1. On the union of cylinders in 3-space Esther Ezra Duke University

2. Problem statement Input: K = {K 1, …, K n } a collection of n infinite cylinders in R 3 of arbitrary radii. Combinatorial problem What is the combinatorial complexity of the boundary of the union? (vertices/edges/faces of the arrangement A(K) of the cylinders that are not contained in the interior of any cylinder).

3. Input: S = {S 1, …, S n } 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(n d ) (tight!). Union of simply-shaped bodies: A substructure in arrangements Combinatorial complexity.

4. Previous results in 2D: Fat objects 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) n convex  -fat objects. Union complexity: O*(n) [Efrat Sharir. 2000]. n  -curved objects. Union complexity: O( s (n) log n) [Efrat Katz. 1999]. Each of the angles   O(n 1+  ), for any  >0. r r’ r’/r  , and   1. r   diam(C), D  C,  < 1 is a constant. r C D depends linearly on 1/ . DS-sequence of order s on n symbols. ( s is a fixed constant). s (n)  O(n).

5. Previous results in 3D: Fat Objects Congruent cubes n arbitrarily aligned (nearly) congruent cubes. Union complexity: O*(n 2 ) [Pach, Safruti, Sharir 2003]. Simple curved objects n congruent inifnite cylinders. Union complexity: O*(n 2 ) [Agarwalm Sharir 2000]. n  -round objects. Union complexity: O*(n 2 ) [Aronov et al. 2006]. Union complexity is ~ “one order of magnitude” smaller than the arrangement complexity! Each of these bounds is nearly-optimal. r C r   diam(C), D  C,  < 1 is a constant. D

6. Previous results in 3D: Fat Objects Fat tetrahedra n  -fat tetrahedra of arbitrary sizes. Union complexity: O*(n 2 ) [Ezra, Sharir 2007]. Special cases: n arbitrary side-length cubes. Union complexity: O*(n 2 ). n  -fat triangular prisms, having cross sections of arbitrary sizes. Union complexity: O*(n 2 ). Each of these bounds is nearly-optimal. fat  

7. The case of cylinders Input: K = {K 1, …, K n } a collection of n infinite cylinders in R 3 of arbitrary radii. Combinatorial problem What is the combinatorial complexity of the boundary of the union? Trivial bound: O(n 3 ). Conjectured by [agarwal, sharir 2000]: Upper bound: O(n 2 ) (?)

8. Quadratic lower bounds R B The number of vertices of the union is Ω(n 2 ). Each blue intersection line of a consecutive pair of cylinders in B intersects all the red cylinders in R.

9. Extend the notion of “fatness” A cylinder is not fat! A wider definition for fatness: We can sweep K with a plane h whose 2D cross section with each K  K is always fat. h h h is the xy - plane.

10. “fatness” in the context of cylinders Theorem: Let K’  K be a subset of K that captures most of the union vertices. There exists a direction d, such that K  h d  is fat, for any K  K’, where h d  is a plane perpendicular to d. The 2D cross section of a cylinder K on h d  is a fat ellipse. If we sweep h d  along K’, the 2D cross section is always fat. hdhd K hdhd d is the z -axis.

11. Envelopes in 3D Input: F = {F 1, …, F n } a collection of n bivariate functions. The lower envelope E F of F is the pointwise minimum of these functions. That is, E F is the graph of the following function: E F (x) = min {F  F} F(x), for x  R 2.

12. The complexity envelopes [Sharir 1994] The combinatorial complexity of the lower envelope of n simple algebraic surfaces in d -space is O*(n d-1 ). For d=3, the complexity of the lower envelope: O*(n 2 )

13. The sandwich region [ Agarwal etal. 1996, koltun sharir 2003] The combinatorial complexity of the sandwich region enclosed between the lower envelope of n simple algebraic surfaces in 3-space and the upper envelope of another such collection is O*(n 2 ).

14. Main idea: Reduce cylinders to envelopes Decompose space into prism cells . Partition the boundary of the cylinders into canonical strips. Show that in each  most of the union vertices v appear on the sandwich region enclosed between the lower envelope of the lower strips and the upper envelope of the upper strips. Apply the bound O*(n 2 ) of [Agarwal, et al. 1996].

15. (1/r)- cutting: (1/r)- cutting: From cylinders to envelopes K is a collection of n cylinders in R 3. 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: a subdivision of space into (openly disjoint) simplicial subcells , s.t., each cell  meets at most n/r elements of the input. 

16. Constructing (1/r)- cuttings: 1.Project all the cylinders in K onto the xy -plane. Let L be the set of the bounding lines of the projections of K. Each cylinder is projected to a strip.

