![O(n 5 ) LR Algorithm Find_LR(Polygon P) area0 Find_LR_0_RC(P) area1 Find_LR_1_RC(P) area2 Find_LR_2_RC(P) area3 Find_LR_3_RC(P) area4 Find_LR_4_RC(P) return maximum(area0, area1, area2, area3, area4) Find_LR_0_RC(P) for i 1 to n [for each edge of P] for j 1 to n for k 1 to n for l 1 to n area area of LR for 0-RC determining set for (i,j,k,l) if LR is empty, then update maximum area return maximum area O(n) O(n 5 )](https://image.slidesharecdn.com/maapr3-1223539405571153-8/85/Maapr3-14-320.jpg)
![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](https://image.slidesharecdn.com/maapr3-1223539405571153-8/85/Maapr3-16-320.jpg)
![O(n log 2 n) LR Algorithm Find_LR(Polygon P) preprocess P H, V horizontal, vertical visibility maps of P P P U internal vertex projections return LR_DivideConquer(P, H, V) LR_DivideConquer(P, H, V) if P.numVertices is “too small”calculate & return LR area P left , P right left, right parts of P L [vertical partitioning line] H left , H left , V left , V right H, V updated for L area left LR_DivideConquer(P left , H left , V left ) area right LR_DivideConquer(P right , H right , V right ) Q U 1<=i<=k Q i [L may contain k partitions] area Q LR_HV_DivideConquer(Q) return maximum(area left , area right , area Q ) O(n log n) U U O(n log 2 n) O(n log n) T(n) < 2T( | n/2 |) + O(nlogn) T( | n/2 |) T( | n/2 |)](https://image.slidesharecdn.com/maapr3-1223539405571153-8/85/Maapr3-20-320.jpg)
![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](https://image.slidesharecdn.com/maapr3-1223539405571153-8/85/Maapr3-24-320.jpg)
Maapr3
- 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
- 5. Given a 2D polygon that: does not intersect itself may have holes has n vertices Find the L argest-Area Axis-Parallel R ectangle How “hard” is it? How “fast” can we find it? What’s the Problem? n 1 n log(n) n log 2 (n) 2 n n 5 n (n) log(n) n (n)
- 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)
- 10. Establish O(n 5 ) upper bound Characterize the Largest Rectangle (LR) examine cases based on polygon/LR contacts Reduce the O(n 5 ) bound to O(n log 2 n) Develop a general framework for dominant case based on rectangular visibility and matrix total monotonicity Use divide-and-conquer: T(n) < 2T( | n/2 |) + O(nlogn) Apply the framework to obtain O(nlogn) for each level Approach
- 11. Characterizing the LR Fixed Contact Independent Sliding Contact Dependent Sliding Contacts Reflex Contact Contacts reduce degrees of freedom
- 12. Characterizing the LR (continued) # RC 4 3 2 1 0 Determining sets of contacts
- 13. Characterizing the LR (continued) 1-parameter: Max. quadratic in 1 variable: O(1) 1 Independent Sliding Contact 2 Dependent Sliding Contacts 3 Dependent Sliding Contacts Maximization Problems for Sliding Contacts 2-parameter: Max. quadratic in 2 variables: O(1) At least one rectangle corner is at an endpoint of polygon edge Reduces to 4 1-parameter problems
- 14. O(n 5 ) LR Algorithm Find_LR(Polygon P) area0 Find_LR_0_RC(P) area1 Find_LR_1_RC(P) area2 Find_LR_2_RC(P) area3 Find_LR_3_RC(P) area4 Find_LR_4_RC(P) return maximum(area0, area1, area2, area3, area4) Find_LR_0_RC(P) for i 1 to n [for each edge of P] for j 1 to n for k 1 to n for l 1 to n area area of LR for 0-RC determining set for (i,j,k,l) if LR is empty, then update maximum area return maximum area O(n) O(n 5 )
- 15. Establish O(n 5 ) upper bound Characterize the Largest Rectangle (LR) examine cases based on polygon/LR contacts Reduce the O(n 5 ) bound to O(n log 2 n) Develop a general framework for dominant case based on rectangular visibility and matrix total monotonicity Use divide-and-conquer: T(n) < 2T( | n/2 |) + O(nlogn) Apply the framework to obtain O(nlogn) for each level Approach
