# Optimal Grid Exploration by Asynchronous Oblivious Robots

## Full text

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### Optimal Grid Exploration by Asynchronous Oblivious Robots

Franck Petit

INRIA/LIP6-CNRS/UPMC

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## Impossibility Results (1)

Remark

Any terminal configuration of any (probabilistic or deterministic) exploration protocol for a grid of n nodes using k < n oblivious robots contains at least one tower.

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## Impossibility Results (1)

Remark

Any terminal configuration of any (probabilistic or deterministic) exploration protocol for a grid of n nodes using k < n oblivious robots contains at least one tower.

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## Impossibility Results (1)

Remark

Any terminal configuration of any (probabilistic or deterministic) exploration protocol for a grid of n nodes using k < n oblivious robots contains at least one tower.

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## Impossibility Results (1)

Remark

Any terminal configuration of any (probabilistic or deterministic) exploration protocol for a grid of n nodes using k < n oblivious robots contains at least one tower.

Initial configuration

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## Impossibility Results (3)

### ! Impossible for (3,3)-grid if k=3

" Tower of size 3 At most one new node is visited

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## Impossibility Results (3)

### ! Impossible for (3,3)-grid if k=3

" Tower of size 3 At most one new node is visited

" Tower of size 2

• The tower remains idle 4 new nodes are visited

(7 distinct nodes are visited in total)

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## Impossibility Results (3)

### ! Impossible for (3,3)-grid if k=3

" Tower of size 3 At most one new node is visited

" Tower of size 2

• The tower remains idle 4 new nodes are visited

(7 distinct nodes are visited in total)

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## Impossibility Results (3)

### ! Impossible for (3,3)-grid if k=3

" Tower of size 3 At most one new node is visited

" Tower of size 2

• The tower remains idle 4 new nodes are visited

(7 distinct nodes are visited in total)

### Impossible since n=6

• The tower moves Multiplicity of size 3

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(0,0)

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## Algorithm

### "Phase 2: Orientation phase

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9)

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9)

(4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8) (4,9)

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## Algorithm

### "Phase 2: Orientation phase

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9)

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9)

(4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8) (4,9)

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## Algorithm

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9)

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9)

(4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8) (4,9)

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## Algorithm

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9)

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9)

(4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8) (4,9)

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

There is exactly one robot at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Choice

There are at least two robots at a corner of the grid

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Algorithm

### "Phase1: Set-Up

! Configuration of type Undefined All the corners of the grid are free

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## Conclusion

! 5 robots are necessary and sufficient to explore (3,3)-grids

! 3 robots are necessary and sufficient to explore (i,j)-grids

### such that j >3

! 4 robots are necessary and sufficient to explore (2,2)-grids

Updating...

## References

Related subjects :