’t step on boxes • not more than k boxes pushed at once • not more than p boxes pulled at once • not more than l boxes lifted at once BMP(k,p,l) Related problems: SOKOBAN = BMP(1,0,0) Push-k = BMP(k,0,0), Push-* = BMP(∞,0,0), PushPush, Push-X

  20. Complexity of BMP(1,0,0) Previous work: Feasibility • P? No non-trivial problem known; open question • NP? NP-hard, but only 1 version (-X) is in NP PSPACE-complete • Optimization: only workload is counted (unlike SBP) Total travel or workload or mix is the same for feasibility • NP-hard in the general setting

  21. Making the Puzzle “more tractable” • k  , p  , l↑ ? • introducing powerful robot “makes it relatively easy to construct intractable puzzles” ([DDO]) • objection: Mosaic Rearranging Problem (“flying” robot) is in P – solved by assignment • Limiting the robot’s capabilities • the exact complexity of most problems (even the “more tractable” ones) is unknown, some are PSPACE-complete • Our optimization problem is in NP for any k, p, l, including infinite values [DDO] E. Demaine, M. Demaine and J. O'Rourke, “PushPush and Push-1 are NP-hard in 2D”, in Proceedings of the12th Annual Canadian Conference on Computational Geometry (CCCG 2000), Fredericton, New Brunswick, Canada, August 16-18, 2000, pages 211-219

  22. Making the Puzzle “more tractable” (contd.) • All blocks movable, no walls, infinite plane • leakage is a problem • l = 0 • k, p >1 – a wall of thickness max(k, p)+1 is rigid • p = 0 – k+1-by-k+1 square is unmovable • k = 0 – p+1-by-p+1 square is unmovable Palliative, what if k, p > 0 or l > 0 ? • In our proof the wall thickness is constant

  23. Another Problem - Crossovers • [DH], [DDHO] contrasted their work to “all previous approaches of building circuits based on graphs, which seem to inherently require [problematic] crossings” • Our construction has no crossings – the reduction is from the HC problem for planar graphs [DH] E. Demaine and M. Hoffman, “Pushing blocks is NP-complete for non-crossing solution paths”, Proc. 13th Canad. Conf. Comput. Geom. (2001), 65-68[DDHO] E. Demaine, M. Demaine, M. Hoffmann, and J. O'Rourke, “Pushing Blocks is Hard”, Computational Geometry: Theory and Applications, Special issue of selected papers from the 13th Canadian Conference on Computational Geometry, 2001.

  24. “Required third dimension to work” [Cul2] [Cul1] J. Culberson, “Sokoban is PSPACE-complete” Proc. Internet Conf. Fun with Algorithms (1998), N. S. E. Lodi, L. Pagli, Ed., Carelton Scientific, 65-76 [Cul2] J. Culberson, Private Communication, 2003 [Cul1]

  25. The Reduction • HC for planar directed graphs with each node v satisfying out(v) + in(v)= 3 [Pl]. [Pl] J. Plesnik, “The NP-completeness of the Hamilton cycle problem for planar digraphs of degree bound two”, Inform. Process. Lett., 8, No. 4(1979), 199-201 [JP] D.S. Johnson and C. H. Papadimitriou, “Computational complexity and the travelling salesman problem”, in ``The Travelling Salesman Problem'' (E. W. Lawler, J. K. Lenstra and A.G. Rinnooy Kan, Eds.), Chap. 3, Wiley, New York, 1982

  26. The Reduction (contd.) Edges – corridors of width 1Nodes – T-intersections

  27. Node gadget Edge gadget The Gadgets Checked box – initial position. Shaded box – target position. Yes, in the edge gadget they coincideThe robot is initially somewhere inside

  28. Analysis • If G is Hamiltonian – the puzzle is solved in 2(n-1) + n = 3n – 2 pushes • If not, then not less than 3n pushes is required So, our problem is NP-complete. BMP(1,0,0) with some boxes fixed to the floor (not all movable) is NP-complete

  29. NP-Completeness Results • BMP(0,1,0) with some boxes fixed • same reduction, just the direction of the edge gadget is reversed • BMP(0,0,1) with some boxes fixed • although the directionality is lost and every edge requires work of 1 unit to pass, the same reduction holds with the bound replaced by 2n – 1 • BMP(k,p,l) with some boxes fixed • nothing changes; the edge is just an “energy waster” • BMP(k,p,l)-X with some boxes fixed • since the proposed solution path is non-crossing • #BMP(k,p,l) with some boxes fixed • could have assigned numbers to the boxes and target locations • #BMP(k,p,l)-X with some boxes fixed

  30. All Blocks Movable Modified node and edge gadgets Breaking through a wall gives no benefit

  31. Main Result All variations of our problem are NP-complete: [#]BMP(k,p,l)-[X] with or without fixed boxes is NP-complete for any (k,p,l) ≠ (0,0,0), including infinite values of k, p, l.

