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

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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

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 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:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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 Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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

SLIDE 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 Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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 Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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 Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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

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Examples: codes and non-codes

1 2 3

NO!

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Examples: codes and non-codes

1 2 3

NO!

1 2 3

NO!

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Examples: codes and non-codes

1 2 3

NO!

1 2 3

NO!

1 2 3

YES!

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Examples: codes and non-codes

1 2 3

NO!

1 2 3

NO!

1 2 3

YES!

◮ Observation: Every path Pn with n ≥ 3 has a code.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Examples: codes and non-codes

1 2 3

NO!

1 2 3

NO!

1 2 3

YES!

◮ Observation: Every path Pn with n ≥ 3 has a code.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Find the right problem

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

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Find the right problem

v u

◮ 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

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Find the right problem

v u

◮ 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

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Find the right problem

v u

◮ 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

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Find the right problem

v u

◮ 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

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Find the right problem

v u

◮ 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

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Infinite graphs

◮ We consider infinite graphs with following properties:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Infinite graphs

◮ We consider infinite graphs with following properties:

◮ twin-free Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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

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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

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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 (GZ) = Z and uv ∈ E(GZ) iff |u − v| = 1 (infinite path)

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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 (GZ) = Z and uv ∈ E(GZ) iff |u − v| = 1 (infinite path)

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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 (GZ) = Z and uv ∈ E(GZ) 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

SLIDE 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 (GZ) = Z and uv ∈ E(GZ) 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

SLIDE 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 (GZ) = Z and uv ∈ E(GZ) 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(GZ)?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 33

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

SLIDE 34

Density of Square and Triangular Grids

◮ Triangular Grid: Karpovsky-Chakrabarty-Levitin (1998)

showed that τ = 1

4. ◮ Square Grid: Cohen-Hongala-Lobstein-Z´

emor (2000) showed that τ ≤ 7

20; and Ben-Haim-Litsyn (2005) showed that

τ ≥ 7

20.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 35

Code for Hex Grid—Upper bound

◮ Cohen-Hongala-Lobstein-Z´

emor (2000) had the following constructions with density 3

7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 36

Code for Hex Grid—Upper bound

◮ Cohen-Hongala-Lobstein-Z´

emor (2000) had the following constructions with density 3

7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 37

Code for Hex Grid—Upper bound

◮ Cohen-Hongala-Lobstein-Z´

emor (2000) had the following constructions with density 3

7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 38

Code for Hex Grid—Upper bound

◮ Cohen-Hongala-Lobstein-Z´

emor (2000) had the following constructions with density 3

7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Code for Hex Grid—Upper bound (Cont.)

◮ Jeff Soosiah has the following construction with density 3 7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Code for Hex Grid—Upper bound (Cont.)

◮ We find more construction with density 3 7:

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Code for Hex Grid—Lower bound

◮ Karpovsky-Chakrabarty-Levitin (1998) showed that τ ≥ 2 5.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Code for Hex Grid—Lower bound

◮ Karpovsky-Chakrabarty-Levitin (1998) showed that τ ≥ 2 5. ◮ Cohen-Hongala-Lobstein-Z´

emor (2000) showed that τ ≥ 16 39 ≈ 0.4102...

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 43

Code for Hex Grid—Lower bound

◮ Karpovsky-Chakrabarty-Levitin (1998) showed that τ ≥ 2 5. ◮ Cohen-Hongala-Lobstein-Z´

emor (2000) showed that τ ≥ 16 39 ≈ 0.4102...

◮ They took a finite portion of the grid, proved a lower bound

for the (finite) graph, and then extended that to infinite grid.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 44

Code for Hex Grid—Lower bound

◮ Karpovsky-Chakrabarty-Levitin (1998) showed that τ ≥ 2 5. ◮ Cohen-Hongala-Lobstein-Z´

emor (2000) showed that τ ≥ 16 39 ≈ 0.4102...

◮ 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 ≥ 12 29 ≈ 0.41379...

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 45

Code for Hex Grid—Lower bound

◮ Karpovsky-Chakrabarty-Levitin (1998) showed that τ ≥ 2 5. ◮ Cohen-Hongala-Lobstein-Z´

emor (2000) showed that τ ≥ 16 39 ≈ 0.4102...

◮ 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 ≥ 12 29 ≈ 0.41379...

◮ We further improve the bounds:

τ ≥ 5 12 ≈ 0.4166667

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 46

Proving a Lower Bound (sketch)

◮ Forget infinite for now

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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).

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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):

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 53

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):

1/1 verypoor.pdf (#15) 2011-05-10 22:13:54

Figure: 1-cluster: 1 − τ − 3 · 1

2τ = 1 − 5 2τ = − 1 24

1/1

pen3.pdf (#16)

2011-05-10 22:21:31

Figure: 3-cluster: 3−3τ −3τ −2· 1

2τ = 3−7τ = 2 24

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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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.

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

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Research Problems

◮ We now know that for infinite hexagon grid, 5 12 ≤ τ ≤ 3 7.

What’s the exact value of τ?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 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)?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid

SLIDE 59

Questions?

Gexin Yu gyu@wm.edu Vertex Identifying Code in Infinite Hexagonal Grid