Shortest Path Queries in a Simple Polygon for 3D Virtual Museum

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

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LOGARITHMIC-TIME LINK PATH QUERIES IN A SIMPLE POLYGON

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195995000234 ◽

1995 ◽

Vol 05 (04) ◽

pp. 369-395 ◽

Cited By ~ 9

Author(s):

ESTHER M. ARKIN ◽

JOSEPH S.B. MITCHELL ◽

SUBHASH SURI

Keyword(s):

Data Structure ◽

Shortest Path ◽

Simple Polygon ◽

Additive Constant ◽

Time And Space ◽

Link Distance ◽

Path Distance ◽

Path Queries ◽

Shortest Path Distance ◽

Small Additive

We develop a data structure for answering link distance queries between two arbitrary points in a simple polygon. The data structure requires O(n3) time and space for its construction and answers link distance queries in O(log n) time, after which a minimum-link path can be reported in time proportional to the number of links. Here, n denotes the number of vertices of the polygon. Our result extends to link distance queries between pairs of segments or polygons. We also propose a simpler data structure for computing a link distance approximately, where the error is bounded by a small additive constant. Finally, we also present a scheme for approximating the link and the shortest path distance simultaneously.

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A new data structure for shortest path queries in a simple polygon

Information Processing Letters ◽

10.1016/0020-0190(91)90064-o ◽

1991 ◽

Vol 38 (5) ◽

pp. 231-235 ◽

Cited By ~ 29

Author(s):

John Hershberger

Keyword(s):

Data Structure ◽

Shortest Path ◽

Simple Polygon ◽

Path Queries

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OPTIMAL PARALLEL ALGORITHMS FOR TRIANGULATED SIMPLE POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s021819599500009x ◽

1995 ◽

Vol 05 (01n02) ◽

pp. 145-170 ◽

Cited By ~ 1

Author(s):

JOHN HERSHBERGER

Keyword(s):

Convex Hull ◽

Parallel Algorithms ◽

Shortest Path ◽

Simple Polygon ◽

Implicit Representation ◽

Shortest Path Tree ◽

Simple Polygons ◽

Pram Model ◽

Crew Pram ◽

Path Queries

We provide optimal parallel solutions to several shortest path and visibility problems set in triangulated simple polygons. Let P be a triangulated simple polygon with n vertices, preprocessed to support shortest path queries. We can find the shortest path tree from any point inside P in O(log n) time using O(n/log n) processors. In the game bounds, we can preprocess P for shooting queries (a query can be answered in O(log n) time by a uniprocessor). Given a set S of m points inside P, we can find an implicit representation of the relative convex hull of S in O(log(nm)) time with O(m) processors. If the relative convex hull has k edges, we can explicitly produce these edges in O(log(nm)) time with O(k/log(nm)) processors. All of these algorithms are deterministic and use the CREW PRAM model.

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Optimal Shortest Path and Minimum-Link Path Queries between Two Convex Polygons inside a Simple Polygonal Obstacle

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195997000077 ◽

1997 ◽

Vol 07 (01n02) ◽

pp. 85-121 ◽

Cited By ~ 4

Author(s):

Yi-Jen Chiang ◽

Roberto Tamassia

Keyword(s):

Data Structure ◽

Shortest Path ◽

Dynamic Environment ◽

Simple Polygon ◽

Efficient Algorithms ◽

Optimal Time ◽

Convex Polygons ◽

Separation Problem ◽

Dynamic Case ◽

Path Queries

We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O( log h + log n) (plus O(k) time for reporting the k edges of the path) using a data structure with O(n) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O( log h + log n) (plus O(k) to report the k links), with O(n3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision [Formula: see text]. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in [Formula: see text]. The data structure uses O(n) space, supports updates in O( log 2 n) time, and performs shortest-path and minimum-link-path queries in times O( log h+ log 2n) (plus O(k) to report the k edges of the path) and O( log h + k log 2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.

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