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Counting and Representing Intersections Among Triangles in R 3

Counting and Representing Intersections Among Triangles in R 3. Esther Ezra and Micha Sharir. The Problem. C. Given a collection of triangles in R 3 Count efficiently their intersections. Represent in a compact form the overall set of intersections, without reporting them explicitly.

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Counting and Representing Intersections Among Triangles in R 3

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  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 to:How 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.)

  4. 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+ε), for any ε > 0.

  5. 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  << n3/2

  6. 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)

  7. 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  << n5/3

  8. Ingredients of the algorithm:Curve-sensitive cutting Input:T = {t1, …, tn} For any rn, there exists a (1/r)-cuttingΞ for T of size O*(r3), s.t. the number of crossings between the edges of the triangles in T and the cells of Ξis 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.

  9. The recursive decomposition Construct a “curve-sensitive” (1/r)-cutting Ξ , and count the intersecting triples in each cell of Ξ separately. Definition: t T t  T ΔΞ ΔΞ t is short in Δ t is long in Δ

  10. Classification of intersecting triples • LLL: All three triangles are long in Δ. • 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 Δ.

  11. 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.

  12. 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 Δ ) in O(NLlog NL) time. Number of clipped long triangles in Δ.

  13. 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(Ns+NL)log (Ns+NL)) A triangle that is short in the parent cell of Δ, but long in Δ .

  14. 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 Δ ) in O(NLlog NL) time. Overall running time: O(NSNL log NL) Number of clipped short triangles in Δ.

  15. 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

  16. The overall recursive algorithm • Initially 0 = R3: NL=0, NS=n. • If NS NL½(Bottom of recursion)Count all LLS, LSS and SSS. • Otherwise, partition 0into smaller subcells , using (1/r)-sensitive cutting. • Count all new LLL intersections within each subcell  of 0. These intersections are counted only once.

  17. 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

  18. Recursion – extra care BUT there are O*(NS r) crossingsin total. The number of short triangles in most of the cells in ΞisO*(NS / r2) . There are only o(r3) cells that are crossed by >> NS / r2 short triangles.

  19. Overall running time O*(NS (NS+NL)) . Initially, NS = n, NL = 0 , The number of intersecting triples in T can be counted in time min{O*(n2) , O*(n8/5+ n1/3)} . The preliminary stage The overall recursive algorithm

  20. Compact representation T = {t1, …, tn} . All intersecting triples among T can be represented bya 3-uniform hypergraph H = (T,E) , E = { {ti, tj, tk} | ti, tj, tk  T , ti, tj,  tk  } A compact representation for H is a collectionH = {Hi = (Vi,Ei)}I of subhypergraphs of H,s.t., • Vi= AiBiCi , Ei = AiBiCi(Hiis a complete tripartite 3-uniform hypergraph.) • E = i Ei . • Ei  Ej =  , for ij . i |Ai|+|Bi|+|Ci|is the overall storage of this representation. The edges are now defined implicitly.

  21. Compact representation: Example C H = (ABC, ABC) A B |A|+|B|+|C| is the overall storage of this representation.

  22. 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

  23. Compact representation subroutines Overall size (and running time) LLL: O(Ns(Ns+NL)log2(Ns+NL)) LLS: O(NsNLlog2 NL) LSS: O*(Ns3 + NsNL) SSS: O(Ns3)

  24. Overall storage and running time Recursive algorithm: O*(NS (NS+NL)) . Initially, NS = n, NL = 0 , The overall size of the compact representation:min{O*(n2) , O*( n1/3)} . Running time: min{O*(n2) , O*(n8/5 +  n1/3)} .

  25. 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 a+b=c ? hc: z = c hb: y = b H: z=x+y (a,b,c)Hiff number of intersections above/belowH < |A||B||C| ha: x = a

  26. 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.

  27. Further research Extend the problem to non-planar (simple) surfaces in R3. Higher dimensions:Count intersecting k-tuples (2  k < d) among n simplices in Rd. ( k=d : O*(nd-1). Optimal? ). LLL Problem: Endpoints do NOT interleave

  28. Representing LLL (LLS) intersections Represent compactly all pairs {ti, tj}, ij, that satisfypi <pj < qi < qj pi lj qj pj t qi li

  29. T2(v) T1 Use a 2-level tree-like structure. The first level structure T1 stores all points piin sorted order. v pj qj Overall size of the output graphs: O(NSNL log2NL) T2(v) stores all the points qiwhose matching points are stored at v. Query with pj

  30. 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 intersections:O(NS3+ δ + NSNL1+δ) , for anyδ > 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).

  31. x+y-1/2  z x+y+1/2 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 a+b=c ? hc: z = c hb: y = b H: z=x+y ha: x = a

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