New and Improved Spanning Ratios for Yao Graphs
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1 New and Improved Spanning Ratios for Yao Graphs Luis Barba 12 Prosenjit Bose 1 Mirela Damian 3 Rolf Fagerberg 4 Wah Loon Keng 5 Joseph O Rourke 6 André van Renssen 1 Perouz Taslakian 7 Sander Verdonschot 1 Ge Xia 5 1 Carleton University 2 Université Libre de Bruxelles 3 Villanova University 4 University of Southern Denmark 5 Lafayette College 6 Smith College 7 American University of Armenia 30th Annual Symposium on Computational Geometry Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
2 Yao-graphs Partition plane into k cones Add edge to closest vertex in each cone Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
3 Geometric Spanners Graphs with short detours between vertices For every u and w, there is a path with length t uw Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
4 Previous Work k > 8 (1 + ε) (Althöfer et al., 1993) k > 8 k > 6 1 cos θ sin θ sin θ 2 (Bose et al., 2004) (Bose et al., 2010) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
5 Previous Work k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k 5 and odd 1/(1 2 sin(3θ/8)) Y 6?? Y 5?? Y 4?? Y 3?? Y 2?? Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
6 Previous Work k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k 5 and odd 1/(1 2 sin(3θ/8)) Y 6?? Y 5?? Y 4?? Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
7 Previous Work k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k 5 and odd 1/(1 2 sin(3θ/8)) Y 6?? Y 5?? Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
8 Previous Work k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k 5 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y 5?? Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
9 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y 5?? Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
10 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
11 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
12 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
13 Odd Yao graphs Basic lemma (used for Yao graphs with k > 6) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
14 Odd Yao graphs Basic lemma (used for Yao graphs with k > 6) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
15 Odd Yao graphs Basic lemma (used for Yao graphs with k > 6) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
16 Odd Yao graphs Even number of cones: Increasing one angle also increases the other Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
17 Odd Yao graphs Even number of cones: Increasing one angle also increases the other Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
18 Odd Yao graphs Odd number of cones: Increasing one angle decreases the other Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
19 Odd Yao graphs Odd number of cones: Worst case occurs for 3θ/4 Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
20 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
21 Our Results First constant upper bound for Y 5 Y k is a constant spanner iff k > 3 Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
22 Our Results First constant upper bound for Y 5 Y k is a constant spanner iff k > 3 Can we do better for Y 5? Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
23 Our Results First constant upper bound for Y 5 Y k is a constant spanner iff k > 3 Can we do better for Y 5? Always apply basic lemma Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
24 Our Results First constant upper bound for Y 5 Y k is a constant spanner iff k > 3 Can we do better for Y 5? Always Strategically apply basic lemma Handle remaining cases Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
25 Improvements for Y 5 What if we only apply the lemma for small angles? Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
26 Improvements for Y 5 If the edge is very short, we re still okay Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
27 Improvements for Y 5 Case 1: Both edges are long and they cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
28 Improvements for Y 5 Case 1: Both edges are long and they cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
29 Improvements for Y 5 Case 1: Both edges are long and they cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
30 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
31 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
32 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
33 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
34 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
35 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
36 Improvements for Y 5 Case 2: Both edges are long and do not cross Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
37 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
38 Spanning ratio of Y 6 Same general idea: Strategically apply basic lemma Handle remaining cases Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
39 Spanning ratio of Y 6 Split cone into center and margins Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
40 Spanning ratio of Y 6 Destination in center Apply basic lemma Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
41 Spanning ratio of Y 6 Closest in center Apply basic lemma Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
42 Spanning ratio of Y 6 Closest in center Apply basic lemma Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
43 Spanning ratio of Y 6 Look from the other side Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
44 Spanning ratio of Y 6 Closest in center Apply (variation of) basic lemma Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
45 Spanning ratio of Y 6 If closest lies in both margins, Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
46 Spanning ratio of Y 6 If closest lies in both margins, consider the other closest Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
47 Spanning ratio of Y 6 If closest lies in both margins, consider the other closest Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
48 Spanning ratio of Y 6 If closest lies in both margins, consider the other closest Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
49 Spanning ratio of Y 6 A few more cases... Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
50 Our Results k > 6 1/(1 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 2 sin(3θ/8)) Y (Damian & Raudonis, 2012) Y Y (Bose et al., 2012) Y 3 (El Molla, 2009) Y 2 (El Molla, 2009) Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
51 Future work Improved lower bounds Competitive routing Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
52 Future work Improved lower bounds Competitive routing Questions? Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG / 15
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