Derrick Stolee. April 12,

Transcription

1 1 / 38 University Of Nebraska Lincoln Computer Science & Mathematics April 12, 2008 s-dstolee1@math.unl.edu

2 2 / 38 Acknowledgments Research Assistance and Funding This research funded by UCARE USDA FCIC/RMA Cooperative Agreement 2IE

3 3 / 38 Motivation Spatial Queries

4 4 / 38 Motivation Spatial Queries

5 5 / 38 Motivation Spatial Queries

6 6 / 38 Motivation Spatial Queries

7 7 / 38 Motivation Spatial Queries

8 Partitions and Layouts Vertices and Chords Consider a set S of solid rectangles with a bounding rectangle B in the plane so that each R S has R B. The rectilinear layout for S in B is denoted L = B, S. Let F = B \ R S R be the free space of L. Definition A rectilinear partition of L is a set P of non-overlapping rectangles so that F = R P R. The rectangles in P are called space rectangles. Note: Non-overlapping does not include borders touching. 8 / 38

9 Analogue in Polygons When the free space F is connected, then the border of F is a rectilinear polygon. If that polygon is simple, we say that the layout L is simple. Partitions and Layouts Vertices and Chords 9 / 38

10 Vertices and Chords Partitions and Layouts Vertices and Chords A corner of a rectangle R S is called a vertex. Definition If a vertex has an angle of 270 degrees in the free space F, then that vertex is a concave vertex. Otherwise, it is a convex vertex. 10 / 38

11 An Example Layout Partitions and Layouts Vertices and Chords 11 / 38

12 An Example Layout Partitions and Layouts Vertices and Chords 12 / 38

13 An Example Layout Partitions and Layouts Vertices and Chords 13 / 38

14 An Example Layout Partitions and Layouts Vertices and Chords 14 / 38

15 Chords and Cogrid Vertices Partitions and Layouts Vertices and Chords Definition Two concave vertices are cohorizontal if there is a horizontal line from one to the other that lies in F. Two concave vertices are covertical if there is a vertical line from one to the other in F. Two vertices are cogrid if either condition holds. Definition A chord is a line between two cogrid vertices. 15 / 38

16 An Example Layout Partitions and Layouts Vertices and Chords 16 / 38

17 on Simple Polygons No Cogrid Vertices Simple Polygons Non-simple Polygons Theorem A simple layout L = B, S with no cogrid concave vertices has a minimum partition of size N + 1, where N is the number of concave vertices. 17 / 38

18 on Simple Polygons With Cogrid Vertices Simple Polygons Non-simple Polygons Define the chord intersection graph as G = (X Y, E), a bipartite graph where X is the set of horizontal chords in L, Y is the set of vertical chords in L, and an edge xy is in E if and only if the chords x and y intersect, possibly at a vertex. Theorem A simple layout L = B, S has a minimum partition of size N α(g) + 1, where N is the number of concave vertices and α(g) is the independence number of the chord intersection graph. 18 / 38

19 19 / 38 on Simple Polygons Simple Polygons Non-simple Polygons

20 20 / 38 on Simple Polygons Simple Polygons Non-simple Polygons

21 21 / 38 on Simple Polygons Simple Polygons Non-simple Polygons

22 on Non-simple Polygons Definition of Holes Definition A hole in a layout L = B, S is a set A S, maximal with respect to R A R connected, which does not touch the border of B. Simple Polygons Non-simple Polygons 22 / 38

23 23 / 38 on Non-simple Polygons Four-Connected Simple Polygons Non-simple Polygons

24 24 / 38 on Non-simple Polygons Four-Connected Simple Polygons Non-simple Polygons

25 25 / 38 on Non-simple Polygons Four-Connected Simple Polygons Non-simple Polygons

26 26 / 38 on Non-simple Polygons Four-Connected Simple Polygons Non-simple Polygons

27 27 / 38 on Non-simple Polygons Four-Connected Simple Polygons Non-simple Polygons

28 on Non-simple Polygons Simple Polygons Non-simple Polygons Theorem A four-connected layout L = B, S has a minimum partition of size N α(g) H + 1, where N is the number of concave vertices, α(g) is the independence number of the chord intersection graph, and H is the number of holes in L. 28 / 38

29 29 / 38 on Non-simple Polygons The Issue with Holes Simple Polygons Non-simple Polygons

30 30 / 38 Algorithmic The First Approach In 1984, Ferrari, Sankar, & Sklansky used an existing algorithm for finding maximum independent sets in bipartite graphs to solve this problem in O(n 2 ).

31 31 / 38 Algorithmic A Better Approach In 1986, Imai & Asano defined an algorithm for maximum independent sets of horizontal and vertical line segments in O(n 3/2 log n). This can be immediately applied to the chord intersection graph.

32 32 / 38 Algorithmic The Fastest Approach (So Far) In 1989, Liou, Tan, & Lee used a branch-and-bound technique to solve the problem for simple polygons in O(n log log n). This required the use of the following lemma: Lemma (Lipski & Preparata) Let G = (X Y, E) be a bipartite graph. Let xy E. If N(x) N(x ) for all x N(y), then there exists a maximum matching containing the edge xy.

33 33 / 38 Covers Removing the non-overlapping requirement from the partition definition results in a cover. The rectilinear cover problem is NP-Complete for polygons with holes. The rectilinear border-cover problem is NP-Complete for polygons with holes, even if the polygon is in general position.

34 34 / 38 Question in Three Dimensions What happens in three dimensions? n dimensions?

35 35 / 38 Question in Three Dimensions Replace vertices with pivots. A 2D slice 270

36 36 / 38 Question in Three Dimensions Chords are 2-dimensional, but are not between just two pivots.

37 37 / 38 Question in Three Dimensions Deadlock conditions arise! (a) Deadlock Condition in 3D. (b) Top View of Deadlock.

38 38 / 38 Question in Three Dimensions What is a hole?

39 39 / 38 References Berman, P., DasGupta, B., Complexities of Efficient Solutions of Cover, Algorithmica Volume 17: pp Ferrari, L., Sankar, V., Sklansky, J., Minimal Rectangular Partitions of Digitized Blobs, Computer Vision, Graphics, and Image Processing, Volume 28 Number 1, October 1984: pp Hsiao, P.Y., Lin, C.Y., Shew, P.W. Optimal Tile partition for space region of integrated circuits geometry, IEEE Proceedings Volume 140 Number 3, May 1993: pp Imai, H., Asano, T. Efficient for Geometric Graph Search, SIAM Journal of Computing, Volume 15 Number 2, May 1986: pp Liou, W.T., Tan, J.J.M., Lee, R.C.T. Simple Polygons n O(nloglogn)-Time, Proc. of the Fifth Annual Symposium on Computational Geometry. 1989: pp Nguyen, V.H., Optimum partitioning of rectilinear layouts, IEEE Proceedings Online Number : 1996 pp