A GEOMETRIC SPANNER OF SEGMENTS

2010 ◽  
Vol 20 (01) ◽  
pp. 43-67
Author(s):  
JINHUI XU ◽  
YANG YANG ◽  
YONGDING ZHU ◽  
NAOKI KATOH
Keyword(s):  
Computational Geometry ◽  
Efficient Algorithm ◽  
Euclidean Distance ◽  
Dominating Set ◽  
Optimal Solution ◽  
Minimum Size ◽  
Steiner Points ◽  
Separation Ratio ◽  
Geometric Spanner ◽  
Relative Separation

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider a generalization of the classical geometric spanner problem (called segment spanner): Given a set S of n disjoint 2-D segments, find a spanning network GS with minimum size so that for any pair of points in S, there exists a path in GS with length no more than t times their Euclidean distance. Based on a number of interesting techniques (such as weakly dominating set, strongly dominating set, interval cover, and imaginary Steiner points), we present an efficient algorithm to construct the segment spanner. Our approach first identifies a set Q of Steiner points in S and then constructs a point spanner for the set of Steiner points. Our algorithm runs in O(|Q| + n2 log n) time and Q is a constant approximation (in terms of its size) of the optimal solution when S has a constant relative separation ratio. The approximation ratio depends on the stretch factor t and the relative separation ratio of S.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

scholarly journals An Efficient Algorithm for Maximizing Range Sum Queries in a Road Network

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tien-Khoi Phan ◽  
HaRim Jung ◽  
Ung-Mo Kim
Keyword(s):  
Efficient Algorithm ◽  
Road Network ◽  
Euclidean Distance ◽  
Optimal Location ◽  
External Memory ◽  
Distance Metric ◽  
Network Database ◽  
Road Network Database

Given a set of positive-weighted points and a query rectangler(specified by a client) of given extents, the goal of a maximizing range sum (MaxRS) query is to find the optimal location ofrsuch that the total weights of all the points covered byrare maximized. All existing methods for processing MaxRS queries assume the Euclidean distance metric. In many location-based applications, however, the motion of a client may be constrained by an underlying (spatial) road network; that is, the client cannot move freely in space. This paper addresses the problem of processing MaxRS queries in a road network. We propose the external-memory algorithm that is suited for a large road network database. In addition, in contrast to the existing methods, which retrieve only one optimal location, our proposed algorithm retrieves all the possible optimal locations. Through simulations, we evaluate the performance of the proposed algorithm.

Inductive graph invariants and approximation algorithms

2021 ◽  
Author(s):  
C. R. Subramanian
Keyword(s):  
Optimization Problems ◽  
Independence Number ◽  
Optimal Solution ◽  
Maximum Size ◽  
Chordal Graphs ◽  
Minimum Size ◽  
Optimal Size ◽  
Induced Subgraph ◽  
Special Cases ◽  
Optimal Formula

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

scholarly journals Performance Improvement on Nonorthogonal Multiple Access without CSIT

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Wenlong Xia ◽  
Yuanping Zhou ◽  
Qingdang Meng
Keyword(s):  
Performance Improvement ◽  
Multiple Access ◽  
Euclidean Distance ◽  
Computational Cost ◽  
Optimal Solution ◽  
The Novel ◽  
Channel State ◽  
State Information ◽  
Virtual Channels ◽  
Constrained Capacity

In this paper, a downlink virtual-channel-optimization nonorthogonal multiple access (VNOMA) without channel state information at the transmitter (CSIT) is proposed. The novel idea is to construct multiple complex virtual channels by jointly adjusting the amplitudes and phases to maximize the minimum Euclidean distance (MED) among the superposed constellation points. The optimal solution is derived in the absence of CSIT. Considering practical communications with finite input constellations in which symbols are uniformly distributed, we resort to the sum constellation constrained capacity (CCC) to evaluate the performance. For MED criterion, the maximum likelihood (ML) decoder is expected at the receiver. To decrease the computational cost, we propose a reduced-complexity bitwise ML (RBML) decoder. Experimental results are presented to validate the superior of our proposed scheme.

scholarly journals Adaptive Scatter Search to Solve the Minimum Connected Dominating Set Problem for Efficient Management of Wireless Networks

Algorithms ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 35 ◽  
Author(s):  
Abdel-Rahman Hedar ◽  
Shada N. Abdulaziz ◽  
Adel A. Sewisy ◽  
Gamal A. El-Sayed
Keyword(s):  
Fitness Function ◽  
Low Cost ◽  
Dominating Set ◽  
Scatter Search ◽  
Optimal Solution ◽  
Connected Dominating Set ◽  
Solution Quality ◽  
Efficient Management ◽  
Minimum Connected Dominating Set ◽  
Test Graph

An efficient routing using a virtual backbone (VB) network is one of the most significant improvements in the wireless sensor network (WSN). One promising method for selecting this subset of network nodes is by finding the minimum connected dominating set (MCDS), where the searching space for finding a route is restricted to nodes in this MCDS. Thus, finding MCDS in a WSN provides a flexible low-cost solution for the problem of event monitoring, particularly in places with limited or dangerous access to humans as is the case for most WSN deployments. In this paper, we proposed an adaptive scatter search (ASS-MCDS) algorithm that finds the near-optimal solution to this problem. The proposed method invokes a composite fitness function that aims to maximize the solution coverness and connectivity and minimize its cardinality. Moreover, the ASS-MCDS methods modified the scatter search framework through new local search and solution update procedures that maintain the search objectives. We tested the performance of our proposed algorithm using different benchmark-test-graph sets available in the literature. Experiments results show that our proposed algorithm gave good results in terms of solution quality.

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