Parallel Computational Geometry visibility CGM complexity: - - PDF document

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Parallel Computational Geometry visibility CGM complexity: - - PDF document

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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

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SLIDE 1

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Frank Dehne www.dehne.net

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

Frank Dehne www.dehne.net

Lower Envelope

  • Given: A set S of n non-intersecting line

segments

S

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SLIDE 2

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Frank Dehne www.dehne.net

Lower Envelope

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

segments

S LE(S)

Frank Dehne www.dehne.net

Lower Envelope

  • Goal: O(1) rounds
  • Problem: Each processor “sees” only O(n/p)

line segments

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SLIDE 3

3

Frank Dehne www.dehne.net

Lower Envelope

S1

  • proc. 1

S2

  • proc. 2

S3

  • proc. 3

S4

  • proc. 4

S = S1∪S2∪S3∪S4

Frank Dehne www.dehne.net

Lower Envelope

  • Step 1: Each proc. Pi computes locally

LE(Si)

S1

  • proc. 1

S2

  • proc. 2

S3

  • proc. 3

S4

  • proc. 4

S = S1∪S2∪S3∪S4

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SLIDE 4

4

Frank Dehne www.dehne.net

Lower Envelope

  • Step 1: Each proc. Pi computes locally

LE(Si)

S1

  • proc. 1

S2

  • proc. 2

S3

  • proc. 3

S4

  • proc. 4

S = S1∪S2∪S3∪S4

  • 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

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SLIDE 5

5

Frank Dehne www.dehne.net

Lower Envelope

  • proc. 1
  • proc. 2
  • proc. 3
  • proc. 4
  • Step 2: Globally sort [∪i=1..p LE(Si)] by

x-coord. of right endpoint

V1 V2 V3 V4

Frank Dehne www.dehne.net

Lower Envelope

  • proc. 1
  • proc. 2
  • proc. 3
  • proc. 4
  • Step 3: Each Pi determines Li and broadcasts

Li to all other Pj (one h-relation)

L1 L2 L3 L4 V1 V2 V3 V4

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SLIDE 6

6

Frank Dehne www.dehne.net

Lower Envelope

  • proc. 1
  • proc. 2
  • proc. 3
  • proc. 4
  • Step 4: Each Pi scans LE(Si) and sends the

segment intersecting Lj to Pj for j=1..p (one h- relation)

L1 L2 L3 L4 V1 V2 V3 V4 R1 R2 R3 R4

Frank Dehne www.dehne.net

Lower Envelope

  • proc. 1
  • proc. 2
  • proc. 3
  • proc. 4
  • Step 5: Each Pi computes LE(Vi ∪ Ri)

L1 L2 L3 L4 V1 V2 V3 V4 R1 R2 R3 R4

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SLIDE 7

7

Frank Dehne www.dehne.net

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