Counting and Representing Intersections Among Triangles in R3

1
Counting and Representing Intersections Among
Triangles in R3

• Esther Ezra and Micha Sharir

2
The Problem
C

• Given a collection of triangles in R3
• Count efficiently their intersections.
• Represent in a compact form the overall set of
  intersections, without reporting them explicitly.

A
B
Running time and storage independent of the
number of intersections.
A?B?C ?(n3) intersections
3
Motivation

• 3-Dimensional slope selectionSelect the k-th
  highest vertex in an arrangement of n triangles
  in R3.(Reduces via parametric search toHow
  many vertices lie above a given plane?)
• Draw a random element out of the entire
  intersection set, without constructing this set
  explicitly.(Used in a randomized solution of 1.)

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Known results in R2

• Problem Running
  time Ref
• Counting intersections O(n4/3)
  Agarwal 91
• (segments)
• Counting intersections O(n3/2)
  Agarwal et al. 93
• (circular arcs)

Use O(n?) to denote O(n?e), for any e gt 0.
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Known results in R3

• Problem Running
  time Ref
• Reporting all ? pairs of O(n4/5 ?4/5)
  de Berg et al. 99
• intersecting triangles
• Counting all intersecting O(n8/5)
  Agarwal et al. 02
• pairs (convex polyhedrons)
• No previous work regarding counting intersecting
  triples among n objects in R3 !

Subquadratic when ? ltlt n3/2
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Our results

• Algorithm that
• Counts all intersecting triples in O(n2) time.
• Constructs a compact representation of the set of
  all intersections using O(n2) time and storage.
• The problem is 3SUM-hard.
• Extensions planar simple shaped objects in R3
  O(n2)

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Preliminary stage(Good for small ?)

• Report all ? pairs of intersecting triangles
  O(n8/5 ?) (using segment intersection queries
  in R3)
• Count on each triangle t the number of
  intersections O(? n1/3) (using planar
  segment-intersection counting.)
• Overall running time O(n8/5 ? n1/3)
• subquadratic when ? ltlt n5/3

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Ingredients of the algorithmCurve-sensitive
cutting

• Input T t1, , tn
• For any r?n, there exists a (1/r)-cutting ? for T
  of size O(r3), s.t. the number of crossings
  between the edges of the triangles in T and the
  cells of ? is O(nr) .
• Use Dobkin-Kirkpatrick hierarchical decomposition
• (Cannot be extended to other objects in R3)
• In general, use a curve-sensitive cutting
  Koltun-Sharir 03

O(nr2) in a standard cutting.
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The recursive decomposition

• Construct a curve-sensitive (1/r)-cutting ? ,
  and count the intersecting triples in each cell
  of ? separately.
• Definition

t ? T
t ? T
? ? ?
? ? ?
t is short in ?
t is long in ?
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Classification of intersecting triples

• LLL All three triangles are long in ?.
• LLS Two triangles are long, and one is short.
• LSS One triangle is long, and two are short.
• SSS All three triangles are short in ?.

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General idea

• Use recursion
• Compute LLL before recursing.
• Compute LLS, LSS, SSS at the bottom of recursion.
• Use the curve-sensitivity of the cutting to
  control the overall performance.

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Counting intersections of type LLL (Agarwal-91)
ql1
ql1
l2
pl2
ql2
pl2
Interleaving
ql2
pl1
t
pl1
l1
Count all intersecting pairs on each clipped
triangle t (within ? ) in O(NLlog NL) time.
Number of clipped long triangles in ? .
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Overall running time

• O(NL2 log NL)
• Important!
• Each LLL intersection has to be counted only
  once.
• Count only intersections that involve at
  least one new long triangle.
• The running time becomes
• O(Ns (NsNL) log (NsNL) )

A triangle that is short in the parent cell of ?,
but long in ? .
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Counting intersections of type LLS

• Use a similar algorithm as in the LLL case.
• Count all intersecting pairs on each clipped
  short triangle t (within ? ) in O(NLlog NL) time.
• Overall running time O(NSNL log NL)

Number of clipped short triangles in ? .
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Counting intersections of type LSS and SSS
(Agarwal-91)

• LSS On each short clipped triangles t, count
  intersecting pairs of long-short segments.
• Passing to the dual plane O(NS2 NL log NL).
• Overall O(NS3 NSNL log NL).
• SSS Brute-force algorithm O(NS3).
• Overall running time O(NS3 NS NL log NL).

t
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The overall recursive algorithm

• Initially ?0 R3 NL0, NSn.
• If NS ? NL½ (Bottom of recursion)Count all LLS,
  LSS and SSS.
• Otherwise, partition ?0 into smaller subcells ?,
  using (1/r)-sensitive cutting.
• Count all new LLL intersections within each
  subcell ? of ?0 .

