Wireless communication has become omnipresent in the world and enables users to have an unprecedented ability to communicate any time and any place. In this article, we propose a mobility model for studying wireless communication. The model incorporates elements such as users, access points, and obstacles so that it faithfully mimics the real environment. Interesting problems that have practical applications are posed and solved. More specifically, we study the complexity of three problems in a grid. The source reachability problem (SRP) models a situation in which we want to determine whether two access points can communicate at a certain time in a mobile environment. When users are involved in this situation, we call this problem the user communication problem (UCP). We show that SRP can be solved in O(max{d , t }m 2 ) time, where d is the number of obstacles, t is the time bound in the statement of the problem, and m is the number of access points; we show that UCP can be solved in O(max{d , t }m 4 ) time. The third problem called the user communication, limited source access problem (UCLSAP) studies a situation where we want to determine whether two users can communicate uninterruptedly during the duration of the model while considering battery-time limits of the access points. In contrast to the first two problems, we demonstrate that UCLSAP is intractable, unless P = NP . In conclusion, we briefly discuss the extension of our model to three dimensions and provide a list of open problems.
We take some parts of a theoretical mobility model in a two-dimension grid proposed by Greenlaw and
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 NPcomplete 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
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