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On a Fire Fighter’s Problem

Klein, Rolf ; Langetepe, Elmar ; Levcopoulos, Christos LU ; Lingas, Andrzej LU and Schwarzwald, Barbara (2019) In International Journal of Foundations of Computer Science 30(02). p.231-246
Abstract
Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed v>1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results.

First, we analyze the natural curve FFv that develops when the fighter keeps building, at speed v, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function (ewZ−sZ)−1, where w and s are real functions of v. For v>vc=2.6144… all zeroes are complex conjugate pairs. If ϕ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ(1/ϕ)... (More)
Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed v>1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results.

First, we analyze the natural curve FFv that develops when the fighter keeps building, at speed v, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function (ewZ−sZ)−1, where w and s are real functions of v. For v>vc=2.6144… all zeroes are complex conjugate pairs. If ϕ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ(1/ϕ) rounds before the fire is contained. As v decreases towards vc these two zeroes merge into a real one, so that argument ϕ goes to 0. Thus, curve FFv does not contain the fire if the fighter moves at speed v=vc. (That speed v>vc is sufficient for containing the fire has been proposed before by Bressan et al. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.)

Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed v is less than 1.618…, the golden ratio. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Motion planning, Dynamic environments, Spiralling strategies, Lower and upper bounds barrier, Reachable set
in
International Journal of Foundations of Computer Science
volume
30
issue
02
pages
16 pages
publisher
World Scientific
external identifiers
  • scopus:85062887663
ISSN
0129-0541
DOI
10.1142/S0129054119500023
language
English
LU publication?
yes
id
42b46030-41f7-4cb6-8a09-e2a6a236f190
date added to LUP
2019-03-14 10:18:07
date last changed
2019-04-23 04:46:41
@article{42b46030-41f7-4cb6-8a09-e2a6a236f190,
  abstract     = {Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed v&gt;1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results.<br>
<br>
First, we analyze the natural curve FFv that develops when the fighter keeps building, at speed v, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function (ewZ−sZ)−1, where w and s are real functions of v. For v&gt;vc=2.6144… all zeroes are complex conjugate pairs. If ϕ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ(1/ϕ) rounds before the fire is contained. As v decreases towards vc these two zeroes merge into a real one, so that argument ϕ goes to 0. Thus, curve FFv does not contain the fire if the fighter moves at speed v=vc. (That speed v&gt;vc is sufficient for containing the fire has been proposed before by Bressan et al. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.)<br>
<br>
Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed v is less than 1.618…, the golden ratio.},
  author       = {Klein, Rolf and Langetepe, Elmar and Levcopoulos, Christos and Lingas, Andrzej and Schwarzwald, Barbara},
  issn         = {0129-0541},
  language     = {eng},
  number       = {02},
  pages        = {231--246},
  publisher    = {World Scientific},
  series       = {International Journal of Foundations of Computer Science},
  title        = {On a Fire Fighter’s Problem},
  url          = {http://dx.doi.org/10.1142/S0129054119500023},
  doi          = {10.1142/S0129054119500023},
  volume       = {30},
  year         = {2019},
}