Approximate Euclidean Shortest Paths in Polygonal Domains (original) (raw)

Given a set mathcalP\mathcal{P}mathcalP of hhh pairwise disjoint simple polygonal obstacles in mathbbR2\mathbb{R}^2mathbbR2 defined with nnn vertices, we compute a sketch Omega\OmegaOmega of mathcalP\mathcal{P}mathcalP whose size is independent of nnn, depending only on hhh and the input parameter epsilon\epsilonepsilon. We utilize Omega\OmegaOmega to compute a (1+epsilon)(1+\epsilon)(1+epsilon)-approximate geodesic shortest path between the two given points in O(n+h((lgn)+(lgh)1+delta+(frac1epsilonlgfrachepsilon)))O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}})))O(n+h((lgn)+(lgh)1+delta+(frac1epsilonlgfrachepsilon))) time. Here, epsilon\epsilonepsilon is a user parameter, and delta\deltadelta is a small positive constant (resulting from the time for triangulating the free space of calP\cal PcalP using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a (2+epsilon)(2+\epsilon)(2+epsilon)-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.