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A Fire Fighter's Problem

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 $\mbox{FF}_v$ 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 $(e^{w Z} - s \, Z)^{-1}$, where $w$ and $s$ are real functions of $v$. For $v>v_c=2.6144 \ldots$ 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 $v_c$ these two zeroes merge into a real one, so that argument $ϕ$ goes to~0. Thus, curve $\mbox{FF}_v$ does not contain the fire if the fighter moves at speed $v=v_c$. (That speed $v>v_c$ is sufficient for containing the fire has been proposed before by Bressan et al. [7], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that any curve that visits the four coordinate half-axes in cyclic order, and in inreasing distances from the origin, needs speed $v>1.618\ldots$, the golden ratio, in order to contain the fire. Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower and upper bounds