Let P be a set of h pairwise-disjoint polygonal obstacles with a total of n vertices in the plane. In this paper, we consider the problem of building a data structure that can quickly compute an L1 shortest obstacle-avoiding path between any two query points s and t. We build a data structure of sizewhere k is the number of edges of the output path. Note that n+h 2 ·log 2 h· 4 √ log h = O(n+h 2+ϵ ) for any constant ϵ > 0. We also extend our techniques to the weighted rectilinear version in which the "obstacles" of P are rectilinear regions with "weights" and allow L1 paths to travel through them with weighted costs. Our algorithm answers each query in O(log n + k) time with a data structure of size O(n 2 · log n · 4 √ log n ) that is built in O(n 2 · log 2 n · 4 √ log n ) time.

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We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon. A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point [Formula: see text] due to the insertion (resp. deletion) of vertex [Formula: see text] to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of [Formula: see text] due to the insertion (resp. deletion) of [Formula: see text]. An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point [Formula: see text] in [Formula: see text] amid vertex insertions and deletions to the simple polygon. If [Formula: see text] is not exterior to the current simple polygon, then the visibility polygon of [Formula: see text] is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of [Formula: see text]. An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment [Formula: see text] due to the insertion of vertex [Formula: see text] to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at [Formula: see text] and [Formula: see text] due to the insertion of [Formula: see text]. An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon. And, the algorithms that maintain the visibility polygon (resp. weak visibility polygon) compute the visibility polygon (resp. weak visibility polygon) with respect to the initial simple polygon during the preprocessing phase.