1. Algorithmic Aspects of Proportional Symbol Maps Sergio Cabello Herman Haverkort Marc van Kreveld Bettina Speckmann IMFM Ljubljana TU Eindhoven Utrecht University TU Eindhoven

2. Proportional symbol maps

3. For quantities on maps; size of symbol is proportional to the quantity –Symbols at points / regions –Classed / unclassed symbol sizes –Symbols are disks / squares /... –Transparent / opaque symbols with regions with points

4. Proportional symbol maps A set of n points with values  convert values to disks / squares of some size 43 17 25 21 12 37 26 5 0—10 11—20 21—30 31—40 41—50 46

5. Proportional symbol maps A set of n points with values  convert values to disks / squares of some size 0—10 11—20 21—30 31—40 41—50 43 17 25 21 12 37 26 5 46

6. Proportional symbol maps A set of n points with values  convert values to disks / squares of some size 0—10 11—20 21—30 31—40 41—50

7. Proportional symbol maps A set of n points with values  convert values to disks / squares of some size 0—10 11—20 21—30 31—40 41—50

8. Proportional symbol maps A set of n points with values  convert values to disks / squares of some size 0—10 11—20 21—30 31—40 41—50

9. Circle outlines and drawings Arrangement of circle outlines; choice which arcs to show –no restrictions  drawing –at least outer boundary  bounded drawing

10. Face correct drawings Bounded + very face appearing in the drawing is correct (locally correct)

11. Physically realizable drawings Face correct + for every face of the arrangement of circles, there is a total order on the disks involved, and for any two faces, the orders do not conflict D1D1 D4D4 D3D3 D2D2 D 2 > D 1 > D 4 D 3 > D 2 > D 4 D 3 > D 2 for each face we see the topmost disk

12. Physically realizable drawings Cyclic overlap can occur D1D1 D5D5 D3D3 D2D2 D4D4

13. Stacking drawings The result of a strict, global (stacking) order of the disks from top to bottom two different stacking drawings of the same disks

14. When is a drawing good? Physically realizable or stacking (bounded and face correct is not sufficient) Maximal visible perimeter of the disks –maximize total perimeter visible (max-total) –maximize least visible perimeter (max-min)... seeing perimeter (length) is better than seeing (disk) area

15. Physically realizable vs. stacking A stacking drawing may need to have a disk that is hardly visible, whereas a physically realizable drawing gives >50% of each perimeter visible stackingphysically realizable

16. Algorithmic results The max-min stacking algorithm improves to –O(n log n) if the max overlap at any point is constant –O(n log 2 n) for unit squares stackingphysically realizable max-min perimeter O(n 2 log n)NP hard (& no PTAS) max-total perimeter openNP hard

17. NP hardness Relatively straightforward reduction from planar 3-SAT with variable gadgets, clause gadgets,... cyclic overlap variable in TRUE state channel 1/8 1/4 least visible perimeter is ¼  planar 3-SAT instance satisfiable

18. Max-min visible perimeter stacking Idea = bottom-up: Try every disk as bottommost and determine visible perimeter for it For efficient implementation, maintain the visible perimeter of every disk if it were bottommost of the remaining disks DiDi disks placed already (lower) not yet placed visible perimeter

19. Max-min visible perimeter stacking Maintain visible perimeter in augmented segment tree Choose “best” bottommost disk  update <n trees in O(log n) time each  in total O(n 2 log n) time DiDi not yet placed visible perimeter TiTi

20. Experimental results Comparison of various stacking methods –quantitatively: least visible, least visible 10, total visible –visually: for artifacts Data sets of various numbers of disks and different distributions of disk sizes –Earthquake magnitude; 602 disks –Earthquake deaths; 602 disks –City population (USA); 156, 538, 1260 disks –...

21. Experimental results Stacking methods –random –left-to-right by leftmost ( ~ bottom-to-top) –max-min optimal –large-to-small ( ~ bottom-to-top) left-to-rightmax-min optimallarge-to-small

23. RND

24. AP

25. B to S

26. LR Boundary

27. LR Center

28. LR Center New

29. Death

30. Magnitude

32. 156 cities, random

33. 156 cities, left-to-right

34. 156 cities, max-min optimal

35. 156 cities, large-to-small

36. Earthquake, max-min optimal

37. Experimental results If the disks have clearly varying sizes, large-to- small and max-min optimal are best If the disks little variation in size, left-to-right and max-min optimal are best Max-min optimal is always best for the least visible disk, or the top-10 (of course) For total visible perimeter, max-min optimal is usually best, but sometimes large-to-small is better Left-to-right has visual artifacts

38. Conclusions Proportional symbol mapping for values at points gives rise to interesting algorithmic questions Whether a max-total visible perimeter stacking order can be computed in poly-time is open The max-min optimal stacking method takes O(n 2 log n) time, but gives better maps than certain O(n log n) time heuristics