the firefighter problem on the grid joint work with rani hod
TRANSCRIPT
The Firefighter Problem• A complete information solitaire positional game.• Played on a graph • Some vertices are “burning”.• Every turn:– a player protects some vertices– The fire spreads to neighboring vertices.Until the fire spreads no more.
Formally:
• A graph the board. • A set of burning vertices. • , Set of fire-proof vertices.• A function , , the firefighter function. • Game step : Player picks a set of vertices in . .
• If is finite:– For every , how many vertices can we save?
• If is infinite:– For which can we ever stop the fire?
• Algorithms.
Questions:
On grids:
• Several grids to consider. Namely , , triangular and hexagonal.
• For periodic , dimension greater then 2 is not relevant.
Finite :• Suggested by Hartnel (‘95)
as a model for spreading phenomena.• Proven algorithmically hard for trees (FKMR ‘07), but approximable (CVY ‘08).Grids :• Wang and Moeller (‘02): , not enough for • Fogarty (’03): , enough for . • Ng and Raff(‘08): enough for .
History:
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Fire
Fire-Proof
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Demonstration:
Proof
• We show that on if , satisfies t , then a square of fire cannot be stopped.
• When we say time :– after the firefighters protected the vertices– before the fire spreads.
• The main concept – Potential
Potential function
}
}• endangered: on , not fireproof, and
adjacent to a burning point. (if it belongs to two fronts – ½ endangered)
• We define as: #endangered on (again corners count as half)
Observation
Potential
}
}• We say the front is frozen at time if
. Otherwise it is active.• We define to be 1 if is active, 0 otherwise.• We will show that at most one fire front is frozen at any
given time.
Observation
Conventions
• When we omit fronts subscripts – we sum over all fronts. (example: )
• When we add * - we sum over all times (example: )
Dealing with firefighters
• Whenever a fireproof vertex is on we say it becomes efficient.
• We denote by the number of fireproof vertices which became efficient, on front , at time .
• This treats inefficient fireproof vertices as movable.
ObservationA fireproof vertex never contributes to more then 1.
Proposition
Proof: Let us examine the process: At turn we have burning vertices. These must have at least neighbors. Any of them which are fireproof increase and
the rest increase .
Key inequality
Relation to length
Summing Lemma 1 over all fronts we get:
Summing over the length relation:
End of the ProofSuppose for all then for all as
well and thus:Proof:Use induction. , thus by lemma 2
No two fire fronts are frozen – that is and thus