17. Constructing (1/r)- cuttings: 2.Choose a random sample R of O(r log r) lines of L ( r is a fixed parameter). 3.Form the planar arrangement A(R) of R : Each cell C of A(R) is a convex polygon. Overall complexity: O(r 2 log 2 r). 4.Triangulate each cell C. Number of simplices: O(r 2 log 2 r) C

18. The cutting property Theorem [Clarkson & Shor] [Haussler & Welzl] : Each simplicial cell is crossed by  n/r lines of L, with high probability. 5.Lift all the simplices in the z -direction into vertical prisms . Obtain a collection  of O(r 2 log 2 r) prisms. Each prism subcell  meets only  n/r silhouette - lines of the cylinders in K.

19. The problem decomposition Construct a (1/r)- cutting  for F as above. Fix a prism-cell  of . Classify each cylinder K that meets  as: large – if the radius r of K satisfies: r  w/2, where w is the width of . small - otherwise. H H’ w 

20. The number of small cylinders in a single prism-cell  K l1l1 l2l2 l1l1 l2l2  2r The silhouette-lines of K do not meet . The projection onto the xy -plane. Claim: A small cylinder K within  must have a silhouette-line crossing . w

21. The problem decomposition Each prism-cell  of  meets At most n large cylinders. At most n/r small cylinders. Next stage: Show that large cylinders behave as functions within . Process in recursion all the small cylinders.

22. Classification of the union vertices Each vertex v of the union that appears in  is classified as: LLL – if all three cylinders that are incident to v are large in . LLS – if two of these cylinders are large and one is small in . LSS - if one of these cylinders is large and the other two are small in . SSS – if all these cylinders are small in .

23. Bounding the number of LLL-vertices Theorem: The number of LLL -vertices in  is O*(n 2 ). Proof sketch: Partition the boundary of the cylinders into M canonical strips . A direction  is good for a strip  if: The angle between  and the normal n to H (or H’ ) is small (in terms of M ). Each line tangent to  forms a large angle (in terms of M ) with . H H’ w  Large constant. n

24. Bounding the number of LLL-vertices A direction  is good for a vertex v of the union, incident upon three strips  1,  2,  3, if it is good for each of  1,  2,  3. Lemma: Each vertex v of the union has at least one good direction , taken from a (small) set of overall O(1) directions. 11 22 33 v  n Depends on M.

25. Bounding the number of LLL-vertices Lemma: Let  be a good direction for a vertex v =  1  2  3 of the union. Then: 1.Any line parallel to  intersects  1 at most once. 2.If we enter into the cylinder K 1 bounded by  1 in the  - direction, we exit  before leaving K 1. H H’  n  w v v’

26. Bounding the number of LLL-vertices The strips  behave as functions in the  - direction inside . Each LLL-vertex appears on the sandwich region enclosed between the upper envelope of the  - upper strips and the lower envelope of the  - lower strips. Overall: O*(n 2 ).

27. The case of congruent cylinders Since all cylinders have equal radii, all cylinders K meeting  are either large or small within . Each vertex v of the union that appears in  is either LLL or SSS ( no LLS, LSS).

28. The case of congruent cylinders Construct a recursive (1/r)- cutting  for K. Number of cells in the cutting:  O(r 2 ). Each cell  meets at most 1.n large cylibders of F. 2.  n/r small cylinders of F. Bound LLL- vertices in each  before applying a new recursive step. Bound SSS- vertices by brute-force at the bottom of the recursion. U(n) = O*(n 2 ) + O*(r) U(n/r) Solution: U(n) = O*(n 2 ). Number of (SSS) vertices on the union boundary.

29. Cylinders with arbitrary radii Theorem: The number of LLS- and LSS- vertices in  is O*(n 2 ). Construct a recursive (1/r)- cutting  for F. Bound LLL-, LLS-, LSS- vertices in each  before applying a new recursive step. Bound SSS- vertices by brute-force at the bottom of the recursion. The overall bound is: O*(n 2 ).

30. Thank you

31. Input: S = {S 1, …, S n } a collection of n simply geometric objects in d -space. The arrangement A(S) is the subdivision of space induced by S. The maximal number of vertices/edges/faces of A(S) is:  (n d ) Arrangement of geometric objects Combinatorial complexity. Each object has a constant description complexity

32. Union of “fat” tetrahedra Input: A set of n fat tetrahedra in R 3 of arbitrary sizes. Result: Union complexity:  O(n 2 ) Almost tight. Special case: Union of cubes of arbitrary sizes. fat thin A cube can be decomposed into O(1) fat tetrahedra.