1. Parallel Computational Geometry
visibility
lower envelope
union of rectangles
3-D convex hull and 2-D Voronoi Diagram
...
CGM complexity:
O(1) rounds
O(Ts(n) / p) local computation
n/p > p

2. Lower Envelope
Given: A set S of n non-intersecting line segments S

3. Lower Envelope Given: A set S of n non-intersecting line segments
LE(S)
Task: Compute the lower envelope LE(S)

4. Lower Envelope Goal: O(1) rounds
Problem: Each processor “sees” only O(n/p) line segments

5. Lower Envelope S1 S2 S3 S4 S = S1S2S3S4 proc. 1 proc. 2 proc. 3

6. Lower Envelope Step 1: Each proc. Pi computes locally LE(Si) S1 S2 S3
S = S1S2S3S4
Step 1: Each proc. Pi computes locally LE(Si)

7. Lower Envelope Step 1: Each proc. Pi computes locally LE(Si) S1 S2 S3
S = S1S2S3S4
Step 1: Each proc. Pi computes locally LE(Si)
proc. 1
proc. 2
proc. 3
proc. 4

8. Lower Envelope
Step 2: Globally sort [i=1..p LE(Si)] by x-coord. of right endpoint

9. Lower Envelope
Step 2: Globally sort [i=1..p LE(Si)] by x-coord. of right endpoint
proc. 1
proc. 2
proc. 3
proc. 4 V1 V2 V3 V4

10. Lower Envelope
Step 3: Each Pi determines Li and broadcasts Li to all other Pj (one h-relation)
proc. 1
proc. 2
proc. 3
proc. 4 V1 V2 V3 V4 L1 L2 L3 L4

11. Lower Envelope
Step 4: Each Pi scans LE(Si) and sends the segment intersecting Lj to Pj for j=1..p (one h-relation)
proc. 1
proc. 2
proc. 3
proc. 4 R1 R2 R3 R4 V1 V2 V3 V4 L1 L2 L3 L4

12. Lower Envelope Step 5: Each Pi computes LE(Vi  Ri) R1 R2 R3 R4 V1 V2
proc. 1
proc. 2
proc. 3
proc. 4 R1 R2 R3 R4 V1 V2 V3 V4 L1 L2 L3 L4

13. Lower Envelope