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Traversing the Machining Graph

Traversing the Machining Graph. Danny Chen, Notre Dame Rudolf Fleischer, Li Jian, Wang Haitao,Zhu Hong, Fudan Sep,2006. 2D-Milling. Example. [Arkin,Held,Smith’00]. Zigzag machining. Example. [Tang,Joneja’03]:. Example. [Tang,Joneja’03]:. The Model. The Model. The Model.

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Traversing the Machining Graph

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  1. Traversing the Machining Graph Danny Chen, Notre Dame Rudolf Fleischer, Li Jian, Wang Haitao,Zhu Hong, Fudan Sep,2006

  2. 2D-Milling

  3. Example [Arkin,Held,Smith’00] Zigzag machining

  4. Example [Tang,Joneja’03]:

  5. Example [Tang,Joneja’03]:

  6. The Model

  7. The Model

  8. The Model Compulsory edge (be traversed exactly once) Non-compulsory edge (be traversed at most once) We are stuck

  9. The Model We are stuck: jump

  10. The Model Goal: minimize jumps

  11. Greedy?

  12. Greedy?

  13. Greedy?

  14. Greedy? 2 jumps

  15. Greedy?

  16. Greedy?

  17. Greedy? 2 jumps

  18. Greedy?

  19. Greedy? 1 jump

  20. Greedy? 1 jump

  21. Greedy? 2 jumps

  22. Greedy? 1 jump

  23. Greedy? 1 jump

  24. Greedy?

  25. Greedy?

  26. Greedy? no jump

  27. Greedy? May be exponential

  28. What is Known Simple polygon: • NP-hard? • Some heuristics [Held’91, Tang,Chou,Chen’98] Polygon with h holes: • NP-hard [Arkin,Held,Smith’00] • 5OPT+6h jumps [AHS’00] • Opt+h+Njumps [Tang,Joneja’03]

  29. What we Show Simple polygon: • NP-hard?No,linear time (DP) • Some heuristics[Held’91, Tang,Chou,Chen’98] Polygon with h holes: • NP-hard [Arkin,Held,Smith’00] • 5OPT+6h jumps[AHS’00] • Opt+h+Njumps[Tang,Joneja’03] • OPT+εh jumps in polynomial time • Opt jumps in linear+O(1)O(h) time (DP)

  30. lemma Lemma [Arkin,Held,Smith’00]: • There exists a optimal solution s.t. (1) every path starts and ends with compulsory edges. (2) No two non-compulsory edges are traversed consecutively. (alternating lemma)

  31. Simple Pocket: The Dual Tree

  32. Simple Pocket:Dynamic Programming start at the leaves

  33. Simple Pocket:Dynamic Programming

  34. Dynamic Programming Does path end here?  5 cases constant time per node

  35. Polygon with h Holes time O(n)+O(1)O(h)

  36. Polygon with h Holes • Identify O(h) pivotal nodes.

  37. Polygon with h Holes • Using arbitrary strategy to cut all the cycles gives a (O(1)^O(h))*O(n) algorithms. • Identify O(h) pivotal node whose removal s.t. 1.break all cycles. 2.each remaining (dual) tree is adjacent to O(1) pivotal nodes. Then, we can do it in (O(1)^O(h))+O(n) time.

  38. Polygon with h Holes:Boundary graph

  39. Polygon with h Holes:Minimum Restrict Path Cover Boundary graph Original Pocket e_1 e_2 e_4 e_3 Forbidden pairs: (e_1,e_4) and (e_2,e_3)

  40. Polygon with h Holes:Minimum Restrict Path Cover • A valid path: no forbidden pairs appear in one path. • MRPC: find min # valid paths cover all vertices.

  41. Polygon with h Holes:Minimum Restrict Path Cover • Graph with Bounded Tree Width (informal) 1 communicaton O(1) communicatons Tree Graph with bound treewidth

  42. Polygon with h Holes:Minimum Restrict Path Cover(MRPC) • It turns out MRPC can be solved in linear time by dynamic programming if the boundary graph has bounded treewidth. (assume its tree-decomposition is given) Remark:If tree-decomposition is not given, find 3-approximation to treewidth in time O(n log n). [Reed’92]

  43. Polygon with h Holes: • k-outerplanar graph:

  44. Polygon with h Holes: • k-outerplanar graph: Peel off the outer layer

  45. Polygon with h Holes: • k-outerplanar graph: Peel off the outer layer Peel again

  46. Polygon with h Holes: • k-outerplanar graph: Peel off the outer layer A 3-outplanar graph Peel again Peel again --nothing left… • Theorem: if a graph is k-outerplanar, it has treewidth 3k-1 . [Bodlaender’88]

  47. Polygon with h Holes • Lemma: (1) If dual graph has a bounded treewidth and bounded degree, its corresponding boundary graph has bounded treewidth. (2) If dual graph is a k-outplanar graph, its corresponding boundary graph is a 2k-outerplanar graph.

  48. Polygon with h Holes • Thus, if the dual graph is (1) a graph with bounded treewidth and bounded degree, or (2)a k-outerplanar graph, MRPC can be solved in polynomial time.

  49. After cut Original Pocket Polygon with h HolesApproximation for general planar graphs • Cut:

  50. After cut Original dual Polygon with h HolesApproximation for general planar graphs • Cut an edge (in the dual):

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