These intersections are counted only once.
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The overall recursive algorithm

• Continue to solve each subproblem in ?
  recursively - with some care.
• Problem
• There are O(r3) new subproblems, each containing
  NL / r long triangles and NS / r short ones.
• With no extra care, the running time may become
  cubic!

The goal of the recursion is only to count
efficiently LSS and SSS
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Recursion extra care

• BUT there are O(NS r) crossings in total.
• The number of short triangles in most of the
  cells in ? is O(NS / r2) .
• There are only o(r3) cells that are crossed by
  gtgt NS / r2 short triangles.

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Overall running time

• O(NS (NSNL)) .
• Initially, NS n, NL 0 ,
• The number of intersecting triples in T can be
  counted in time minO(n2) , O(n8/5 ? n1/3) .

The preliminary stage
The overall recursive algorithm
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Compact representation

• T t1, , tn . All intersecting triples among
  T can be represented by a 3-uniform hypergraph H
  (T,E) ,
• E ti, tj, tk ti, tj, tk ? T , ti,? tj, ?
  tk ??
• A compact representation for H is a collection H
  Hi (Vi,Ei)I of subhypergraphs of H, s.t.,
• Vi Ai?Bi?Ci , Ei Ai?Bi?Ci (Hi is a complete
  tripartite 3-uniform hypergraph.)
• E ?i Ei .
• Ei ? Ej ? , for i?j .
• ?i AiBiCi is the overall storage of this
  representation.

The edges are now defined implicitly.
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Compact representation Example
C
H (A?B?C, A?B?C)
A
B
ABC is the overall storage of this
representation.
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Algorithm

• Use a similar recursive mechanism as in the
  counting problem.
• Modify the four simple subroutines (for the
  planar case), so they construct a compact
  representation in similar running times.
• Modify the preliminary stage, so that it produces
  the compact representation.

Uses the four modified subroutines
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Compact representation subroutines

• Overall size (and running time)
• LLL O(Ns (NsNL) log2(NsNL) )
• LLS O(Ns NL log2 NL )
• LSS O(Ns3 NsNL)
• SSS O(Ns3)

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Overall storage and running time

• Recursive algorithm O(NS (NSNL) ) .
• Initially, NS n, NL 0 ,
• The overall size of the compact
  representationminO(n2) , O(? n1/3) .
• Running time
• minO(n2) , O(n8/5 ? n1/3) .

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Counting all intersecting triples is a 3SUM-hard
problem

• 3SUM Given 3 sets of integers A, B, and C, of
  total size n, are there a ? A, b ? B, and c ? C
  with abc ?

hc z c
hb y b
H zxy
(a,b,c)?H iff number of intersections above/below
H lt ABC
ha x a
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Extensions Planar simply shaped objects

• S s1, , sn .
• s ? S is bounded by a closed curve of constant
  description complexity.
• Algorithm Use a similar recursive mechanism as
  in the case of triangles. Modify the four
  simple subroutines (in the triangle case), so
  they count all intersecting triples in similar
  running times.

Use curve-sensitive cutting for S.
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Further research

• Extend the problem to non-planar (simple)
  surfaces in R3.
• Higher dimensionsCount intersecting k-tuples (2
  ? k lt d) among n simplices in Rd. ( kd
  O(nd-1). Optimal? ).

LLL
Problem Endpoints do NOT interleave
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Representing LLL (LLS) intersections
pi
lj
qj
pj
t
qi
li

• Represent compactly all pairs ti, tj, i?j, that
  satisfy pi lt pj lt qi lt qj

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T2(v)
T1
Use a 2-level tree-like structure.
The first level structure T1 stores all points pi
in sorted order.
v
pj
qj
Overall size of the output graphs O(NSNL
log2NL)
T2(v) stores all the points qi whose matching
points are stored at v.
Query with pj
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Representing LSS intersections
Idea Construct a (1/r)-cutting ? for St and
locate the points of Lt in the cells of ?
. Overall size of the representation of all
LSS intersectionsO(NS3 d NSNL1d) , for any
d gt 0 .
t
The set of the double-wedges (dual to short
segments within t).
The set of the points (dual to long segments
within t).
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Counting all intersecting triples is a 3SUM-hard
problem

• 3SUM Given 3 sets of integers A, B, and C, of
  total size n, are there a ? A, b ? B, and c ? C
  with abc ?

hc z c
hb y b
H zxy
ha x a