1. Open problems for  -skeletons in R 2 and R 3. Miroslaw Kowaluk University of Warsaw EuroGIGA meeting, Lugano 2011

2.  = 0,8 For a given set P containing n points in R m and the parameter  we define  -skeleton as a graph (P, E), in which xy  E iff any point from P\{x,y} doesn’t belong to an area R(x,y,  ), where 2. For 0 <  < 1, R(x,y,  ) is the intersection of two spheres with the radius d(x,y)/2 , which boundaries contain the both points x i y. 1. For  = 0, R(x,y,  ) is the segment xy. yx

3.  = 1 3. For 1   < , R(x,y,  ) is a intersection of two spheres with radius  d(x,y)/2 and centered in points (1-  /2)x+(  /2)y and (  /2)x+(1-  /2)y resp. 4. For  = , R(x,y,  ) is an unbounded strip between two lines containing x and y resp. And perpendicular to the segment xy.  =  yx  = 2 lune-based definition

4. 4. For  = , R(x,y,  ) is the whole plane. yx  =  =  sphere-based definition 3. For 1   < , R(x,y,  ) is the union of two spheres, whose boundaries contain the both points x i y.  = 1,25 yx

5. Properties of  -skeletons. The  -skeleton for a set of points P and  = 1 is called Gabriel Graph (GG(P)) (Gabriel,Sokal 69), and for  = 2 is called Relative Neighbourhood Graph (RNG(P)) (Toussaint 80). Theorem (Kirkpatrick,Radke 85).  GG(P)MST(P)  RNG(P)  DT(P) x y z x y z w

6. Theorem. For 0   < 1 the  –skeleton can have  (n 2 ) edges. Theorem (Hurtado, Liotta, Meijer, 2003) For 0   < 1 the  –skeleton can be computed in the optimal time O(n 2 ). Theorem (Jaromczyk, Kowaluk, Yao) For 1    2 the  –skeleton in R 2 in L p (for 1 < p <  ) can be computed from DT(P) in linear time.

7. Open problem. Is there algorithm computing  –skeleton for 2 <    that requires o(n 2 ) time ?

8. The  –spectrum defines for each pair of points in P the maximum value of  for which area R(x,y,  ) is empty. Theorem (Hurtado, Liotta, Meijer, 2003) The  –spectrum for all pairs of points in P can be found in time O(n 2 ).

9. Open problem. Is there an algorithm finding  –spectrum only for edges which belong to Delaunay triangulation of the given set of points P that requires o(n 2 ) time ?

10. Let L(u,v) be a length of the shortest path between vertices u and v in a connected graph G in R 2, and D(u,v) be a distance between u i v. Spanning ratio S of the graph G is defined as follows S = max (u,v)  G L(u,v)/D(u,v). Theorem (Keil,Gutwin 92). Spanning ratio of the DT(P), where |P| = n, is O(1). Theorem (Bose,Devroye,Evans,Kirkpatrick 02). Spanning ratio of the RNG(P), where |P| = n, is  (n). Spanning ratio GG(P), where |P| = n, is  (n 1/2 ). L(u,v) D(u,v) vu

11. Open problem. Find a better estimation of the spanning ratio for the  – skeleton where 1    2.

12. Theorem (Agarwal, Matoušek 92). RNG(P) in R 3 in general position of points in L p (for 1 < p <  ) has O(n 4/3 ) edges and can be found in expected time O(n 3/2+  ). Theorem (Chazelle,Edelsbrunner,Guibas,Hershberger,Seidel,Sharir 90). GG(P) in R 3 can have  (n 2 ) edges.

13. Open problem. What is the graph complexity for the  –skeleton in R 3 where 1    2 ?

14. Thank you for your attention