Parallel Computational Geometry visibility CGM complexity: - PDF document

Parallel Computational Geometry • visibility • CGM complexity: • lower envelope • O(1) rounds • union of rectangles • O(T s (n) / p) local • 3-D convex hull and 2- computation D Voronoi Diagram • n/p > p ... Frank Dehne www.dehne.net Lower Envelope • Given: A set S of n non-intersecting line segments S Frank Dehne www.dehne.net 1

Lower Envelope • Given: A set S of n non-intersecting line segments S LE(S) • Task: Compute the lower envelope LE ( S ) Frank Dehne www.dehne.net Lower Envelope • Goal: O(1) rounds • Problem: Each processor “sees” only O(n/p) line segments Frank Dehne www.dehne.net 2

Lower Envelope proc. 1 proc. 2 proc. 3 proc. 4 S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 S 1 S 2 S 3 S 4 Frank Dehne www.dehne.net Lower Envelope proc. 1 proc. 2 proc. 3 proc. 4 S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 S 1 S 2 S 3 S 4 • Step 1: Each proc. P i computes locally LE(S i ) Frank Dehne www.dehne.net 3

Lower Envelope proc. 1 proc. 2 proc. 3 proc. 4 S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 S 1 S 2 S 3 S 4 • Step 1: Each proc. P i computes locally LE(S i ) proc. 1 proc. 2 proc. 3 proc. 4 Frank Dehne www.dehne.net Lower Envelope • Step 2: Globally sort [ ∪ i=1..p LE(Si)] by x-coord. of right endpoint Frank Dehne www.dehne.net 4

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 V 1 V 2 V 3 V 4 Frank Dehne www.dehne.net Lower Envelope • Step 3: Each P i determines L i and broadcasts L i to all other P j (one h-relation) proc. 1 proc. 2 proc. 3 proc. 4 V 1 V 2 V 3 V 4 L 1 L 2 L 3 L 4 Frank Dehne www.dehne.net 5

Lower Envelope • Step 4: Each P i scans LE(S i ) and sends the segment intersecting L j to P j for j=1..p (one h- relation) proc. 1 proc. 2 proc. 3 proc. 4 R 1 R 2 R 3 R 4 V 1 V 2 V 3 V 4 L 1 L 2 L 3 L 4 Frank Dehne www.dehne.net Lower Envelope • Step 5: Each P i computes LE(V i ∪ R i ) proc. 1 proc. 2 proc. 3 proc. 4 R 1 R 2 R 3 R 4 V 1 V 2 V 3 V 4 L 1 L 2 L 3 L 4 Frank Dehne www.dehne.net 6

Lower Envelope # rounds local comp. Step 1 - O(n/p log n) Step 2 5 O(n/p log n) Step 3 1 O(n/p) Step 4 1 O(n/p) Step 5 - O(n/p log n) TOTAL 7 O(n/p log n) Frank Dehne www.dehne.net Frank Dehne www.dehne.net 7