Square Grid Points Coveraged by Connected Sources with Coverage Radius of One on a Two-Dimensional Grid

TL;DR

This paper introduces the Square Grid Points Coverage problem in a 2D grid, proves its NP-completeness, and proposes an approximation algorithm with practical extensions for coverage with mobile or static sources.

Contribution

It defines the SGPC problem, proves its NP-completeness, and presents the ASGC approximation algorithm with analysis and potential extensions.

Findings

SGPC is NP-complete.

ASGC achieves a specific approximation ratio.

Coverage can be achieved in eight steps with movement or instantly with increased radius.

Abstract

We take some parts of a theoretical mobility model in a two-dimension grid proposed by Greenlaw and Kantabutra to be our model. The model has eight necessary factors that we commonly use in a mobile wireless network: sources or wireless signal providers, the directions that a source can move, users or mobile devices, the given directions which define a user's movement, the given directions which define a source's movement, source's velocity, source's coverage, and obstacles. However, we include only the sources, source's coverage, and the obstacles in our model. We define Square Grid Points Coverage (SGPC) problem to minimize number of sources with coverage radius of one to cover a square grid point size of p with the restriction that all the sources must be communicable and proof that SGPC is in NP-complete class. We also give an Approx-Square-Grid-Coverage (ASGC) algorithm to compute the approximate solution of SGPC. ASGC uses the rule that any number can be obtained from the addition of 3, 4 and 5 and then combines 3-gadgets, 4-gadgets and 5-gadgets to specify the position of sources to cover a square grid point size of p. We find that the algorithm achieves an approximation ratio of . Moreover, we state about the extension usage of our algorithm and show some examples. We show that if we use ASGC on a square grid size of p and if sources can be moved, the area under the square grid can be covered in eight-time-steps movement. We also prove that if we extend our source coverage radius to 1.59, without any movement the area under the square gird will also be covered. Further studies are also discussed and a list of some tentative problems is given in the conclusion.