Static and Dynamic Boundary Labeling

Transcription

1 Static and Dynamic Boundary Labeling Hsu-Chun Yen Dept. of Electrical Engineering National Taiwan University

2 Map Labeling Question: where to put the labels? Basic requirements: No overlaps No ambiguity Optimizing quality measure etc 2

3 Classification Where? Internal vs. External When? Static vs. Dynamic How? Manual vs. Automated 3

4 Internal vs. External Labeling Internal labeling Labels placed next to objects Easier to associate labels with objects External labeling Labels placed in the external region Utilize space more efficiently 4

5 Static vs. Dynamic Labeling Scale=1 Fixed label size Scale=0.5 5

6 Dynamic Internal Labeling Zooming/panning (Been et al., IEEE TVCG 2006) Rotating (Gemsa et al., EuroCG 2013) Trajectory-based (Gemsa et al., WADS 2011) 6

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8 Boundary Labeling (Bekos et al., GD 2004) Given a set P = {p1,..., pn} of points and an axisparallel rectangle R that contains P. p i is associated with an axis-parallel rectangular open label. labels placed and connected to their corresponding sites by polygonal lines (leaders). no two labels/leaders intersect labels lie outside R. R P.8 8

9 Boundary Labeling (Bekos et al., GD 2004) Type-opo leaders Type-po leders Type-s leaders site leader label Min (total leader length) s.t. #(leader crossing) = 0 1-side, 2-side, 4-side (Bekos & Symvonis, GD 2005) 9

10 Objectives LEGAL : find a legal label placement TLLM : find a legal label placement such that the total leader length is minimum TLHM : find a legal label placement such that the total label height is minimum TBM : find a legal label placement such that the total number of bends is minimum or, equivalently, the number of type-o leaders is maximum LSM : find the maximum label size for which a legal label placement is possible P.10 10

11 Some Known Results (Bekos et al., GD 2004; GD 2005, ) Model opo, 1-s, var-labels opo, 4-s, uni-labels po, 1-s, uniform opo, 2-s, uni-max-labels po, 2-s, uni-max-labels opo, 4-s, uni-sqr-labels opo, 2-s, var-labels s, 1-s, uni-labels s, 1-s, uni-labels s, 4-s, uni-sqr-labels Objective legal # bends legal legal Total Leader Length (TLL) TLL TLL TLL legal TLL TLL Complexity O(n log n) O(n 2 ) O(n log n) O(n 2 ) O(n 2 ) O(n 2 ) O(n 5 ) O(nH 2 ) O(n log n) O(n 2 ) O(n 2 ) 11

12 Some Variants Polygons labeling (Bekos et. al, APVIS 2006) Multi-stack boundary labeling (Bekos et. al, FSTTCS 2006) Octilinear leader: (Bekos et. al, SWAT 2008) 12

13 Many-to-one Boundary Labeling (Lin, Kao,Yen, JGAA 2008) Distribution of some animals in Taiwan: Brown booby Taiwan hill partridge Masked palm civet Hawk Melogale moschata Bamboo partridge Chinese pangolin Mallard 13

14 2-sided May-to-one BL (Lin, Kao,Yen, JGAA 2008) Taiwan hill partridge Brown booby Melogale moschata Masked palm civet Bamboo partridge Hawk Mallard Chinese pangolin 14

15 Algorithmic Analysis of Some Many-to-One BL Problems (Lin, Kao, Yen, JGAA 2008) objective Min #(crossing) Min Total leader length # of sides leader type complexity solution 1-side opo NP-complete 3-approx. 2-side opo NP-complete 3(1+.301/c)- approx. 1-side po NP-complete heuristic 2-side po NP-complete heuristic any any polynomial time 15

16 Variants of Many-to-one BL Original many-to-one BL (Lin, Kao,Yen, JGAA 2008) M-t-O BL with hyperleaders & dummy labels (Lin, PacificVis 2010) PS2 Port 8 DIMMs 2 CPUs 6 DIMMs USB Port COM Port VGA Port 4 CPUs 6 Expansion IDE Slot ATX Power Supply 2 LAN Ports Battery BIOS 2 Chipsets 6 SATA PS2 Port 6 DIMMs COM Port VGA Port 2 LAN Ports 6 Expansion IDE Slot 2 CPUs 2 DIMMs ATX Power Supply Battery BIOS 2 Chipsets 6 SATA M-t-O BL with backbones (Bekos et al., GD 2013) PS2 Port USB Port COM Port VGA Port 8 DIMMs ATX Power Supply 2 LAN Ports Battery 4 CPUs BIOS 6 Expansion 2 Chipsets IDE Slot 6 SATA 16

