Thick non-crossing paths and minimum-cost flows in polygonal domains
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Maximum thick paths in static and dynamic environments
Proceedings of the twenty-fourth annual symposium on Computational geometry
Shortest non-crossing walks in the plane
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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We study the problem of finding shortest non-crossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zero-thickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple n-gon, we compute in O(n + K) time a representation of the set of K shortest non-crossing thick paths joining the terminal pairs; using the representation, any particular path can be output in time proportional to its complexity. In a polygonal domain with h holes we compute K shortest thick non-crossing paths in O (( K + 1)hh! poly(n,K)) time, using an efficient method to compute any one of the K thick paths if the "threadings" of all paths amidst the holes are specified. We show that if h is not constant, the problem is NP-hard; we also show the hardness of approximation. We give a pseudopolynomial-time algorithm for some rectilinear versions of the problem. We apply our thick paths algorithms to obtain the first algorithmic results for the minimum-cost continuous flow problem—an extension of the standard discrete minimum-cost network flow problem to continuous domains. The results are based on showing a continuous analog of the Network Flow Decomposition Theorem. We investigate the problem of finding the maximum number of thick paths that can be routed in a polygonal domain. Using a modification of the "continuous uppermost path" algorithm, we give a constructive proof of a continuous Menger-type result: the maximum number of paths equals to the length of a shortest path in the "thresholded critical graph" of the domain. The algorithm computes (a representation of) the paths in O( nh + n log n) time. We show how to use the algorithm to find maximum monotone flows and paths. For simple polygons we give faster, linear-time algorithms. The non-crossing thick paths problems, as well as the continuous flow problems, arise in the air traffic management problem of optimally routing air traffic lanes in Flow Constrained Areas while avoiding weather hazards, no-fly zones, and other constraints. The other motivations for studying the problems come from wire routing for circuits and from information propagation in sensor networks. We also consider other motion planning problems: the touring problems, shortest paths with bounded number of links in rectilinear domains, and optimal tours in "pixelated" environments and grids.