Substructures in Geometric Arrangements Esther Ezra Advisor: Micha Sharir

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Substructures in Geometric ArrangementsEsther
Ezra Advisor Micha Sharir
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Problem IUnion of simply-shaped bodies

• Input
• S S1, , Sn a collection of n simply-shaped
• bodies in d-space of constant description
  complexity.
• The problem
• What is the maximal number of vertices/edges/faces
  
• that form the boundary of the union of the bodies
  in S ?
• Trivial bound O(nd) (tight!).

Combinatorial complexity.
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Problem II A single cell in an arrangement of
geometric objects

• Input
• S S1, , Sn a collection of n geometric
  objects in
• d-space of constant description complexity.
• A(S) The arrangement induced by S .
• The problem
• What is the maximal number of
• vertices/edges/faces that form the
• boundary of a single cell of A(S) ?
• Trivial bound O(nd) (NOT tight!).

A hole in the complement of the union.
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• Union of Geometric Objects

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Previous results in 2D

• The case of pseudodisks
• A collection of n compact connected sets, s.t.,
• the boundaries of any pair of sets intersect at
  most twice.
• Arise in the case of a convex translating robot R
  
• amid convex pairwise disjoint obstacles.
• Union complexity 6n-12 O(n)
• Kedem et al. 1986.

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Previous results in 2DFat objects
Each of the angles ? ?

• A set of n ?-fat triangles.
• Number of holes in the union O(n) .
• Union complexity O(n loglog n) . Matousek et
  al. 1994
• Fat curved objects (of constant description
  complexity)
• A set of n convex ?-fat objects.
• Union complexity O(n) Efrat Sharir. 2000.
• A set of n ?-curved objects.
• Union complexity O(?s(n) log n) Efrat Katz.
  1999.

r/r ? ? , and ? ?1.
r
O(n1?) , for any ?gt0 .
r
r ? ?? diam(C) , D ? C, ? lt 1 is a constant.
DS-sequence of order s on n symbols. (s is a
fixed constant). ?s(n) ? O(n) .
C
r
D
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Previous results in 3D

• Translational motion planning
• P is a set of k convex polyhedra with n facets
  (overall)
• that arise in the case of a convex translating
  robot R
• amid convex (pairwise disjoint) obstacles.
• Each P ? P is the Minkowski sums of (-R) and an
  obstacle.
• Union complexity O(nk log k) Aronov, Sharir
  1997 .
• The general problem
• P is any set of k convex polyhedra
• with n facets (overall).
• Union Complexity O(k3 nk log k) Aronov,
  Sharir, Tagansky 1997 .

Cannot be applied when R is non-convex.
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Previous results in 3DFat Objects

• Congruent cubes
• A set of n arbitrarily aligned (nearly) congruent
  cubes.
• Union complexity O(n2) Pach, Safruti, Sharir
  2003 .
• Simple curved objects
• A set of n congruent inifnite cylinders.
• Union complexity O(n2) Agarwal Sharir 2000.
• A set of n ?-round objects.
• Union complexity O(n2) Aronov et al. 2006.
• Each of these bounds is nearly-optimal.

r ? ?? diam(C) , D ? C, ? lt 1 is a constant.
C
r
D
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Special case Fat tetrahedra

• Input
• T T1, , Tn a collection of n fat tetrahedra
  in R3.
• Combinatorial problem
• What is the combinatorial complexity of
• the boundary of the union?
• Trivial bound O(n3) .

fat
A cube can be decomposed into O(1) fat
tetrahedra.
thin
It is sufficient to bound the number of
intersection vertices.
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Classification of the intersection vertices

• Outer vertex
• The intersection of an edge of a
• tetrahedron with a facet of
• another tetrahedron.
• Overall O(n2) .
• Inner vertex
• The intersection of three facets of
• three distinct tetrahedra.
• Overall O(n3) .
• Reduce the problem to
• How many inner vertices appear
• on the boundary of the union?

v
u
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?-fat dihedral/trihedral wedges
?-fat dihedral wedge W W is the intersection of
two halfspaces.The dihedral angle ? ? .
?
The dihedral angle.
?-fat trihedral wedge W W is the intersection
of three halfspaces.The solid angle ? ? . W is
(?,?)-substantially fat if the sum of the angles
of its three facets ? ?, and ? gt 4?/3 .
?
The solid angle.
?
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?-fat tetrahedron

• A tetrahedron T is ?-fat if
• Each pair of its facets define an ?-fat dihedral
  wedge.
• Each triple of its facets define an ?-fat
  trihedral wedge.

