2019/05/06 by Sang-Sub Kim, Kim, Sang-Sub, Rolf Klein +7
Computer Science · #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #F.2.2 #FOS: Computer and information sciences #cs.CG
paper · pdf · doi:10.48550/arxiv.1905.02067
15 pages, 10 figures, pre-print of an article published in WADS 2019
arxiv created 2019/05/06 · openalex publication_date 2019/05/06 · arxiv updated 2019/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2006, Alberto Bressan suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed t ⋅ v, where v is a speed constant. How large a speed v is needed? He proved that speed v>2 is sufficient, and that v>1 is necessary. This gap of (1,2] is still open. The crucial question seems to be the following. \em When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire's way? We study the situation where the fire must be contained in the upper L1 half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers. Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed v=1.8772 is sufficient, while v >1.66 is necessary.