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

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

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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

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SLIDE 1

Geometric Firefighting

Rolf Klein

University of Bonn

HMI, June 19, 2018

Rolf Klein Geometric firefighting

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SLIDE 2

Firefighting techniques

Extinguish the fire

Rolf Klein Geometric firefighting

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SLIDE 3

Firefighting techniques

Extinguish the fire Prevent fire from spreading further

Rolf Klein Geometric firefighting

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SLIDE 4

Firefighting techniques

Extinguish the fire Prevent fire from spreading further

Rolf Klein Geometric firefighting

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SLIDE 5
  • A. Bressan, 2006

Problem circular fire spreads in the plane at unit speed

Rolf Klein Geometric firefighting

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SLIDE 6
  • 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

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SLIDE 7
  • 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

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SLIDE 8
  • 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

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SLIDE 9

Speed v > 2 is sufficient

p

Rolf Klein Geometric firefighting

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Speed v > 2 is sufficient

1

p

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SLIDE 11

Speed v > 2 is sufficient

Rolf Klein Geometric firefighting

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SLIDE 12

Speed v = 1 is not sufficient

s f EB DB

Rolf Klein Geometric firefighting

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SLIDE 13

Speed v = 1 is not sufficient

s f EB DB

Rolf Klein Geometric firefighting

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SLIDE 14

Speed v = 1 is not sufficient

s f πx

s

EB s0 DB

Rolf Klein Geometric firefighting

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SLIDE 15

Speed v = 1 is not sufficient

s f πf

s

EB s0 DB πf

s0

πf

s ≤ πf s0 ≤ EB+2DB 2

< EB + DB

Rolf Klein Geometric firefighting

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SLIDE 16

Speed v = 1 is not sufficient

s f πx

s

EB s0 DB

Rolf Klein Geometric firefighting

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SLIDE 17
  • A. Bressan: conjecture v = 2 necessary

Rolf Klein Geometric firefighting

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SLIDE 18
  • A. Bressan: conjecture v = 2 necessary

500 USD reward (2011)

Rolf Klein Geometric firefighting

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SLIDE 19
  • A. Bressan: conjecture v = 2 necessary

500 USD reward (2011) gap (1, 2] still open

Rolf Klein Geometric firefighting

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SLIDE 20

Difficulty: delaying barriers

s f πf

s

EB s0 DB πf

s0

πf

s ≤ πf s0 ≤ EB+2DB 2

< EB + DB

Rolf Klein Geometric firefighting

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SLIDE 21

Can delaying barriers be useful?

Rolf Klein Geometric firefighting

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SLIDE 22

Parallel processes ?

Rolf Klein Geometric firefighting

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SLIDE 23

Consumption ratio

Rolf Klein Geometric firefighting

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Consumption ratio

sup

t≥t0

barriers consumed by fire at time t t

Rolf Klein Geometric firefighting

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SLIDE 25

Hitting a vertical barrier

Rolf Klein Geometric firefighting

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Hitting a vertical barrier

Rolf Klein Geometric firefighting

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SLIDE 27

Hitting a vertical barrier

Rolf Klein Geometric firefighting

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SLIDE 28

Hitting a vertical barrier

Rolf Klein Geometric firefighting

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Hitting a vertical barrier

Rolf Klein Geometric firefighting

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Hitting a vertical barrier

Rolf Klein Geometric firefighting

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Hitting a vertical barrier

Rolf Klein Geometric firefighting

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Different case

Rolf Klein Geometric firefighting

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Different case

Rolf Klein Geometric firefighting

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Different case

Rolf Klein Geometric firefighting

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Different case

Rolf Klein Geometric firefighting

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Different case

Rolf Klein Geometric firefighting

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Fact 1

zi zi+1

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Fact 1

zi

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Fact 1

zi ≥ 2 consumption points for duration ≥ zi

Rolf Klein Geometric firefighting

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Fact 2

zi zi+1

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Fact 2

zi zi+1

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Fact 2

zi zi+1 consumption ≥ zi + zi+1

Rolf Klein Geometric firefighting

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SLIDE 43

RHS consumption ratio

zi zi+1 wi wi+1

Rolf Klein Geometric firefighting

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SLIDE 44

RHS consumption ratio

zi zi+1 wi wi+1

1

Rolf Klein Geometric firefighting

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SLIDE 45

RHS consumption ratio

zi zi+1 wi wi+1

1

1 2 Rolf Klein Geometric firefighting

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SLIDE 46

Low minima

zi zi+1 wi wi+1

1

1 2 + ǫ 1 2

assuming total consumption ratio < 1.5 + ǫ

Rolf Klein Geometric firefighting

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SLIDE 47

Inequality zi+1 <

1+2ǫ 1−2ǫ wi

Rolf Klein Geometric firefighting

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Lower bound proof

zi zi+1 wi

vj−1 yj vj

Rolf Klein Geometric firefighting

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Lower bound proof

zi zi+1 wi

vj−1 yj vj

Rolf Klein Geometric firefighting

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Lower bound proof

zi zi+1 wi

vj−1 yj vj

Rolf Klein Geometric firefighting

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Lower bound proof

zi zi+1 wi

vj−1 yj vj

> >

Rolf Klein Geometric firefighting

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Lower bound proof

zi zi+1 wi

vj−1 yj vj

> > = ⇒ zi > zi+1

Rolf Klein Geometric firefighting

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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

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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

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Construction

1 17 34 34 · 4 1020 34 · 42 34 · 4i−1 1 34 238 51 · 4i−1 34 · 4 51 · 43 51 · 42 34 · 4i 51 · 4i−1

Rolf Klein Geometric firefighting

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Consumption ratios

1 1 1 1 3 RHS LHS 34 · 4i−1 34 · 4i 34 · 4i

Rolf Klein Geometric firefighting

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Consumption ratios

1 1 1 1 3 RHS LHS 34 · 4i−1 34 · 4i 34 · 4i max 1.11 min 0.6266 sum1.8

Rolf Klein Geometric firefighting

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Open

line plus perpendicular segments, L1 : [1.53069, 1.8)

Rolf Klein Geometric firefighting

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Open

line plus perpendicular segments, L1 : [1.53069, 1.8) closed curve with forest inside, L2 : (1, 2] (Bressan conjecture: 2)

Rolf Klein Geometric firefighting