17 1.5-sided Boundary Labeling 1.5-sided Boundary Labeling (Lin, Poon, Takahashi, Wu, Yen, COCOA 2011) type-opo: direct leader vs. indirect leader #1 #2 #3 #4 #5 indirect leader #1 #6 #2 #3 #4 #6 Annotation system for wordprocessing S/W #5 direct leader 17

18 1.5-sided Boundary Labeling (labelsize, labelport, Objective) (labelsize, labelport, Objective) #1 #2 #3 #1 #2 #1 #2 #3 longer length #3 #1 #2 shorter length #1 #2 #3 #4 #4 #1 #2 #3 #(bends) = 6 #(bends) = 2 Uniform label Nonuniform label Min (total leader length) (TLLM for short) Min (total bend num) (TBM for short) (labelsize, labelport, Objective) #1 #2 #1 #2 #1 #2 Fixed-ratio port Fixed-position port Sliding port 18

19 Algorithmic Problem Setting Analysis (LabelSize, LabelPort, Objective) time (uniform, FR/sliding, TLLM) O(n log n) (uniform, FP, TLLM) O(n 5 ) (uniform, FR/FP/sliding, TBM) O(n 5 ) (nonuniform, FR/FP/sliding, TLLM) NP-complete* (nonuniform, FR/FP/sliding, TBM) NP-complete* * Pseudo-polynomial time algorithms and fixed-parameter algorithms are designed for those intractable problems. 19

20 Other Variants of BL BL with free spaces Convex Boundary (Priesner, 2013) (Jami, 2012) Panorama images (Gemsa, et al., ACM SIGSPATIAL GIS 2011) 20

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22 Dynamic External (Boundary) Labeling Zooming/panning Rotating Trajectory-based 22

23 Dynamic One-sided Boundary Labeling ( Nollenburg, Polishchuk, Sysikaski, ACM SIGSPATIAL GIS 2010) Label Size Invariance: Labels on screen have fixed size, invariant under zooming Minimize total leader length Optimal label placement as a function of z (zoom level) Single stack model Clustered model Zooming f(z)? 23

24 Dynamic Boundary Labeling Viewing window moves along some trajectory Maintain mental map Maximize total # of label assignments Solution? 24

25 Necklace Map (Speckmann, Verbeek, IEEE TVCG 2010) Map Labeling + Cartogram? Regions have sizes The surrounding necklace is given The center of each circle lies in the wedge Circles do not overlap Maximize circles 25

26 Cartograms Given a weighted planar graph G=(V, E, w), find a contact representation of G s.t. the area of each region equals the weight of the vertex. Weight: geographic size, population, trade amount 26

27 Area Universality Eevery possible area assignment can be realized by a combinatorially equivalent one

28 Area-universal Rectilinear Cartograms For rectangular cartograms, onesidedness is sufficient and necessary (Eppstein et al. 2012) Simple characterization for rectilinear cartograms is not known Linear time algorithm for 10-sided rectilinear cartograms (Alam et al., 2011) 8-sided rectilinear cartograms are sufficient; but a constructive algorithm is not known 28

29 Cartogram + Labeling? Label placement under dynamic environment, i.e., weight assignment of the map changes Label Label Label Label Label Label Label Label Label Label A challenging problem! 29

30 Contact Representations Contact Representation Disk contact Rectilinear polygons are popular in cartograms Planar graph Rectangular dual Triangle contact 30

31 Simplicity Issues Issue # 1: Shape of bounding rectilinear region? Issue # 2: Shape of internal rectilinear region? 31

32 Orthogonally Convex Floorplan (Chang, Yen, GD 2013) Issue # 1 Simple, connected, int. triangulated plane graph G dual Orthogonally convex polygon Q Q-floorplan of G dual 32

33 Issue # 2 Polygonal complexity I-, L-, and T-modules are sufficient for floorplans of maximal plane graphs. (Liao, Lu, Yen, J. Algorithm 2003) 33

34 T-free Rectilinear Duals How many sides ae needed for rectilinear duals without T-shape modules? (Non-rotated) Staircase module: 34

35 T-free Rectangular Duals (Chang, Yen, 2014) For maximal plane graphs, T-free rectilinear duals have polygonal complexity of 12. A linear time algorithm that constructs nonrotated 12-sided staircase rectilinear duals Exists one such that every non-rotated staircase rectilinear dual is not area-universal. 35

36 Conclusion and future work Boundary labeling raises many interesting and challenging algorithmic problems with real-world applications Future work includes: More on dynamic boundary labeling Other variants of boundary labeling Cartogram design + (boundary) labeling 3D boundary labeling 36

37 Thank You for Your Attention P.37 37