Output-Sensitive Construction of the Union of Triangles

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Output-Sensitive Construction of the Union of
Triangles

• Esther Ezra and Micha Sharir

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Definition Union

• T ?1,, ?n - collection of n triangles in the
  plane.
• The union U?i ?I is defined as all regions in
  the plane that are covered by the triangles of T.

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Constructing the Union Motivation

• Robot motion planning
• Construct the forbidden portion of the
  configuration space.
• Ray shooting amid semi-algebraic sets in R3
• Construct the union of 4-dimensional regions.

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Computing the union

• Constructing the arrangement of the triangles
  too slow! O(n2)
• Output-sensitive algorithm
• (in terms of the number of edges on the
  boundary)?
• unlikely to exist!
• 3SUM HOLE-IN-UNION
• The best known solutions to problems from the
  3SUM-hard family require ?(n2) time in the worst
  case.

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Union of triangles in R2Known results

• Special cases Union size
  Ref
• Fat triangles O(n loglog n)
  MPSSW-94
• Pseudodiscs O(n)
  KLPS-86
• General triangles
• Algorithm Running time
  Ref
• RIC O(n log n T1)
  AH-01
• DC O(n2)
  EHS-02

?i Vi / i Vi the set of vertices in depth i
Performs well in practice
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Output-sensitive union construction

• Given a collection T ?1,, ?n of n triangles
  in the plane, such that there exists a subset S
  ?T (unknown to us),
• of ? ltlt n triangles with
• ?? ?S ? ?? ?T ? ,
• construct efficiently the union of the triangles
  in T.

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Output Sensitivity Example I
? 2
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Output Sensitivity Example II
Only 4 triangles determine the boundary of the
union
? 6
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Our Result

• We show that when there exists a subset S ?T of
  ? ltlt n triangles, such that ?? ?S ? ?? ?T ? ,
  the union can be constructed in O(n4/3 n?)
  time.
• Subquadratic when ? o(n)

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A Set Cover in a Set System

• Use a variant of the method of Bronnimann and
  Goodrich for finding a set cover in a set system
  of finite VC-dimension
• Our set system (V, T)

The set of vertices of A(T) that lie inside the
union
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Hitting Set in a Set System

• Dual set system (T, V)
• V Tv v ? V
• Tv consists of all the triangles in T that
  contain v in their interior.
• A hitting set a subset H ? T , s.t. H has a non
  empty intersection with every subset member of
  V.
• A hitting set H for (T,V) is a set cover for
  (V,T).
• ? H ? T

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Finding a Hitting Set general
scheme(Bronnimann and Goodrich)

1. Assign weights to the triangles in T.Initially,
   all weights are 1.
2. Net finder Construct a (1 / 2?)-net N for
   (T,V) .( guess of a hitting set for (T, V)
   with N O(? log ?) ).
3. Verifier If there exists a subset Tv that is
   not hit by N (there exists a vertex v?V outside
   ?N) , double the weights of the triangles in
   Tv. Goto 2.Else N is a hitting set for
   (T,V)

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Performance of the algorithm

• A hitting set of size O(? log ?) is found after
  O(? log (n/?)) iterations.

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Ideal Setting

• Problem The algorithm requires the knowledge of
  V.
• (But we cannot afford to compute V explicitly).
• Solution Variant of the algorithm Use a random
  subset R? V instead.
• R r ?(t log n)
• Lemma A subset H that covers R, covers most of
  the vertices of V , with high probability
• The number of remaining uncovered vertices ? ?/t

H and R are independent!
? V
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The residual cost of the B-G algorithm

• How many vertices at positive-depth (in V) are
  constructed by the B-G algorithm?
• O(?2 log2? ?/t ), with high probability.

All remaining uncovered vertices
The O(? log ?) triangles in H (over pessimistic)
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Implementation

• Net finder
• Drawing a sample R
• Verifier
• The actual construction of the union

O(? log (n/?)) times
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Simple Implementation
If ? O(n4/3), construct the entire
arrangement, and report the union

• Sampling R O(r n2 / ?) pairs of triangle edges
  O(r) real vertices.
• Net-finder O(n)
• Verifier (brute-force) O(r? n)

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The actual construction of the union

• Divide the process into two stages
• Construct the union of all the triangles in H.
• Insert all the remaining triangles (covering ?
  ?/t positive-depth vertices).

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U is the union of the triangles in H.
t1, t2, t3 are the remaining triangles.
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The actual construction of the union

• Divide the set of the remaining triangles into
  n/(? log ?) subsets, each of size O(? log ? ) .
• Construct ? H in O(?2 log2?) time.
• For each such subset S, construct A(S).
• Report all intersections between S and ? H in an
  output-sensitive manner.
• O(n? ? / t)

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Overall running time

• O(n4/3 ? r n2 / ? r?2 n? ? / t)
• r O(t log n)
• Choose t k1/2 / ?
• O(n4/3 n?)

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Extensions Simple objects in R2

• Implementation generic and simple.
• The algorithm can be easily extended to other
  simple geometric objects in R2 O(n4/3 n?)

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Extensions Simple objects in R3
If ? O(n2), construct the entire arrangement,
and report the union

• Sampling R O(r n3 / ?)
• Net-finder O(n)
• Verifier (brute-force) O(r? n)
• Actual construction of the union construct the
  union, in an output-sensitive manner, on every
  facet of the input objects separately. O(n2 ?
  ? / t).
• Overall running time O(n2 ?).

Similar routines as before
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Extensions Simple objects in Rd

• Apply the (d-1)-approach on each facet of the
  input objects, using induction on d.
• Overall running time O(nd-1 ?).

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Earlier variant

• We used the Disjoint-Cover (DC) algorithmEzra,
  Halperin, Sharir 2002
• Implementation
• more complex.
• heavily relies on the geometry of the input
  objects Less generic.
• Less efficient (subquadratic only for a smaller
  range of ?).

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Further research

• Simpler efficient alternative approaches?
• RIC fails!