?
?
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The union of fat wedges

• Pach, Safruti, Sharir 2003
• The combinatorial complexity of the union
• of n ?-fat dihedral wedges O(n2) .

The bound depends linearly on 1/? .
Thin dihedral wedges (almost half-planes) create
a grid with O(n3) vertices.
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The union of fat wedgesA quadratic lower bound
construction
R
W
B
Merge the wedges in R and in B so that they form
a 2D-grid on W.
The right facet of W shaves the edges of the
wedges in R and in B.
The number of vertices of the union is O(n2).
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The union of fat trihedral wedges An almost
quadratic upper bound

• Pach, Safruti, Sharir 2003
• The union of n (?,?)-substantially fat trihedral
  wedges O(n2).
• The combinatorial complexity of the union of n
  congruent arbitrarily aligned cubes is O(n2).

Apply a reduction from cubes to wedges.
Each cube intersects only O(1) cells of the grid.
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More general union problems

• Union of arbitrary side-length cubes
• Use the grid reduction? Does not work!
• Need to apply a more elaborate partition
  technique of space, so as to reduce cubes to
  wedges.
• Union of fat tetrahedra
• The grid reduction does not work even when the
  tetrahedra are congruent!

?
Each tetrahedron induces at least one
non-substantially fat trihedral wedge.
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Our results
Almost tight.

• The combinatorial complexity of the union of n
  ?-fat tetrahedra, of arbitrary sizes, is O(n2) .
• The combinatorial complexity of the union of
  polyhedra, of arbitrary sizes, that can be
  decomposed or covered by n fat tetrahedra, is
  O(n2) .
• The combinatorial complexity of the union of n
  ?-fat trihedral wedges is O(n2) .
• The combinatorial complexity of the union of n
  ?-fat triangular prisms, having cross sections
  of arbitrary sizes is O(n2) .

Special case Union of cubes of arbitrary sizes.
Follows easily by our analysis.
Follows easily by our analysis.
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From tetrahedra to wedges

• T is a collection of n ?- fat tetrahedra in R3.
• Use (1/r)-cutting in order to partition space.
• (1/r)-cutting A useful divide conquer
  paradigm.
• Fix a parameter 1 ? r ? n .
• (1/r)-cutting is a subdivision of
• space into openly disjoint simplicial
• subcells ?, s.t., each cell ? meets at
• most n/r tetrahedra facets of T .

?
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Constructing (1/r)-cuttings

• Choose a random sample R of O(r log r) of the
  planes containing tetrahedra facets in T (r is a
  fixed parameter).
• Form the arrangement A(R) of REach cell C of
  A(R) is a convex polyhedron.Overall complexity
  O(r3 log3r).
• Triangulate each cell C, and obtain a collection
  ? of O(r3 log3r) simplices.
• Theorem
• Clarkson Shor Haussler Welzl
• Each cell ? of ? is crossed by ? n/r
• tetrahedra facets of T , with high probability.

Use the hierarchical decomposition of Dobkin
Kirkpatrick
C
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Triangulating a cell The hierarchical
decomposition of Dobkin Kirkpatrick

• Hierarchical representation of a convex
  polyhedron C (An informal description)
• Construct a (large) independent set V1 of
  vertices of CC1 .
• Remove the vertices in V1 from C1Fill each hole
  with simplicial subcells, and peel them off C1.
• Obtain a new polyhedron C2 ? C1 .
• Apply steps 13 recursively.Bottom of recursion
  The new polyhedron Ck is a simplex.

C1C
C3
C2
k O(log r) number of levels in the recursion.
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The DK-hierarchical decomposition

• Claim
• There exists a hierarchical representation for C
  that satisfies
• k O(log r) .
• .
• Each line l that stabs C, crosses only O(log r)
  of its simplices.

l
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Properties of the overall decomposition ?

• The DK-decomposition properties imply
• The overall number of cells ? of ? is O(r3 log3r)
  .
• Each tetrahedron edge crosses at most O(r log2r)
  simplices ? of ? .
• Another crucial property to follow.

There are O(r log r) planes in R.
Due to the stabbing line property.
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The problem decomposition

• Construct an edge-sensitive (1/r)-cutting ?
  for T as above.
• Fix a cell ? of ?.
• Some of the tetrahedra in T may become
  half-spaces/?-fat dihedral wedges inside ? .
• Classify each (inner) vertex v of the union that
  appears in ? as
• Good - if all three tetrahedra that are incident
  to v are half-spaces/?-fat dihedral wedges in ?.
  