  32. Open Problems • An interesting problem in P (since [DH], [DDHO]) • maybe, an optimization problem? • if the initial and target can be separated by a line – greedy [SBARG] • More optimization problems in NP • PushPush version (feasibility is hard) [DH] E. Demaine and M. Hoffman, “Pushing blocks is NP-complete for non-crossing solution paths”, Proc. 13th Canad. Conf. Comput. Geom. (2001), 65-68[DDHO] E. Demaine, M. Demaine, M. Hoffmann, and J. O'Rourke, “Pushing Blocks is Hard”, Computational Geometry: Theory and Applications, Special issue of selected papers from the 13th Canadian Conference on Computational Geometry, 2001[SBARG] The Stony Brook Algorithms Reading Group, 2003

  33. Related Problems • Lawn-mowing, Milling • SBP • BMP

  34. Geometric Algorithms for Air Traffic Management • Air space: free space and no-fly zones • K flights. (sk, tk) • Route the flights Simplifying assumptions No time dependence 2D

  35. Constrained Path Model for a “flyable” path • Thick path • Link-constrained path – discrete model for curvature constrained path • Rectilinear • Monotone

  36. Paths • Non-crossing • Short

  37. Objective Function • Length of individual path • L1, L2 • link length • Length of all paths • sum of lengths of individual paths • VLSI: “wire routing” – rectilinear version • the length of the longest path • hasn’t been studied earlier

  38. Minsum vs. Minmax • Minmax tend to be harder • in graphs minsum s-t paths – min cost flow, minmax – NP-hard • May be different • K-approximation to each other • NP-hard if K is not constant [BP] [BP] O. Bastert, S.P. Fekete. "Geometrische Verdrahtungsprobleme." Technical Report ZPR 96-247. 1996

  39. K Short Non-Crossing Constrained Paths Connect the pairs of points inside a polygon by non-crossing constrained paths so that the length of the longest path is as small as possible Notation

  40. Constraints/Restrictions • Constraint – condition on a path • Restriction – condition on P or on placement of (sk, tk) inside P Restrictions make the problem easier. Constraints generally make the problem harder Constraints – considered earlier

  41. Restrictions h = 0 – P is a simple polygon • further restricted to be monotone • further restricted to be convex (sk, tk) placement inside P • sk aligned • si and sj coincide for some i and j • sk are on the boundary of P • all sk are on one edge of P – combination of the above two • Same restrictions on tk placement

  42. Problem Formulation In a polygon P, such that P is [restrictions on P], K pairs of points (sk, tk) [restrictions on (sk, tk) placement] are given. Also given is a bound B. Find K non-crossing [constrained] polygonal paths in P connecting sk and tk, such that the [length definition] length of the longest path is not more than B.

  43. A Special Case • sk, tk are on the boundary of P • Unconstrained paths • No path is allowed to “go around” a terminal of another path • monotone

  44. Solution • Ordering around the boundary of P determines feasibility • not s1, s2 … sK, tK, tK-1,…, t1 – not feasible • o.w. – route in any order, all shortest paths will be found

  45. All sk, tk are on the Boundary of P O(n+K), [Pap] [Pap] E. Papadopoulou, “K-Pairs non-crossing shortest Paths in a Simple Polygon”, ISAAC 1996: 305-314 Restriction on sk, tk placement is “saved for future use”: • polynomial algs work with no restriction • hardness results work with all restrictions “active”

  46. K = 1 • Solved • Unconstrained • Rectilinear • Monotone • Thick • new_obst = old_obst + [0 ,1]2 • Find unconstrained path avoiding new_obst

  47. K = 1, Link-Constrained Path • Link-Constrained Path: • the length of each link is ≥ L • the angle between consecutive links is ≥ θ • Robotics • given links • the angle between consecutive links is ≤θ • What is the region reachable by such robot arm? • in a polygon?

  48. Reachable Region • All links are 1-links • simulations • Different link lengths (1,2,4,8,…) • at least exponential complexity

  49. Wobbly Link Lengths • Link length is 1 ±δ • fuzzy contours

  50. Wobbly Path • Link length is 1 ±δ • Angle between links is θ±γ

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