The Visibility Problem and Binary Space Partition

1. The Visibility ProblemandBinary Space Partition

2. The Visibility Problem b a c e d

3. Z-buffering • Draw objects in arbitrary order • For each pixel, maintain depth (“z”) • Only draw pixel if new “z” is closer Instead: draw objects in order, from back to front (“painter’s algorithm”)

4. 1 3 Binary Planar Partitions 0 0 b 2 1 ab cde a c 3 de a b c 2 e d e d

5. 1 3 Painter’s Algorithm 0 0 b 2 1 a c 3 a b c 2 e d e d

6. Binary Planar Partitions b1 b b2 a c

7. Auto-partitions a b1 b2,c c b1 b b2 b2 a c

8. Auto-partitions a b1 b2,c b2 b1 b2 c1 c2 a c1 c c2

9. Auto-partitions 1+2+3+∙∙∙+n = n(n+1)/2 = O(n2)

10. Binary Planar Partitions Goal: Find binary planer partition, with small number of fragmentations

11. Random Auto-Partitions Choose random permutation of segments (s1, s2, s3,…, sn) While there is a region containing more than one segment, separate it using first si in the region

12. Random Auto-Partitions u can cut v4 only if u appears before v1,v2,v3,v4 in random permutation P(u cuts v4) ≤ 1/5 u v1 v4 v3 v2

13. t3 t1 t2 Random Auto-Partitions E[number of cuts u makes] = E[num cuts on right] + E[num cuts on left] = E[Cv1+Cv2+∙∙∙] + E[Ct1+Ct2+∙∙∙] = E[Cv1]+ E[Cv2]+∙∙∙ + E[Ct1]+ E[Ct2]+∙∙∙ ≤ 1/2+1/3+1/4+∙∙∙+1/n + 1/2+1/3+1/4+∙∙∙+1/n = O(log n) E[total number of fragments] = n + E[total number of cuts] = n + uE[num cuts u makes] = n+nO(log n) = O(n log n) u v1 v4 v3 v2 Cv1=1 if u cuts v1, 0 otherwise

14. Random Auto-Partitions Choose random permutation of segments (s1, s2, s3,…, sn) While there is a region containing more than one segment, separate it using first si in the region O(n log n) fragments in expectation

15. Free cuts Use internal fragments immediately as “free” cuts

16. Binary Space Partitions • Without free cuts: O(n3) • With free cuts: O(n2)