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

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

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

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

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

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)

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

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

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

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

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

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

Lower Envelope