k-PAIRS NON-CROSSING SHORTEST PATHS IN A SIMPLE POLYGON

1999 ◽  
Vol 09 (06) ◽  
pp. 533-552 ◽  
Author(s):  
EVANTHIA PAPADOPOULOU
Keyword(s):  
Outer Boundary ◽  
Shortest Paths ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Polygonal Domain ◽  
Total Cost

This paper presents a simple O(n+k) time algorithm to compute the set of knon-crossing shortest paths between k source-destination pairs of points on the boundary of a simple polygon of n vertices. Paths are allowed to overlap but are not allowed to cross in the plane. A byproduct of this result is an O(n) time algorithm to compute a balanced geodesic triangulation which is easy to implement. The algorithm extends to a simple polygon with one hole where source-destination pairs may appear on both the inner and outer boundary of the polygon. In the latter case, the goal is to compute a collection of non-crossing paths of minimum total cost. The case of a rectangular polygonal domain where source-destination pairs appear on the outer and one inner boundary12 is briefly discussed.

SEARCHING A ROOM BY TWO GUARDS

2002 ◽  
Vol 12 (04) ◽  
pp. 339-352 ◽  
Author(s):  
SANG-MIN PARK ◽  
JAE-HA LEE ◽  
KYUNG-YONG CHWA
Keyword(s):  
Laser Beam ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Polygonal Region

We consider the problem of searching for mobile intruders in a polygonal region with one door by two guards. Given a simple polygon [Formula: see text] with a door d, which is called a room [Formula: see text], two guards start at d and walk along the boundary of [Formula: see text] to detect a mobile intruder with a laser beam between the two guards. During the walk, two guards are required to be mutually visible all the time and eventually meet at one point. We give a characterization of the class of rooms searchable by two guards and an O(n log n)-time algorithm to test if a given room admits a walk, where n is the number of the vertices in [Formula: see text].

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.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

scholarly journals A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon

2016 ◽  
Vol 56 (4) ◽  
pp. 836-859 ◽  
Author(s):  
Hee-Kap Ahn ◽  
Luis Barba ◽  
Prosenjit Bose ◽  
Jean-Lou De Carufel ◽  
Matias Korman ◽  
...  
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Linear Time Algorithm

scholarly journals Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2001
Author(s):  
Konstantin Gorbunov ◽  
Vassily Lyubetsky
Keyword(s):  
Linear Time ◽  
Exact Algorithm ◽  
Time Algorithm ◽  
Additive Constant ◽  
Weighted Graphs ◽  
Total Cost ◽  
Operation Costs ◽  
Double Cut And Join ◽  
Vertex Degrees ◽  
Directed Weighted Graphs

We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.

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