Applications of Arrangements in Computational Geometry

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Applications of Arrangements in Computational
Geometry

• Supervisor Prof. Micha Sharir
• Esther Ezra

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Problems
Triangles in R2 Fat objects in R2 ,R3

• Union of geometric objects
• Counting and representing intersections among
  triangles in R3
• Minimum Hausdorff distance under translation
• Medial axis and union of balls

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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

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, such that
• ?? ?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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Computing the union

• Constructing the arrangement of the triangles
  too slow! O(n2)
• Output-sensitive algorithm
• (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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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 subquadratic
  time.
• We use the Disjoint-Cover (DC) algorithmEzra,
  Halperin, Sharir 2002

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The DC algorithm General idea

• Suppose we are given the set V of all vertices
  of the arrangement A(T), that are contained in
  the interior of the union.
• Run a greedy algorithm that inserts the triangles
  into the union by their weights.The weight of a
  triangle ? ? TThe number of (uncovered)
  vertices inside the triangle.

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Disjoint-Cover (DC) algorithm

• Suppose we have inserted (?1,,?j).
• For each remaining triangle ?, we temporarily set
  S? to be the set of all vertices of V in the
  interior of ? that are not covered by ? iltj ?i .
• Set W(?) S? for each remaining triangle.
• Set ?j1 to be the triangle with the maximum
  weight.
• Update the union to include ?j1 .
• Proceed until all triangles have been chosen.

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Preprocessing stage Example step1
Temporary initial weights W(?1)7 W(?2)5 W(?3)
3 W(?4)0
?3
?1
?4
?2
DC Chooses ?1
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Preprocessing stage Examplestep2
Temporary current weights W(?2)3 W(?3)1 w(?4)0
?3
?1
?4
?2
DC chooses ?2
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Preprocessing stage Examplestep3
Temporary current weights W(?3)1 w(?4)0
?3
?1
?4
?2
DC chooses ?3
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Preprocessing stage Examplestep4 (final)
Temporary current weights w(?4)0
Final weights W(?1)7 W(?2)3 W(?3)1 w(?4)0
?3
?1
?4
?2
(?1, ?2, ?3, ?4)
DC chooses ?4
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The residual cost of the DC algorithm

• How many vertices at positive-depth (in V) are
  constructed by the DC algorithm?

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Set Cover in Hypergraphs

• H(V,E) V V E T
• A set cover a subset of E whose union covers V.
• The DC algorithm is a greedy algorithm for
  finding a set cover.

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Finding a set cover

• There are ? triangles that cover V.
• The (greedy) DC algorithm will find a cover of
  size O(? logn).
• The residual cost of the DC algorithm is
  O(?2log2n) .

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The approximate DC algorithm

• We replace V with a (small) random subset R? V
  .
• R r ?(?t log n)
• We resample R at every iteration of the DC
  algorithm.

t is a parameter of the quality of the sampling
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Approximate weights

• Approximate weight of ? number of uncovered
  vertices of R inside ?.
• The algorithm proceeds as above, using
  approximate weights to determine the insertion
  order.

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Analysis General idea

• The approximate weights of the heavy triangles
  (W(?) gt , ? V) reflect their actual
  weights.
• The approximate DC algorithm chooses at the j-th
  iteration a triangle whose real weight does not
  differ significantly from the largest real weight
  at this step.

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The residual cost of the approximate DC algorithm

• Theorem
• Let T ?1,,?n be a given collection of n
• triangles in the plane, such that ? of them
• determine the union of the triangles in T.
• Let ? V, and r ?(?t log n) .
• Then the residual cost of the approximate DC
  algorithm is
• O(?2 log2t ?/t) , with high probability.

All remaining uncovered vertices
The first q O(? log t) heavy triangles.
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Implementation

• Sampling R compact representation
• Computing the insertion orderrange-searching
  machinery
• The actual construction

q times
q O(? log t) is the number of the first
triangles leaving lt ?/t uncovered vertices.
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The actual construction of the union

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

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U is the union of the first q triangles.
t1, t2, t3 are the remaining triangles.
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Running time

• Careful implementation of all steps (using
  range-searching, and other techniques) yields an
  efficient (subquadratic) algorithm.

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New and improved solution!

• 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)

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Hitting Set in Hypergraphs

• 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 of H ? T , s.t. H has a
  nonempty intersection with every subset in V.
• A hitting set H for (T,V) is a set cover for
  (V,T),
• U H U T

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Finding a Hitting Set (Bronnimann and Goodrich)

• Assign weights to the triangles in T.Initially,
  all weights are 1.
• Net finder Construct a (1 / 2?)-net N for
  (T,V) .( guess of a hitting set for (T, V) ).
• Verifier If the there exits a subset Tv that is
  not hit by 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.
• Variant of the algorithm Consider 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

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

• Sampling R ?O(t n2 / ?)
• Net-finder ? O(n)
• Verifier ? O(n)
• The actual construction similar procedure as
  before ? O(n? ? / t)
• Overall running time ? O(n4/3 n?)

