Vertex Identifying Code in Infinite Hexagonal Grid

Transcription

1 Gexin Yu College of William and Mary Joint work with Ari Cukierman

2 Definitions and Motivation Goal: put sensors in a network to detect which machine failed

3 Definitions and Motivation Goal: put sensors in a network to detect which machine failed A solution: put a sensor on each node

4 Definitions and Motivation Goal: put sensors in a network to detect which machine failed A solution: put a sensor on each node

5 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:

6 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

7 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

8 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

9 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.

10 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

11 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

12 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.

13 Examples: codes and non-codes NO!

14 Examples: codes and non-codes NO! NO!

15 Examples: codes and non-codes NO! NO! YES!

16 Examples: codes and non-codes NO! NO! YES! Observation: Every path P n with n 3 has a code.

17 Examples: codes and non-codes NO! NO! YES! Observation: Every path P n with n 3 has a code.

18 Find the right problem u v

19 Find the right problem u v Obstacle: N[u] = N[v], so N[u] D = N[v] D for any D.

20 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].

21 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.

22 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.

23 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.

24 Infinite graphs We consider infinite graphs with following properties:

25 Infinite graphs We consider infinite graphs with following properties: twin-free

26 Infinite graphs We consider infinite graphs with following properties: twin-free locally finite (every vertex has finite degree)

27 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)

28 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)

29 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)

30 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.

31 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).

32 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 )?

33 Density of Square and Triangular Grids Triangular Grid: Karpovsky-Chakrabarty-Levitin (1998) showed that τ = 1 4.

34 Density of Square and Triangular Grids Triangular Grid: Karpovsky-Chakrabarty-Levitin (1998) showed that τ = 1 4. Square Grid: Cohen-Hongala-Lobstein-Zémor (2000) showed that τ 7 20 ; and Ben-Haim-Litsyn (2005) showed that τ 7 20.

35 Code for Hex Grid Upper bound Cohen-Hongala-Lobstein-Zémor (2000) had the following constructions with density 3 7 :

36 Code for Hex Grid Upper bound Cohen-Hongala-Lobstein-Zémor (2000) had the following constructions with density 3 7 :

37 Code for Hex Grid Upper bound Cohen-Hongala-Lobstein-Zémor (2000) had the following constructions with density 3 7 :

38 Code for Hex Grid Upper bound Cohen-Hongala-Lobstein-Zémor (2000) had the following constructions with density 3 7 :

39 Code for Hex Grid Upper bound (Cont.) Jeff Soosiah has the following construction with density 3 7 :

40 Code for Hex Grid Upper bound (Cont.) We find more construction with density 3 7 :

41 Code for Hex Grid Lower bound Karpovsky-Chakrabarty-Levitin (1998) showed that τ 2 5.

42 Code for Hex Grid Lower bound Karpovsky-Chakrabarty-Levitin (1998) showed that τ 2 5. Cohen-Hongala-Lobstein-Zémor (2000) showed that τ

43 Code for Hex Grid Lower bound Karpovsky-Chakrabarty-Levitin (1998) showed that τ 2 5. Cohen-Hongala-Lobstein-Zémor (2000) showed that τ They took a finite portion of the grid, proved a lower bound for the (finite) graph, and then extended that to infinite grid.

44 Code for Hex Grid Lower bound Karpovsky-Chakrabarty-Levitin (1998) showed that τ 2 5. Cohen-Hongala-Lobstein-Zémor (2000) showed that τ They took a finite portion of the grid, proved a lower bound for the (finite) graph, and then extended that to infinite grid. Cranston and Y. (2009) used a cake-sharing idea and proved τ

45 Code for Hex Grid Lower bound Karpovsky-Chakrabarty-Levitin (1998) showed that τ 2 5. Cohen-Hongala-Lobstein-Zémor (2000) showed that τ They took a finite portion of the grid, proved a lower bound for the (finite) graph, and then extended that to infinite grid. Cranston and Y. (2009) used a cake-sharing idea and proved τ We further improve the bounds: τ

46 Proving a Lower Bound (sketch) Forget infinite for now

47 Proving a Lower Bound (sketch) Forget infinite for now Suppose that D is a code for G. Put a cake at each v D and redistribute so that each u V (G) get at least t cake (0 < t < 1).

48 Proving a Lower Bound (sketch) Forget infinite for now Suppose that D is a code for G. Put a cake at each v D and redistribute so that each u V (G) get at least t cake (0 < t < 1). Then D t V (G), or τ(g) = D V (G) t.

49 Proving a Lower Bound (sketch) Forget infinite for now Suppose that D is a code for G. Put a cake at each v D and redistribute so that each u V (G) get at least t cake (0 < t < 1). Then D t V (G), or τ(g) = D V (G) t. The same idea works for infinite graphs.

50 Proving a Lower Bound (sketch) Forget infinite for now Suppose that D is a code for G. Put a cake at each v D and redistribute so that each u V (G) get at least t cake (0 < t < 1). Then D t V (G), or τ(g) = D V (G) t. The same idea works for infinite graphs. Key: how should we share the cake?

51 Proving a Lower Bound (Cont.) To start with, we assume that v D gets enough charge evenly from its neighbors in D: a vertex v D gets 1 k τ from each of its k neighbor in D.

52 Proving a Lower Bound (Cont.) To start with, we assume that v D gets enough charge evenly from its neighbors in D: a vertex v D gets 1 k τ from each of its k neighbor in D. We analyze the remaining charges (in the worst case) of k-clusters (components of k vertices in D):

53 :13:54 1/1 verypoor.pdf (#15) :21:31 1/1 open3.pdf (#16) Proving a Lower Bound (Cont.) To start with, we assume that v D gets enough charge evenly from its neighbors in D: a vertex v D gets 1 k τ from each of its k neighbor in D. We analyze the remaining charges (in the worst case) of k-clusters (components of k vertices in D): Figure: 1-cluster: 1 τ τ = τ = 1 24 Figure: 3-cluster: 3 3τ 3τ τ = 3 7τ = 2 24

54 Proving a Lower Bound (Cont.) The key observation is that in the worst cases for 1-cluster, we can always find a so-called Type-1 paired 3-clusters or Type-2 paired 3-clusters near the 1-cluster, which can be used to supply enough charges for the 1-cluster.

55 Research Problems We now know that for infinite hexagon grid, 5 12 τ 3 7. What s the exact value of τ?

56 Research Problems We now know that for infinite hexagon grid, 5 12 τ 3 7. What s the exact value of τ? How about 3-dimensional grids? How about n-dimension hypercube?

57 Research Problems We now know that for infinite hexagon grid, 5 12 τ 3 7. What s the exact value of τ? How about 3-dimensional grids? How about n-dimension hypercube? It is NP-hard to find the minimum ID-code for a given graph, even a given connected planar graph with maximum degree 4 and girth at least k 3. Can we find any good bounds for such graphs?

58 Research Problems We now know that for infinite hexagon grid, 5 12 τ 3 7. What s the exact value of τ? How about 3-dimensional grids? How about n-dimension hypercube? It is NP-hard to find the minimum ID-code for a given graph, even a given connected planar graph with maximum degree 4 and girth at least k 3. Can we find any good bounds for such graphs? How about 2 + -identifying code (that is, use more powerful sensors)?

59 Questions?