Fashion, apparel, textile, merchandising, garments

1. Finding the Largest Area Axis-Parallel Rectangle in a Polygon in O (n log 2 n) Time MATHEMATICAL SCIENCES COLLOQUIUM Prof. Karen Daniels Wednesday, October 18, 2000

2. Computational Geometry in Context Applied Computer Science Geometry Theoretical Computer Science Applied Math Computational Geometry Efficient Geometric Algorithms Design Analyze Apply

3. Taxonomy of Problems Supporting Apparel Manufacturing Ordered Containment Geometric Restriction Distance-Based Subdivision Maximum Rectangle Limited Gaps Minimal Enclosure Column-Based Layout Two-Phase Layout Lattice Packing Core Algorithms Application-Based Algorithms Containment Maximal Cover

4. A Common (sub)Problem Find a Good (and Convex) Approximation Outer Inner

6. Related Work n 1 n log(n) n log 2 (n) 2 n n 5 n  (n) log(n) n  (n)

7. Related Work (continued) n 1 n log(n) n log 2 (n) 2 n n 5 n  (n) log(n) n  (n)

8. Summary of Algorithmic Results for a Variety of Polygon Types Karen Daniels Victor Milenkovic Dan Roth   (n log(n)) this talk n 1 n log(n) n log 2 (n) 2 n n 5 n  (n) log(n) n  (n)

9. Establishing an Upper Bound of O(n log 2 n)

11. Characterizing the LR Fixed Contact Independent Sliding Contact Dependent Sliding Contacts Reflex Contact Contacts reduce degrees of freedom

16. A General Framework for the 2-Contact Case Definition : M is totally monotone if, for every i<i’ and j<j’ corresponding to a legal 2x2 minor, m ij’ > m ij implies m i’j’ > m i’j Theorem [Aggarwal,Suri87]: If any entry of a totally monotone matrix of size mxn can be computed in O(1) time, then the row-maximum problem for this matrix can be solved in  (m+n) time. A General Framework for the 2-Contact Case Area Matrix M for “empty corner rectangles” LR is the Largest Empty Corner Rectangle (LECR) 24 15 27 24 b c a 1 2 3 10 14 6 20 15 a b c 1 2 3

19. Theorem : The LR in an n-vertex general polygon can be found in O(n log 2 n) time. Partitioning the polygon with a vertical line produces a vertically separated, horizontally convex polygon for the merge step of divide-and-conquer. LR Algorithm for a General Polygon

21. Establishing a Lower Bound of  (n log n)

22. Lower Bounds in Context n 1 n log(n) n log 2 (n) 2 n n 5 SmallestOuterRectangle :   (n) SmallestOuterCircle :   (n) LargestInnerRectangle :   (n log n) LargestInnerCircle :   (n log n) point set, polygon point set, polygon point set polygon LargestInnerRectangle :   (n log 2 (n)) polygon

23. Establishing a Lower Bound of  (n log n) MAX-GAP instance : given n real numbers { x 1 , x 2 , ... x n } find the maximum difference between 2 consecutive numbers in the sorted list. O(n) time transformation LR algorithm must take as least as much time as MAX-GAP. But, MAX-GAP is already known to be in  (n log n). LR algorithm must take  (n log n) time for self-intersecting polygons. self-intersecting, orthogonal polygon x 2 x 4 x 3 x 1 LR area is a solution to the MAX-GAP instance

24. Establishing a Lower Bound of  (n log n) Extend to non-degenerate holes using symbolic perturbation. EVEN-DISTRIBUTION : given n real numbers { x 1 , x 2 , ... x n } check if there exist adjacent x i , x j in the sorted list s.t. x j - x i > 1 LR must take as least as much time as EVEN-DISTRIBUTION . But, EVEN-DISTRIBUTION is already known to be in  (n log n). LR algorithm must take  (n log n) time for polygons with degenerate holes. [McKenna et al. (85)] O(n) time transformation orthogonal polygon with degenerate holes x 2 x 4 x 3 x 1 LR area is a solution to the EVEN-DISTRIBUTION instance