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We consider the problem of finding an obstacle-avoiding path between two points s and t in the plane, amidst a set of disjoint polyg- onal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle-avoiding s-t path measured in the Lv-metric. Such an approxi- mate shortest path is called a c-short path, or a short path with stretch )actor c. The goal is to preprocess the obstacle-scattered plane by creat- ing an efficient data structure that enables fast reporting of a c-short path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting trade-off between the stretch factor, the query time and the preprocessing bounds. Our main results are al- gorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing.

Planar spanners and approximate shortest path queries among obstacles in the plane