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

1997 ◽  
Vol 07 (01n02) ◽  
pp. 85-121 ◽  
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.

LOGARITHMIC-TIME LINK PATH QUERIES IN A SIMPLE POLYGON

1995 ◽  
Vol 05 (04) ◽  
pp. 369-395 ◽  
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.

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 249-261 ◽  
Author(s):  
XUEHOU TAN
Keyword(s):  
Hierarchical Structure ◽  
Shortest Path ◽  
Line Segment ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Search Trees ◽  
Bridge Problem ◽  
Path Queries

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.

EFFICIENT ALGORITHMS FOR SOLVING DIAGONAL VISIBILITY PROBLEMS IN A SIMPLE POLYGON

1995 ◽  
Vol 05 (04) ◽  
pp. 433-458 ◽  
Author(s):  
SOO-HWAN KIM ◽  
SUNG YONG SHIN ◽  
KYUNG-YONG CHWA
Keyword(s):  
Data Structure ◽  
Simple Polygon ◽  
Efficient Algorithms

A diagonal guard is a guard capable of moving along an edge or an internal diagonal in a polygon. A polygon which can be guarded by a diagonal guard is called diagonal-visible. We consider the following three problems concerning the diagonal visibility in a polygon P. First, determine whether or not a guard diagonal exists in P, i.e., P is diagonal-visible. Second, compute all guard diagonals of P. Third, given a query diagonal, determine whether or not it is a guard diagonal. For these problems, we construct a data structure for keeping all guard diagonals in O(n log n) time and O(n) space. Using this data structure, we show that these problems can be solved in O(n log n), O(n log n+k), and O(1) time, respectively, where k is the number of guard diagonals.

Walking an Unknown Street with Limited Sensing

2019 ◽  
Vol 33 (13) ◽  
pp. 1959042
Author(s):  
Qi Wei ◽  
Xuehou Tan ◽  
Yonggong Ren
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Mobile Robot ◽  
Shortest Path ◽  
Competitive Strategy ◽  
Visible Region ◽  
Simple Polygon ◽  
Limited Sensing

This paper studies a searching problem in an unknown street. A simple polygon [Formula: see text] with two distinguished vertices, [Formula: see text] and [Formula: see text], is called a street if the two boundary chains from [Formula: see text] to [Formula: see text] are mutually weakly visible. We use a mobile robot to locate [Formula: see text] starting from [Formula: see text]. Assume that the robot has a limited sensing capability that can only detect the constructed edges (also called gaps) on the boundary of its visible region, but cannot measure any angle or distance. The robot does not have knowledge of the street in advance. We present a new competitive strategy for this problem and prove that the length of the path generated by the robot is at most 9-times longer than the shortest path. We also propose a matching lower bound to show that our strategy is optimal. Compared with the previous strategy, we further relaxed the restriction that the robot should take a marking device and use the data structure S-GNT. The analysis of our strategy is tight.

scholarly journals SHORTEST PATH QUERIES IN RECTILINEAR WORLDS

1992 ◽  
Vol 02 (03) ◽  
pp. 287-309 ◽  
Author(s):  
MARK DE BERG ◽  
MARC VAN KREVELD ◽  
BENGT J. NILSSON ◽  
MARK OVERMARS
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Simple Solution ◽  
Single Shot ◽  
Fixed Target ◽  
Higher Dimensional ◽  
Axis Parallel ◽  
Fixed Constant ◽  
Path Queries

In this paper, a data structure is given for two and higher dimensional shortest path queries. For a set of n axis-parallel rectangles in the plane, or boxes in d-space, and a fixed target, it is possible with this structure to find a shortest rectilinear path avoiding all rectangles or boxes from any point to this target. Alternatively, it is possible to find the length of the path. The metric considered is a generalization of the L1-metric and the link metric, where the length of a path is its L1-length plus some (fixed) constant times the number of turns on the path. The data structure has size O((n log n)d−1), and a query takes O( log d−1 n) time (plus the output size if the path must be reported). As a byproduct, a relatively simple solution to the single shot problem is obtained; the shortest path between two given points can be computed in time O(nd log n) for d≥3, and in time O(n2) in the plane.

OPTIMAL PARALLEL ALGORITHMS FOR TRIANGULATED SIMPLE POLYGONS

1995 ◽  
Vol 05 (01n02) ◽  
pp. 145-170 ◽  
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.

scholarly journals An external memory data structure for shortest path queries

2003 ◽  
Vol 126 (1) ◽  
pp. 55-82 ◽  
Author(s):  
David Hutchinson ◽  
Anil Maheshwari ◽  
Norbert Zeh
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
External Memory ◽  
Path Queries
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