IOEfficient Construction of Constrained Delaunay Triangulations

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I/O-Efficient Construction of Constrained
Delaunay Triangulations

• Pankaj K. Agarwal, Lars Arge, and Ke Yi
• Duke University

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DT vs. Constrained DT
Delaunay Triangulation
Constrained Delaunay Triangulation As much
Delaunay as possible while keeping the
constrained edges
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Constrained DT Definition

• A set of points P
• A set of non-intersecting obstacle segments S
  with endpoints in P
• CDT(P,S) consists of all segments in S and all
  edges connecting pairs of points p,q of P such
  that
• (1) p and q can see each other, and
• (2) there exists a circle passing through p and
  q, that contains only points of P that cannot see
  both p and q

p
q
A valid edge in DT
p
q
A valid edge in CDT
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Constrained DT Definition (2)

• A set of points P
• A set of non-intersecting obstacle segments S
  with endpoints in P
• Or equivalently, CDT(P,S) consists of all
  triangles pqr such that
• (1) any two of p, q, and r can see each other or
  are connected by a segment of S
• (2) their circumcircle contains only points that
  cannot see the interior of pqr.

r
p
q
A valid triangle in DT
r
p
q
A valid triangle in CDT
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I/O Model
CPU
N elements Scan O(N/B) I/Os (linear) Sort
O(N/B logMN) I/Os
Main memory M
Disk size B
Disk
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Previous Results

• Internal memory algorithms
• DT many algorithms Aurenhammer and Klein,
  practical
• CDT Chew 1987, Wang and Schubert 1987, runs
  in O(N log N) time but impractical Use
  incremental construction in practice
• External memory algorithms
• DT Crauser et al. 2001 runs in expected O(N/B
  logMN) I/Os
• CDT not known

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Our Results

• The first external memory algorithm for building
  the constrained Delaunay triangulation
• Runs in expected O(N/B logMN) I/Os when segments
  lt M
• Implementation

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Algorithm

• Take a sample R of size M including all segment
  endpoints
• Build CDT(R,S)
• Construct the conflict list of each edge of
  CDT(R,S)

The conflict list of an edge e consists of all
points in two adjacent circumcircles and visible
from e
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Algorithm

• Build CDT of each conflict list (recurse if still
  too large to fit in memory)
• Assemble results together (discussed later)

Linear I/Os (in the total size of conflict
lists) if ignore recursion
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Analysis

• Lemma Expected size of each conflict list
  O(N/M)
• Following Clarkson and Shor framework
• Difference Only a constant fraction of the
  samples are random
• O(logMN) levels of recursion
• Total O(N/B logMN)

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Kernels
Kernel
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Assemble Results Together
Kernel

• Lemma 1 A triangle is a valid triangle of
  CDT(P,S) iff its circumcenter lies inside the
  kernel

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Assemble Results Together
Kernel

• Lemma 2 The circumcenter of each triangle of
  CDT(P,S) lies inside exactly one kernel

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Implementation Experiments

• Using TPIE
• Compared with Shewchuks Triangle program
• Incremental construction
• Delete all edges intersected by s
• Retriangulate two polygons on both sides of s
• Pre-sort points and segments along some space
  filling curve (Hilbert curve)

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Experiment Uniform Data
10 million points Varying segments Ave segment
length 0.003 Memory 128M
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Experiment Segment Length
10 million points 10,000 segments Varying
segment length Memory 128M
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Experiments Real World Data
points LIDAR points from the Neuse River Basin
Segments TIGER roaddata Largest dataset 0.5
billion points, 1 million segment
(input file 80GB)
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Experiments Real World Data
7.5 hours
Couldnt run internalalgorithm
16M 28M 44M 59M 91M 116M 163M 257M 503M
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Open Questions

• External memory algorithms for arbitrary
  segments
• Analysis of the (internal memory) randomized
  incremental algorithm
• O(n log2 n)
• O(n log n)

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Thank you!
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Extending the Plane
Secondary sheet
Primary sheet
Seidel, 1989
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Extending the Plane
Secondary sheets
Primary sheet
Seidel, 1989
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Traveling Rule
x
Secondary sheets
z
y
Primary sheet
Seidel, 1989
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Extended Voronoi Diagram
Secondary sheets
Primary sheet
p
Seidel, 1989
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Extended Voronoi Diagram
Secondary sheets
Primary sheet
CDT(P,S) is dual to EVD(P,S)
Seidel, 1989
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Kernels
Secondary sheets
Primary sheet