On the Maximal Shortest Paths Cover Number

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S⊆V(G) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ(G). We show that it is NP-hard to determine ξ(G). We establish a connection between ξ(G) and several other graph parameters. We present a linear time algorithm that computes exact value for ξ(T) of a tree T.

Download Full-text

AN APPROXIMATE MORPHING BETWEEN POLYLINES

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195905001658 ◽

2005 ◽

Vol 15 (02) ◽

pp. 193-208 ◽

Cited By ~ 1

Author(s):

SERGEY BEREG

Keyword(s):

Shortest Path ◽

Optimization Problem ◽

Linear Time ◽

Medial Axis ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Geodesic Path ◽

Minimum Width ◽

Approximation Factor ◽

Previous Algorithm

We consider the problem of continuously transforming or morphing one simple polyline into another polyline so that every point p of the initial polyline moves to a point q of the final polyline using the geodesic shortest path from p to q. The width of a morphing is defined as the longest geodesic path between corresponding points of the polylines. The optimization problem is to compute a morphing that minimizes the width. We present a linear-time algorithm for finding a morphing with width guaranteed to be at most two times the minimum width of a morphing. This improves the previous algorithm10 by a factor of log n. We develop a linear-time algorithm for computing a medial axis separator. We also show that the approximation factor is less than two for κ-straight polylines.

Download Full-text

PRACTICAL EFFICIENCY OF THE LINEAR-TIME ALGORITHM FOR THE SINGLE SOURCE SHORTEST PATH PROBLEM

Journal of the Operations Research Society of Japan ◽

10.15807/jorsj.43.431 ◽

2000 ◽

Vol 43 (4) ◽

pp. 431-447 ◽

Cited By ~ 8

Author(s):

Yasuhito Asano ◽

Hiroshi Imai

Keyword(s):

Shortest Path ◽

Linear Time ◽

Time Algorithm ◽

Shortest Path Problem ◽

Linear Time Algorithm ◽

Single Source

Download Full-text

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195993000063 ◽

1993 ◽

Vol 03 (01) ◽

pp. 85-105 ◽

Cited By ~ 19

Author(s):

SVANTE CARLSSON ◽

BENGT J. NILSSON ◽

SIMEON NTAFOS

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Np Hard ◽

Art Gallery ◽

Art Galleries ◽

Art Gallery Problem ◽

Natural Connection ◽

Route Problem ◽

Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

Download Full-text

GUARD PLACEMENT FOR MAXIMIZING L-VISIBILITY EXTERIOR TO A CONVEX POLYGON

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195909003003 ◽

2009 ◽

Vol 19 (04) ◽

pp. 357-370

Author(s):

DEBABRATA BARDHAN ◽

SANSANKA ROY ◽

SANDIP DAS

Keyword(s):

Shortest Path ◽

Convex Polygon ◽

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Placement Problem ◽

Maximum Area

Two points a and b are said to be L-visible among a set of polygonal obstacles if the length of the shortest path from a to b avoiding these obstacles is no more than L. For a given convex polygon P with n vertices, Gewali et al.1 addressed the guard placement problem on the boundary of P that covers the maximum area outside to the polygon under L-visibility with P as obstacle. Their proposed algorithm runs in O(n) time if [Formula: see text], where π(P) denotes the perimeter of P. They conjectured that if [Formula: see text], then the problem can be solved in subquadratic time. In this paper, we settle the conjecture in the affirmative sense, by proposing an easy to implement linear time algorithm for any arbitrary value of L.

Download Full-text

A linear time algorithm for minimum equitable dominating set in trees

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-4552 ◽

2021 ◽

Vol 40 (4) ◽

pp. 805-814

Author(s):

Sohel Rana ◽

Sk. Md. Abu Nayeem

Keyword(s):

Linear Time ◽

Dominating Set ◽

Domination Number ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Minimum Cardinality ◽

Np Complete ◽

General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

Download Full-text

AN OPTIMAL DATA STRUCTURE FOR SHORTEST RECTILINEAR PATH QUERIES IN A SIMPLE RECTILINEAR POLYGON

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195996000149 ◽

1996 ◽

Vol 06 (02) ◽

pp. 205-225 ◽

Cited By ~ 12

Author(s):

SVEN SCHUIERER

Keyword(s):

Data Structure ◽

Shortest Path ◽

Linear Time ◽

Time Algorithm ◽

Query Time ◽

Linear Time Algorithm ◽

Link Length ◽

N Storage ◽

Path Queries ◽

Rectilinear Polygon

We present a data structure that allows to preprocess a rectilinear polygon with n vertices such that, for any two query points, the shortest path in the rectilinear link or L1-metric can be reported in time O( log n+k) where k is the link length of the shortest path. If only the distance is of interest, the query time reduces to O( log n). Furthermore, if the query points are two vertices, the distance can be reported in time O(1) and a shortest path can be constructed in time O(1+k). The data structure can be computed in time O(n) and needs O(n) storage. As an application we present a linear time algorithm to compute the diameter of a simple rectilinear polygon w.r.t. the L1-metric.

Download Full-text

Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

Download Full-text