• Bad - otherwise.

Apply the nearly-quadratic bound of Pach,
Safruti, Sharir 2003.
At least one of these tetrahedra has three (or
more) facets that meet ? .
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Bounding the number of bad vertices
? meets all four facets of T.

• Fix a tetrahedron T ? T
• A cell ? is called bad for T,
• if it meets at least three facets of T.
• Goal
• For each fixed tetrahedron T ? T,
• the number of bad cells is small.
• Lemma
• There are only O(r) bad cells for T.

?
Immediate when the cells meet an edge of T
(stabbing line property).
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Goal Bad cells are scarce
The trivial bound is O(r2).
The 2D cross-sections of all cells intersecting F
is a 2D arrangement of lines. Overall number of
cells O(r2) .
F
Our bound improves the trivial bound by roughly
an order of magnitude.
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The frequency of bad cells
F1
F2
Two facets of T can meet O(r2) cells.
The construction is impossible for three facets
of T !
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The overall analysis

• Construct a recursive edge-sensitive
  (1/r)-cutting ? for T .
• Most of the vertices of the union become good at
  some recursive step.
• Bound the number of bad vertices by brute
  forceat the bottom of the recursion.
• The overall bound is O(n2) .

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SpecializationUnion of fat triangles

• Input
• T T1, , Tn a collection of n ?-fat triangles
  in the plane.
• Construct a (1/r)-cutting ? for T.
• Number of cells in the cutting ?O(r2) .
• Each cell ? meets at most n/r triangles edges of
  T .
• Fix a cell ? of ?.
• Classify each triangle T ? T as
• W-triangle in ? , if ? meets at most two edges
  of T.
• T-triangle in ? , otherwise.

T behaves as a half-plane or a wedge inside ?.
T behaves as a triangle inside ?.
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The classification of the vertices of the union

• Each vertex v of the union that appears in ? is
  classified as
• WW if the two triangles that are incident to v
  are W-tetrahedra in ?.
• WT if one of these triangles is a W-triangle
  and the other is a T-trianglein ?.
• TT - if both of these triangles are T-triangles
  in ?.

?
w
v
u
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Bounding the number of intersection vertices of
the union

• WW vertices
• Easy reduce to the case of
• ?-fat wedges.
• WT, TT vertices
• More involved.
• In particularWT verticesObtain a
  nearly-linear bound.

Apply the linear bound of Efrat, Rote, Sharir
1994.
Use a non-trivial variant of the construction.
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T-triangles are scarce
Only a single triangle ? can meet all three edges
of T.
?
T
On average, each cell ? meets at most ?n/r2
triangle edges of T-triangles.
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The recursive scheme

• Construct a (1/r)-cutting ? for T.
• Number of cells in the cutting ?O(r2) .
• Each cell ? meets at most
• n/r W-triangles edges of T .
• ? n/r2 T-triangles edges of T.
• Bound WW and WT vertices before applying a new
  recursive step.
• Bound TT vertices by brute-force at the bottom of
  the recursion.
• U(n) O(n) O(r2)?U(n/r2)
• Solution U(n) O(n) .

Number of vertices on the union boundary.
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• A single cell in geometric arrangements

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A single cell in 2D arrangements

• Arrangement of lines
• Complexity of arrangement O(n2) .
• Complexity of a single cell O(n) .
• Arrangement of segments/triangles
• Complexity of arrangement O(n2) .
• Complexity of a single cell O(n) ?
• Actual bound ?(n?(n)) Guibas et al. 1989

C
The inverse Ackermann function.
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Previous results
Some constant.

• 2D arrangements
• n Jordan arcs, each pair of arcs intersect in at
  most s point.
• Single cell complexity O(?s2(n))? O(n) Guibas
  et al. 1989.
• k convex polygons with n edges in total.
• Single cell complexity T(n?(k)) Aronov Sharir
  1997.
• 3D arrangements
• A set T of n triangles.
• The complexity of all non-convex cells in A(T)
  O(n7/3 log n)
• Aronov Sharir 1990.
• A set S of n low degree algebraic surface
  patches.
• Single cell complexity O(n2) Halperin Sharir
  1995.

