On a Fire Fighter’s Problem
(2019) In International Journal of Foundations of Computer Science 30(02). p.231246 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 halfaxes 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:
http://lup.lub.lu.se/record/42b4603041f74cb68a09e2a6a236f190
 author
 Klein, Rolf ; Langetepe, Elmar ; Levcopoulos, Christos ^{LU} ; Lingas, Andrzej ^{LU} and Schwarzwald, Barbara
 organization
 publishing date
 2019
 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
 01290541
 DOI
 10.1142/S0129054119500023
 language
 English
 LU publication?
 yes
 id
 42b4603041f74cb68a09e2a6a236f190
 date added to LUP
 20190314 10:18:07
 date last changed
 20190423 04:46:41
@article{42b4603041f74cb68a09e2a6a236f190, 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.<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>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.)<br> <br> Second, we show that for any curve that visits the four coordinate halfaxes 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 = {01290541}, language = {eng}, number = {02}, pages = {231246}, 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}, }