Geometric Firefighting Rolf Klein University of Bonn HMI, June 19, - PowerPoint PPT Presentation

Geometric Firefighting Rolf Klein University of Bonn HMI, June 19, 2018 Rolf Klein Geometric firefighting

Firefighting techniques Extinguish the fire Rolf Klein Geometric firefighting

Firefighting techniques Extinguish the fire Prevent fire from spreading further Rolf Klein Geometric firefighting

Firefighting techniques Extinguish the fire Prevent fire from spreading further Rolf Klein Geometric firefighting

A. Bressan, 2006 Problem circular fire spreads in the plane at unit speed Rolf Klein Geometric firefighting

A. Bressan, 2006 Problem circular fire spreads in the plane at unit speed build, in real time, barrier curves to contain the fire Rolf Klein Geometric firefighting

A. Bressan, 2006 Problem circular fire spreads in the plane at unit speed build, in real time, barrier curves to contain the fire such that, at each time t , total length of curves built ≤ v · t Rolf Klein Geometric firefighting

A. Bressan, 2006 Problem circular fire spreads in the plane at unit speed build, in real time, barrier curves to contain the fire such that, at each time t , total length of curves built ≤ v · t How large a speed v is needed? Rolf Klein Geometric firefighting

Speed v > 2 is sufficient p Rolf Klein Geometric firefighting

Speed v > 2 is sufficient 1 p Rolf Klein Geometric firefighting

Speed v > 2 is sufficient Rolf Klein Geometric firefighting

Speed v = 1 is not sufficient EB s f DB Rolf Klein Geometric firefighting

Speed v = 1 is not sufficient EB s f DB Rolf Klein Geometric firefighting

Speed v = 1 is not sufficient EB π x s s f DB s 0 Rolf Klein Geometric firefighting

Speed v = 1 is not sufficient EB π f s s f π f DB s 0 s 0 π f s ≤ π f s 0 ≤ EB +2 DB 2 < EB + DB Rolf Klein Geometric firefighting

Speed v = 1 is not sufficient EB π x s s f DB s 0 Rolf Klein Geometric firefighting

A. Bressan: conjecture v = 2 necessary Rolf Klein Geometric firefighting

A. Bressan: conjecture v = 2 necessary 500 USD reward (2011) Rolf Klein Geometric firefighting

A. Bressan: conjecture v = 2 necessary 500 USD reward (2011) gap (1 , 2] still open Rolf Klein Geometric firefighting

Difficulty: delaying barriers EB π f s s f π f DB s 0 s 0 π f s ≤ π f s 0 ≤ EB +2 DB 2 < EB + DB Rolf Klein Geometric firefighting

Can delaying barriers be useful? 0 Rolf Klein Geometric firefighting

Parallel processes ? 0 Rolf Klein Geometric firefighting

Consumption ratio 0 Rolf Klein Geometric firefighting

Consumption ratio 0 barriers consumed by fire at time t sup t t ≥ t 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Hitting a vertical barrier 0 Rolf Klein Geometric firefighting

Different case 0 Rolf Klein Geometric firefighting

Different case 0 Rolf Klein Geometric firefighting

Different case 0 Rolf Klein Geometric firefighting

Different case 0 Rolf Klein Geometric firefighting

Different case 0 Rolf Klein Geometric firefighting

Fact 1 z i +1 z i Rolf Klein Geometric firefighting

Fact 1 z i Rolf Klein Geometric firefighting

Fact 1 z i ≥ 2 consumption points for duration ≥ z i Rolf Klein Geometric firefighting

Fact 2 z i +1 z i Rolf Klein Geometric firefighting

Fact 2 z i +1 z i Rolf Klein Geometric firefighting

Fact 2 z i +1 z i consumption ≥ z i + z i +1 Rolf Klein Geometric firefighting

RHS consumption ratio z i +1 w i +1 z i w i Rolf Klein Geometric firefighting

RHS consumption ratio 1 z i +1 w i +1 z i w i Rolf Klein Geometric firefighting

RHS consumption ratio 1 1 2 z i +1 w i +1 z i w i Rolf Klein Geometric firefighting

Low minima 1 1 2 + ǫ 1 2 w i z i +1 w i +1 z i assuming total consumption ratio < 1 . 5 + ǫ Rolf Klein Geometric firefighting

Inequality 1+2 ǫ z i +1 < 1 − 2 ǫ w i Rolf Klein Geometric firefighting

Lower bound proof w i z i +1 z i v j − 1 v j y j Rolf Klein Geometric firefighting

Lower bound proof w i z i +1 z i v j − 1 v j y j Rolf Klein Geometric firefighting

Lower bound proof w i z i +1 z i v j − 1 v j y j Rolf Klein Geometric firefighting

Lower bound proof w i > z i +1 z i v j − 1 v j > y j Rolf Klein Geometric firefighting

Lower bound proof w i > z i +1 z i v j − 1 v j > y j = ⇒ z i > z i +1 Rolf Klein Geometric firefighting

Theorem (S.-S. Kim, D. K¨ ubel, E. Langetepe, B. Schwarzwald, R.K.) A horizontal line with vertical line segments attached has consumption ratio ≥ 1 . 53069 . Rolf Klein Geometric firefighting

Theorem (S.-S. Kim, D. K¨ ubel, E. Langetepe, B. Schwarzwald, R.K.) A horizontal line with vertical line segments attached has consumption ratio ≥ 1 . 53069 . But a ratio of v = 1 . 8 can be attained. Rolf Klein Geometric firefighting

Construction 34 · 4 i 34 · 4 i − 1 34 · 4 2 34 · 4 34 · 4 34 17 51 · 4 i − 1 51 · 4 2 51 · 4 3 51 · 4 i − 1 34 1020 238 1 1 Rolf Klein Geometric firefighting

Consumption ratios 3 1 1 0 RHS 34 · 4 i 34 · 4 i − 1 0 1 1 LHS 34 · 4 i Rolf Klein Geometric firefighting

Consumption ratios 3 1 1 0 RHS 34 · 4 i 34 · 4 i − 1 0 1 1 LHS 34 · 4 i max 1 . 11 sum1 . 8 min 0 . 6266 Rolf Klein Geometric firefighting

Open line plus perpendicular segments, L 1 : [1 . 53069 , 1 . 8) Rolf Klein Geometric firefighting

Open line plus perpendicular segments, L 1 : [1 . 53069 , 1 . 8) closed curve with forest inside, L 2 : (1 , 2] (Bressan conjecture: 2) Rolf Klein Geometric firefighting