The 5 -graph is a spanner Prosenjit Bose, Pat Morin, Andr e van - PowerPoint PPT Presentation

The θ 5 -graph is a spanner Prosenjit Bose, Pat Morin, Andr´ e van Renssen and Sander Verdonschot Carleton University June 20, 2013 Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 1 / 19

θ -graphs Partition plane into k cones Add edge to ‘closest’ vertex in each cone Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 2 / 19

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) The θ 5 -graph is a spanner June 20, 2013 3 / 19

Previous Work Clarkson 1987 θ -graphs with k > 8 are (1 + ε )-spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with k > 6 have spanning ratio 1 1 − 2 sin( θ/ 2) Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 4 / 19

Previous Work Clarkson 1987 θ -graphs with k > 8 are (1 + ε )-spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with k > 6 have spanning ratio 1 1 − 2 sin( θ/ 2) El Molla 2009 θ 2 and θ 3 are not spanners Bonichon et al. 2010 θ 6 is a planar 2-spanner Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 4 / 19

Previous Work Clarkson 1987 θ -graphs with k > 8 are (1 + ε )-spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with k > 6 have spanning ratio 1 1 − 2 sin( θ/ 2) El Molla 2009 θ 2 and θ 3 are not spanners Bonichon et al. 2010 θ 6 is a planar 2-spanner What about θ 4 and θ 5 ? Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 4 / 19

θ 5 Challenges Asymmetric Steps can get further away w v u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 5 / 19

θ 5 Challenges Asymmetric Steps can get further away w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 5 / 19

Connectedness Induction on size of canonical triangle w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 6 / 19

Connectedness Base case: smallest canonical triangle w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case1: w lies near the bisector w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w v w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w v w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Connectedness Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector w v w u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 7 / 19

Spanning Ratio - Strategy Find a vertex v with A path w � v shorter than a · |△ uw | w u v Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 8 / 19

Spanning Ratio - Strategy Find a vertex v with A path w � v shorter than a · |△ uw | A canonical triangle smaller than b · |△ uw | w u v Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 8 / 19

Spanning Ratio - Strategy Find a vertex v with A path w � v shorter than a · |△ uw | A canonical triangle smaller than b · |△ uw | Then there is a path u � w shorter than c · |△ uw | w u v Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 8 / 19

Spanning Ratio - Cases w 3 4 2 u 1 Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 9 / 19

Spanning Ratio - Case 1 w Case 1 u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 10 / 19

Spanning Ratio - Case 1 w Case 1 w � v ≤ a · |△ uw | u v w Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 10 / 19

Spanning Ratio - Case 1 w Case 1 w � v ≤ a · |△ uw | |△ uv | ≤ b · |△ uw | u v w Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 10 / 19

Spanning Ratio - Case 1 w Case 1 w � v ≤ a · |△ uw | |△ uv | ≤ b · |△ uw | Done! u v w Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 10 / 19

Spanning Ratio - Case 2 & 3 w Works for Case 2 and 3. 3 2 u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 11 / 19

Spanning Ratio - Case 4 w Case 4 v u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4 w Case 4 Our strategy doesn’t v work everywhere u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4 w Case 4 Our strategy doesn’t work everywhere v But it does work in a large part u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4 w Case 4 Our strategy doesn’t work everywhere But it does work in a large part Left with a small region that we can’t solve u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4 w Case 4 Our strategy doesn’t work everywhere But it does work in a large part Left with a small region that we can’t solve u What about v u ? Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4 w Case 4 e b Our strategy doesn’t d work everywhere c But it does work in a large part Left with a small region that we can’t solve u What about v u ? Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 12 / 19

Spanning Ratio - Case 4b Case 4b w � v ≤ a · |△ uw | w v |△ uv | ≤ b · |△ uw | Done! u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 13 / 19

Spanning Ratio - Case 4c w Case 4c Convert to worst-case v u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 14 / 19

Spanning Ratio - Case 4c w Case 4c Convert to worst-case v u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 14 / 19

Spanning Ratio - Case 4c w Case 4c Convert to worst-case w � v ≈ 0 |△ uv | ≈ |△ uw | Done! u ≈ v Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 14 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19

Spanning Ratio - Case 4d w Case 4d Convert to worst-case v w v u u Sander Verdonschot (Carleton University) The θ 5 -graph is a spanner June 20, 2013 15 / 19