- 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
- 17. A General Framework for the 2-Contact Case Property I Polygonal regions P and P’ satisfy P’ P and each vertex-edge rectangle for P, V, and E is a vertex-edge rectangle for P’, V’, and E’. Property II For every vertex v V’ and every edge e E’: if any point q interior(e) is rectangularly visible from v inside P’, then all of e is rectangularly visible from v. Property III If vertex v V’ and a point q E’ are rectangularly visible with respect to vertices(P’), then v and q are rectangularly visible with respect to P’. P V E P’ V’ E’ Given vertically separated, y-monotone chains V, E of P, “orthogonalize” them Goal : reduce to the Largest Empty Corner Rectangle (LECR) problem U
- 18. Monotonicity and Aggarwal et al.’s matrix searching-based O(nlogn) algorithm for the LECR problem lead to the following: LR Algorithm for a General Polygon Lemma : The LR in an n-vertex vertically separated, horizontally convex polygon can be found in O(n log n) time. Goal : Produce a vertically separated, horizontally convex polygon for the merge step of divide-and-conquer. Lemma : If V’ and E’ are y-monotone, then M defined by our LR-measure (“area”) is totally monotone.
- 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
- 20. O(n log 2 n) LR Algorithm Find_LR(Polygon P) preprocess P H, V horizontal, vertical visibility maps of P P P U internal vertex projections return LR_DivideConquer(P, H, V) LR_DivideConquer(P, H, V) if P.numVertices is “too small”calculate & return LR area P left , P right left, right parts of P L [vertical partitioning line] H left , H left , V left , V right H, V updated for L area left LR_DivideConquer(P left , H left , V left ) area right LR_DivideConquer(P right , H right , V right ) Q U 1<=i<=k Q i [L may contain k partitions] area Q LR_HV_DivideConquer(Q) return maximum(area left , area right , area Q ) O(n log n) U U O(n log 2 n) O(n log n) T(n) < 2T( | n/2 |) + O(nlogn) T( | n/2 |) T( | n/2 |)
- 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
- 25. Establish O(n log 2 n) upper bound for LR Establish O(n 5 ) upper bound Characterize the Largest Rectangle (LR) examine cases based on polygon/LR contacts Reduce the O(n 5 ) bound to O(n log 2 n) Develop a general framework for dominant case based on rectangular visibility and matrix total monotonicity Use divide-and-conquer: T(n) < 2T( | n/2 |) + O(nlogn) Apply the framework to obtain O(nlogn) for each level Establish (n log n) lower bound for LR Summary
- 26. For More Information Computational Geometry: Graduate CS course in Computational Geometry to be offered at UMass Lowell in Spring ‘01 Introductory texts: Computational Geometry in C (O’Rourke) Computational Geometry: An Introduction (Preparata & Shamos) Bibliography: ftp://ftp.cs.usask.ca/pub/geometry/ Software:http://www.geom.umn.edu/software/cglist/ My research: http://www.cs.uml.edu/~ kdaniels Journal paper : “Finding the largest area axis-parallel rectangle in a polygon” ( Computational Geometry: Theory and Applications ) Prof. Victor Milenkovic: Frequent co-author and former PhD advisor http://www.cs.miami.edu/~vjm
Editor's Notes
- #2: I joined the UMass Lowell Computer Science faculty this summer. This collection of slides is intended to familiarize the reader/viewer with my field of research (Computational Geometry), summarize my previous research results in this field and outline my plan for Computational Geometry research at UMass Lowell.