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Extensions

• Implementation generic and simple.
• The algorithm can be easily extended to other
  simple geometric objects in R2 ? O(n4/3
  n?)Simplices in R3 (details)
• ? O(n2 ?).

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Current research

• Simpler efficient alternative approaches?
• RIC fails!
• Extensions to simple three-dimensional objects.

Seems possible after the simplification of the
sampling procedure
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Union of Fat objects (R2)known results

• Input Union size Ref
  
• Fat triangles O(n loglog n)
  MPSSW-94
• Pseudodiscs O(n)
  KLPS-86
• a-fat (convex) O(n1e)
  ES-97
• (a,ß)-covered (simple) O(?s2(n) log2n loglog n)
  Efrat-99

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Union of Fat objects (R3)known results

• Input Union size Ref
  
• k convex polyhedra (R3) O(k3 nk log k)
  AST-97
• with total n faces.
• The case of Minkowski sums O(nk log k)
  AS-97
• Minkowski sums of n O(n2e)
  AS-00
• p.d. triangles with a ball (R3)
• n nearly congruent cubes O(n2e)
  PSS-03
• (arbitrarily aligned in R3)
• n ?-round objects (R3) O(n2e)
  AEKS-03

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Proposed research

• Study the complexity of the union of fat
  polyhedra.
• Simplest cases
• n arbitrary cubes in R3. O(n2e) ?
• n regular simplices in R3 (near congruent or of
  arbitrary size) . O(n2e) ?
• Higher dimensions?

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Counting and representing intersections among
triangles in R3
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Matching Problems

• A, B ? Rd
• The one-directional Hausdorff distance from A to
  B
• h(A,B) max a?A min b? B ?(a,b)
• The bidirectional Hausdorff distance from A to B
• H(A,B) max(h(A,B), h(B,A))

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Minimum Hausdorff distance under translation

• We wish to compute
• D(A,B) min t H(A, B?t)
• t is a vector in Rd, and B?t bt b ? B .
• Problem
• Compute D(A,B) , where A, B are finite point sets
  in Rd .

A
A
B
B? t
D(A,B)
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Known results Bidirectional

• Input running time Ref
• Points in R2 (L2) O(n3 log n)
  HKS-93
• Points in R2 (L?) O(n2 log2n)
  CK-92
• Points on a line O(n log n)
  Rote-91
• approximation running time Ref
• O(1) O(n)
  Alt-97
• (use reference point)
• PTAS (1e) O(n/ e2 log n) Schirra-88

Alternatively, use parametric search
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Known results Lower bound

• (algorithm dependent)
• Points in R2 (L? ,L2) O(n3)
  Rucklidge-96

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Known results One directional Points in R2
(L2)

• Cardoze-Schulman-98
• Integer values O(bn log n logO(1)s)
• Approximation of (1e) to the threshold problem
• Given d, identify all translations t, s.t.
• h(A, B?t ) lt d(1e)
• Running time O(b n/eO(d) log (n/ e) logO(1)(s/
  ed))

Error probability of 1/nb
The diameter of the data set
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Proposed research
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Medial axis of a union of balls

• Definition
• The locus of all points inside the union that
  have more than one nearest boundary point

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Motivation
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?-shape
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Known results

• Attali Monatanvert (R2 and R3)
• Let U be a union of balls, and let V denote its
  vertex set. Then the medial axis of U consists of
• The singular faces of the ?-shape of U.
• The subset of the Voronoi diagram Vor(V) whose
  closest point on ?U is a vertex of V.

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Known results

• Amenta Kulluri (Rd)
• Let U be a union of balls, and let V denote its
  vertex set. Then the medial axis of U consists of
• The singular faces of the ?-shape of U.
• The subset of the Voronoi diagram Vor(V) which
  intersects the regular components of the ?-shape
  of U

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Proposed research

• Establish a tight bound on the size of the medial
  axis M of a union of balls in R3
• n2 ? M ? n4

Best known lower bound
The number of vertices on U is O(n2), and the
size of their Voronoi diagram is at most quadratic
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Union of Balls in R3known results

• Input Union size Ref
• Unit balls, all ?(n2)
  BD-99
• containing the origin

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Our result

• A construction that yields ?(n2) vertices on the
  boundary of union, when the centers are located
  on two perpendicular axes.
• Naïve equi-distant placement
• of centers does not work!

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Every circle (an intersection of two consecutive
balls) contributes ?(n) vertices to the union.