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Output-Sensitive Construction of the Union of Triangles

Output-Sensitive Construction of the Union of Triangles. Esther Ezra and Micha Sharir. 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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Output-Sensitive Construction of the Union of Triangles

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  1. Output-Sensitive Construction of the Union of Triangles Esther Ezra and Micha Sharir

  2. Definition: Union T ={1,…, n} - collection of n triangles in the plane. The union U=iIis defined as all regions in the plane that are covered by the triangles of T.

  3. 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.

  4. 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.

  5. 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 + Θ1) AH-01 DC O(n2) EHS-02 i|Vi| / i Vi – the set of vertices in depthi Performs well in practice

  6. “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  << n triangles with S = T, construct efficiently the union of the triangles in T.

  7. Output Sensitivity: Example I  = 2

  8. Output Sensitivity: Example II Only 4 triangles determine the boundary of the union  = 6

  9. Our Result We show that when there exists a subset S T of  << n triangles, such that S = T, the union can be constructed in O*(n4/3 + n) time. Subquadratic when = o*(n)

  10. 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

  11. 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

  12. Finding a Hitting Set – general scheme(Bronnimann and Goodrich) • Assign weights to the triangles in T.Initially, all weights are 1. • Net finder:Construct a (1 / 2)-netN for (T,V*) .( guess of a hitting set for (T, V*) with |N| = O( log ) ). • 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*)

  13. Performance of the algorithm A hitting set of size O( log ) is found after O( log (n/)) iterations.

  14. Ideal Setting Problem: The algorithm requires the knowledge of V+. (But we cannot afford to computeV+explicitly). Solution: Variant of the algorithmUse a random subsetR 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+|

  15. 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)

  16. Implementation • Net finder • Drawing a sample R • Verifier • The actual construction of the union O( log (n/))times

  17. Simple Implementation Sampling R: O*(r n2 / ) pairs of triangle edges O(r) real vertices. Net-finder: O(n) Verifier: (brute-force) O*(r + n) If = O(n4/3), construct the entire arrangement, and report the union

  18. 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-depthvertices).

  19. U is the union of the triangles in H. t1, t2, t3 are the remaining triangles.

  20. 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)

  21. Overall running time O*(n4/3 + r n2 /  + r2 + n +  / t) r = Ω(t log n) Choose t = k1/2 /  O*(n4/3 + n)

  22. 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)

  23. Extensions: Simple objects inR3 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

  24. Extensions: Simple objects inRd Apply the (d-1)-approach on each facet of the input objects, using induction on d. Overall running time: O*(nd-1).

  25. Earlier variant We used the Disjoint-Cover (DC) algorithm[Ezra, 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 ).

  26. Further research Simpler efficient alternative approaches? RIC : fails!

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