Parallel Computational Geometry

1
ParallelComputational 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 gt p

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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

S
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
proc. 1
proc. 2
proc. 3
proc. 4
S1
S2
S3
S4
S S1?S2?S3?S4
6
Lower Envelope
proc. 1
proc. 2
proc. 3
proc. 4
S1
S2
S3
S4
S S1?S2?S3?S4

• Step 1 Each proc. Pi computes locally LE(Si)

7
Lower Envelope
proc. 1
proc. 2
proc. 3
proc. 4
S1
S2
S3
S4
S S1?S2?S3?S4

• Step 1 Each proc. Pi computes locally LE(Si)

proc. 1
proc. 2
proc. 3
proc. 4
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Lower Envelope

• Step 2 Globally sort ?i1..p LE(Si) by
  x-coord. of right endpoint

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Lower Envelope

• Step 2 Globally sort ?i1..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 j1..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)

proc. 1
proc. 2
proc. 3
proc. 4
R1
R2
R3
R4
V1
V2
V3
V4
L1
L2
L3
L4
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Lower Envelope
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