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Chapter 5 Guillotine Cut

Chapter 5 Guillotine Cut. Ding-Zhu Du. (1) Rectangular Partition. Rectangular Partition. Given a rectilinear area bounded by rectilinear polygon with rectilinear holes , partition it into rectangles with minimum total edge length. Canonical Partition.

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Chapter 5 Guillotine Cut

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  1. Chapter 5 Guillotine Cut Ding-Zhu Du (1) Rectangular Partition

  2. Rectangular Partition • Given a rectilinear area bounded by rectilinearpolygon with rectilinear holes, partition it into rectangles with minimum total edge length.

  3. Canonical Partition There is a minimum partition such that every maximal segment passes through a concave vertex of the boundary.

  4. Without any hole In any canonical partition, there exists a maximal segment cuttingthe area into two parts.

  5. Without any hole Dynamic programming can find optimal partition in time O(n ) 5

  6. An NP-hard special case

  7. Guillotine cut

  8. Guillotine Partition A sequence of guillotine cuts Canonical one: every cut passes a hole.

  9. Minimum Guillotine Partition Dynamic programming In time O(n ): 5 Each cut has at most 2n choices. 4 There are O(n ) subproblems. Minimum guillotine partition can be a polynomial-time approximation.

  10. Analysis (idea) • Consider a minimum rectangular partition. • Modify it into a guillotine partition by adding some segments. • Estimate the total length of added segments.

  11. Add vertical segment There is a vertical segment > 0.5 length of vertical edge rectangle. Charge to > 0.5 a Charge added length to the vertical segment. a

  12. Add horizontal segment when vertical segment cannot Be added. Charge 0.5 to all horizontal segments facing the added line.

  13. Conclusion • Minimum guillotine partition is 2-approximation for minimum rectangular partition.

  14. 1-guillotine cut 1-guillotine cut

  15. 1-guillotine partition A sequence of 1-guillotine cuts Canonical one: every cut passes through a hole. 2 1

  16. Minimum 1-guillotine partition Dynamic programming: 3 Each cut has O(n ) choices. 12 There are O(n ) subproblems. 15 Time = O(n ) Minimum 1-guillotine partition is a polynomial-time Approximation.

  17. Subproblems result from 1-guillotine cuts Each edge of rectangle has O(n ) choices. 3 12 There are O(n ) subproblems.

  18. Analysis (idea) • Consider a minimum rectangular partition. • Modify it into a 1-guillotine partition by adding some segments. • Estimate the total length of added segments.

  19. 1-dark point A vertical 1-dark point is a point between two horizontal cuts. In a vertical line, all vertical 1-dark points form a 1-guillotine cut. Similarly, define horizontal 1-dark points.

  20. Area V of vertical 1-dark points

  21. Area H of horizontal 1-dark points

  22. V<H There exists a vertical line such that length (vertical 1-dark points) < length (horizontal 1-dark points) Do vertical 1-gullotine cut. Charge 0.5 to vertical segment directly facing to horizontal 1-dark points on the cut line. Similarly, if V > H.

  23. V<H There exists a vertical line such that length (vertical 1-dark points) < length (horizontal 1-dark points) Do vertical 1-gullotine cut. Charge 0.5 to vertical segment directly facing to horizontal 1-dark points on the cut line. Similarly, if V > H.

  24. Conclusion • Minimum 1-guillotine partition is 2-approximation for minimum rectangular partition.

  25. Rectangular Partition There is a PTAS for minimum rectangular partition.

  26. m-guillotine cut m m-guillotine cut m

  27. m-guillotine partition A sequence of m-guillotine cuts Canonical one: every cut passes through a hole. 2 1

  28. Minimum m-guillotine partition Dynamic programming: 2m+1 Each cut has O(n ) choices. 4(2m+1) There are O(n ) subproblems. 10m+5 Time = O(n ) Minimum m-guillotine partition is a polynomial-time Approximation.

  29. Subproblems result from m-guillotine cuts Each edge of rectangle has O(n ) choices. 2m+1 4(2m+1) There are O(n ) subproblems.

  30. Analysis (idea) • Consider a minimum rectangular partition. • Modify it into a m-guillotine partition by adding some segments. • Estimate the total length of added segments.

  31. m-dark point A vertical m-dark point is a point between two groups of m horizontal cuts. In a vertical line, all vertical m-dark points form a m-guillotine cut. Similarly, define horizontal m-dark points.

  32. Area V of vertical m-dark points

  33. Area H of horizontal m-dark points

  34. V<H There exists a vertical line such that length (vertical m-dark points) < length (horizontal m-dark points) Do vertical m-gullotine cut. Charge 0.5/m to m vertical segment directly facing to horizontal m-dark points on the cut line. Similarly, if V > H.

  35. Conclusion • Minimum m-guillotine partition is (1 + 1/m)-approximation for minimum rectangular partition. • Choose m = 1/ε. A PTAS is obtained. • Time = n O(1/ε)

  36. Rectilinear Steiner Tree • Given a set of points in the rectilinear plane, find a minimum length tree interconnecting them. • Those given points are called terminals.

  37. Initially

  38. Thanks, End

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