Vertex Identifying Code in Infinite Hexagonal Grid Gexin Yu - PowerPoint PPT Presentation

Vertex Identifying Code in Infinite Hexagonal Grid Gexin Yu gyu@wm.edu College of William and Mary Joint work with Ari Cukierman Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit ◮ a sensor at v can see v and its neighbors Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit ◮ a sensor at v can see v and its neighbors ◮ Problem: Find a subset D ⊂ V ( G ) s.t. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit ◮ a sensor at v can see v and its neighbors ◮ Problem: Find a subset D ⊂ V ( G ) s.t. ◮ for all v ∈ V ( G ), N [ v ] ∩ D � = ∅ , and Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit ◮ a sensor at v can see v and its neighbors ◮ Problem: Find a subset D ⊂ V ( G ) s.t. ◮ for all v ∈ V ( G ), N [ v ] ∩ D � = ∅ , and ◮ ∀ u , v ∈ V ( G ) if u � = v then N [ u ] ∩ D � = N [ v ] ∩ D Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Definitions and Motivation ◮ Goal: put sensors in a network to detect which machine failed ◮ A solution: put a sensor on each node (Too expensive) ◮ Assumptions: ◮ machines fail one at a time ◮ each sensor only sends one bit ◮ a sensor at v can see v and its neighbors ◮ Problem: Find a subset D ⊂ V ( G ) s.t. ◮ for all v ∈ V ( G ), N [ v ] ∩ D � = ∅ , and ◮ ∀ u , v ∈ V ( G ) if u � = v then N [ u ] ∩ D � = N [ v ] ∩ D ◮ Definition: We call such a set D a (vertex identifying) code. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Examples: codes and non-codes 1 3 2 NO! Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Examples: codes and non-codes 1 3 2 NO! 1 3 2 NO! Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Examples: codes and non-codes 1 3 2 NO! 1 3 2 NO! 1 3 2 YES! Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Examples: codes and non-codes 1 3 2 NO! 1 3 2 NO! 1 3 2 YES! ◮ Observation: Every path P n with n ≥ 3 has a code. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Examples: codes and non-codes 1 3 2 NO! 1 3 2 NO! 1 3 2 YES! ◮ Observation: Every path P n with n ≥ 3 has a code. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v ◮ Obstacle: N [ u ] = N [ v ], so N [ u ] ∩ D = N [ v ] ∩ D for any D . Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v ◮ Obstacle: N [ u ] = N [ v ], so N [ u ] ∩ D = N [ v ] ∩ D for any D . ◮ Fact: G has a code iff for all u � = v we have N [ u ] � = N [ v ]. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v ◮ Obstacle: N [ u ] = N [ v ], so N [ u ] ∩ D = N [ v ] ∩ D for any D . ◮ Fact: G has a code iff for all u � = v we have N [ u ] � = N [ v ]. ◮ Definition: We call such a graph twin-free. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v ◮ Obstacle: N [ u ] = N [ v ], so N [ u ] ∩ D = N [ v ] ∩ D for any D . ◮ Fact: G has a code iff for all u � = v we have N [ u ] � = N [ v ]. ◮ Definition: We call such a graph twin-free. ◮ New problem: If G is twin-free, find a smallest code. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Find the right problem u v ◮ Obstacle: N [ u ] = N [ v ], so N [ u ] ∩ D = N [ v ] ∩ D for any D . ◮ Fact: G has a code iff for all u � = v we have N [ u ] � = N [ v ]. ◮ Definition: We call such a graph twin-free. ◮ New problem: If G is twin-free, find a smallest code. ◮ We are most interested in infinite grids. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) ◮ Ex. V ( G Z ) = Z and uv ∈ E ( G Z ) iff | u − v | = 1 (infinite path) Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) ◮ Ex. V ( G Z ) = Z and uv ∈ E ( G Z ) iff | u − v | = 1 (infinite path) Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) ◮ Ex. V ( G Z ) = Z and uv ∈ E ( G Z ) iff | u − v | = 1 (infinite path) ◮ Definition: Rather than the smallest size code, we want the lowest density (fraction) code. Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) ◮ Ex. V ( G Z ) = Z and uv ∈ E ( G Z ) iff | u − v | = 1 (infinite path) ◮ Definition: Rather than the smallest size code, we want the lowest density (fraction) code. ◮ We call this the density of G , τ ( G ). Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Infinite graphs ◮ We consider infinite graphs with following properties: ◮ twin-free ◮ locally finite (every vertex has finite degree) ◮ vertex transitive (graph looks the same from each vertex) ◮ Ex. V ( G Z ) = Z and uv ∈ E ( G Z ) iff | u − v | = 1 (infinite path) ◮ Definition: Rather than the smallest size code, we want the lowest density (fraction) code. ◮ We call this the density of G , τ ( G ). ◮ Question: what is τ ( G Z )? Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

Density of Square and Triangular Grids ◮ Triangular Grid: Karpovsky-Chakrabarty-Levitin (1998) showed that τ = 1 4 . Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid