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Finding the Largest Area Axis-Parallel Rectangle in a Polygon in O (n log 2 n) Time

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 . Geometry. Computational Geometry Efficient Geometric Algorithms. Design. Analyze. Theoretical

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Finding the Largest Area Axis-Parallel Rectangle in a Polygon in O (n log 2 n) Time

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  1. Finding the Largest Area Axis-Parallel Rectangle in a Polygon in O(n log2 n) Time MATHEMATICAL SCIENCES COLLOQUIUM Prof. Karen Daniels Wednesday, October 18, 2000

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

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

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

  5. n a(n) n n log2(n) n log(n) n a(n) log(n) n5 2n 1 What’s the Problem? • Given a 2D polygon that: • does not intersect itself • may have holes • has n vertices • Find the Largest-Area Axis-Parallel Rectangle • How “hard” is it? • How “fast” can we find it?

  6. Related Work n a(n) n n log2(n) n log(n) n5 2n n a(n) log(n) 1

  7. Related Work(continued) n a(n) n n log2(n) n log(n) n a(n) log(n) n5 2n 1

  8. Summary of Algorithmic Results for a Variety of Polygon Types n a(n) n n log2(n) n log(n) n a(n) log(n) n5 2n 1 Karen Daniels Victor Milenkovic Dan Roth this talk W(n log(n))

  9. Establishing an Upper Bound of O(n log2 n)

  10. Approach • Establish O(n5) upper bound • Characterize the Largest Rectangle (LR) • examine cases based on polygon/LR contacts • Reduce the O(n5) bound to O(n log2 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

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

  12. Characterizing the LR (continued) Determining sets of contacts # RC 4 3 2 1 0

  13. Characterizing the LR (continued) Maximization Problems for Sliding Contacts 1-parameter: • Max. quadratic in 1 variable: O(1) • 1 Independent Sliding Contact • 2 Dependent Sliding Contacts • 3 Dependent 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(n5) 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(n5) O(n)

  15. Approach • Establish O(n5) upper bound • Characterize the Largest Rectangle (LR) • examine cases based on polygon/LR contacts • Reduce the O(n5) bound to O(n log2 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

  16. a A General Framework for the 2-Contact Case b c b c a 1 2 3 24 27 1 10 2 15 24 14 3 20 15 6 A General Framework for the 2-Contact Case Area Matrix M for “empty corner rectangles” Definition: M is totally monotone if, for every i<i’ and j<j’ corresponding to a legal 2x2 minor, mij’ > mij implies mi’j’ > mi’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 Q(m+n) time. LR is the Largest Empty Corner Rectangle (LECR)

  17. U A General Framework for the 2-Contact Case P Goal: reduce to the Largest Empty Corner Rectangle (LECR) problem E’ V’ • 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 e V’ and every edge e e E’: if any point q e interior(e) is rectangularly visible from v inside P’, then all of e is rectangularly visible from v. • Property III • If vertex v e V’ and a point q e E’ are rectangularly visible with respect to vertices(P’), then v and q are rectangularly visible with respect to P’. P’ Given vertically separated, y-monotone chains V, E of P, “orthogonalize” them E V

  18. LR Algorithm for a General Polygon Lemma: If V’ and E’ are y-monotone, then M defined by our LR-measure (“area”) is totally monotone. Monotonicity and Aggarwal et al.’s matrix searching-based O(nlogn) algorithm for the LECR problem lead to the following: 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.

  19. LR Algorithm for a General Polygon Partitioning the polygon with a vertical line produces a vertically separated, horizontally convex polygon for the merge step of divide-and-conquer. Theorem: The LR in an n-vertex general polygon can be found in O(n log2 n) time.

  20. O(n log2 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 Pleft , Pright left, right parts of P L [vertical partitioning line] Hleft , Hleft , Vleft , Vright H, V updated for L arealeft LR_DivideConquer(Pleft , Hleft , Vleft) arearight LR_DivideConquer(Pright, Hright , Vright) Q U1<=i<=k Qi[L may contain k partitions] areaQ LR_HV_DivideConquer(Q) return maximum(arealeft , arearight, areaQ ) O(n log n) O(n log2 n) T(n) < 2T( | n/2 |) + O(nlogn) U U T( | n/2 |) T( | n/2 |) O(n log n)

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

  22. point set, polygon point set, polygon SmallestOuterCircle: Q(n) LargestInnerRectangle: O(n log2(n)) polygon point set SmallestOuterRectangle: Q(n) LargestInnerCircle: Q(n log n) polygon LargestInnerRectangle: W(n log n) 2n 1 n n log(n) n log2(n) n5 Lower Bounds in Context

  23. LR area is a solution to the MAX-GAP instance self-intersecting, orthogonal polygon x2 x4 x1 x3 Establishing a Lower Bound of W(n log n) MAX-GAP instance: given n real numbers { x1, x2, ... xn } 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 W(n log n). LR algorithm must take W(n log n) time for self-intersecting polygons.

  24. EVEN-DISTRIBUTION: given n real numbers { x1, x2, ... xn } check if there exist adjacent xi, xj in the sorted list s.t. xj- xi > 1 O(n) time transformation LR area is a solution to the EVEN-DISTRIBUTION instance orthogonal polygon with degenerate holes x2 x4 x1 x3 LR must take as least as much time as EVEN-DISTRIBUTION. But, EVEN-DISTRIBUTION is already known to be in W(n log n). LR algorithm must take W(n log n) time for polygons with degenerate holes. [McKenna et al. (85)] Establishing a Lower Bound of W(n log n) Extend to non-degenerate holes using symbolic perturbation.

  25. Summary • Establish O(n log2 n) upper bound for LR • Establish O(n5) upper bound • Characterize the Largest Rectangle (LR) • examine cases based on polygon/LR contacts • Reduce the O(n5) bound to O(n log2 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 W(n log n) lower bound for LR

  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

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