Segments s1.
Many components
Curved simply-shaped regions
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Previous results

• d-dimensional arrangements
• A set of n (d-1)-simplices.
• Single cell complexity O(nd-1log n) Aronov
  Sharir 1994.
• A set S of n low degree algebraic surface
  patches.
• Generalizing the bound of Halperin Sharir
  1995.
• Single cell complexity O(nd-1) Basu 2003.
• All bounds are almost tight in the worst case.
• The case of k convex polyhedra with n facets in
  R3
• Use Aronov Sharir 1994 O(n2 log n) .
• This bound does not depend on k.
• Conjecture Aronov, Sharir, Tagansky 1997
• The actual upper bound is close to O(nk) .

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Lower bounds for the unbounded cell in 3D
O(nk) vertices
O(k2) vertices
Can be modified to O(nk?(k)) vertices
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Our result

• The combinatorial complexity of a single cell of
  A(? ) is O(nk1?) ,? ? gt 0 .
• We use a variant of the technique of Halperin
  Sharir 1995 .
• We present a deterministic algorithm that
  constructs a single cell in O(nk1? log3 n) time,
  ? ? gt 0 .

The bound depends on the number k of polyhedra
Crucial The input regions are of constant
description complexity
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Classification of the intersection vertices

• Outer vertex The intersection of an edge of a
  polyhedron with a facet of another
  polyhedron.Overall number O(nk) .
• Inner vertexThe intersection of three facets of
  three distinct polyhedra.Overall number O(nk2)
  .
• Reduce the problem to
• How many inner vertices appear
• on the boundary of a single cell?

u
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Analysis Exposed convex chains
Classify each vertex v by How long can we freely
go from v when alternating out-of/into the
unbounded cell.
Not meeting any polyhedra
?
1 step
?
After the removal of P 4 steps
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Analysis Continue

• We trace this way Exposed convex chains.
• Number of steps length of the chain
• V(j)(? ) the number of inner vertices of the
  unbounded cell of A(? ) with ? j steps.

5 steps
?
?
V(0)(? ) bounds the overall number of inner
vertices of the unbounded cell.
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The overall complexity of exposed chains

• Exposed chains of length ? 4
• Use recurrence V(j)(? ) ? V(j1)(? )
• Exposed chains of length 4 or 5
• Lemma
• The number of vertices on exposed chains of
  length ? 5 is O(nk) .
• The number of vertices on exposed closed chains
  (of length 4) is O(nk) .

Multiply by O(k?).
?
This is the only interesting case.
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Solving the recurrence

• V(j)(? ) O(nk1?) ,? ? gt 0, 0 ? j ? 4
• The combinatorial complexity of a single cell of
  A(? ) is O(nk1?) ,? ? gt 0 .

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• Thanks!

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Exposed chains of length ? 5
?
?
?M ? P_3
?M ? P
?
?
MF_1 ? P_2
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Motivation Translational motion planning
The workspace.

• Input
• Robot R , a set A A1, , Ak of k disjoint
  obstacles.
• The robot and the obstacles are
• (not necessarily convex) polyhedra in 3-space.
• The free space
• The set of all legal placements of R,
• while translating R in 3-space.

collision .
R does not intersect any of the obstacles in A.
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The configuration space
reference point.

• The robot R is mapped to a point.
• Each obstacle Ai is mapped to the set
• Pi (x,y,z) R(x,y,z) ? Ai ? ?
  Ai?(-R(0,0,0))
• A point p in Pi corresponds to an illegal
  placement of R and vice versa.

The forbidden placements of R .
The Minkowski sum.
The expanded obstacle.
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The free space

• The free space is

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Union of convex polyhedra

• Input
• P P1, , Pk a collection of k convex
  polyhedra in 3-space with n facets in total (k
  n).
• Combinatorial problem
• What is the maximal number of vertices/edges/faces
  that form the boundary of the union of the
  polyhedra in P ?Trivial bound O(n3) (tight!).

It is sufficient to bound the number of
intersection vertices.
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Union of convex polyhedra

• General goal
• Obtaining (natural) special cases where the union
  has subcubic/nearly-quadratic complexity.
• Computational problem
• An algorithm that constructs the union?
• Not efficient when the complexity of the whole
  union is high (cubic).

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RestrictionA single component of the free space

• A single component of
• The subset of all placements reachable from a
  given initial free placement of R via a
  collision-free motion.

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Restatement A single component in the
complement of the union

• Input
• ? P1, , Pk a collection of k convex
  polyhedra in 3-space with n facets in total.
• The problem
• What is the maximal number of vertices/edges/face
  s that formthe boundary of a single component
  of ?

Minkowski sum of a convex obstacle with a convex
part of R.
It is sufficient to bound the number of
intersection vertice.
A single cell of A(? ).
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Single (bounded) cell in 2D