Maximum Covered Path within a Region * Manuel Abellanas † Antonio Bajuelos ‡ Inˆ es Matos ‡ Abstract Let S be a set of sensors with a given transmission range r ∈ R + . The objective of this paper is to calculate the maximum coverage of a path between two points p and q on a region R. This problem is solved for four cases: R is the line segment pq, R is a planar graph and p and q are two of its nodes, R is a polygonal region containing p and q and, to conclude, R is the whole plane. A maximum covered path between p and q is also found on the second case. 1 Introduction and Related Work Problems related to wireless ad-hoc networks (or just sensor networks) have emerged in the last few years given the fast development of technology. Sensor networks are used to solve a great diversity of problems that range from battlefield monitoring to weather detection, museums’ security and even wildlife protection [5, 12]. The problems introduced in the following are related to coverage. Since each sensor can be located anywhere within a specific region, coverage measures the quality of this placement. Let R be a region that is surveilled by a given wireless sensor network. Following the approach to coverage studied by Huang and Tseng [8], the objective of the next set of problems is to decide whether a path within R is covered. If so, find the path’s maximum coverage provided by such network. Coverage can be looked at from two opposite perspectives. In the worst-case coverage there is an attempt to locate the regions of R that are hidden from the sensors, that is, not surveilled. The best-case coverage is characterised as an attempt to locate the areas that are highly surveilled, thus identifying the “best” surveilled regions of R. The objective above clearly is included in this second case. s 2 s 4 s 3 s 1 q 1 r q 2 s 5 q 3 Figure 1: Set S with range r is represented by dots. Point q 1 has MC S (q 1 ) = 3 since it is a point of D(s 1 ,r) ∩ D(s 2 ,r) ∩ D(s 5 ,r); q 2 is 1-covered by S since it is only interior to D(s 2 ,r). Point q 3 has MC S (q 3 ) = 4 since it is interior to four disks. Let S be a set of n points on the plane that represent the location of n devices which are able to send or receive some sort of wireless signal, like sensors. Also suppose that the devices of S are homogeneous, that is, they all have the same power transmission range r ∈ R + which is fixed. Assuming * When this paper was finished, the third author was supported by a FCT fellowship, grant SFRH/BD/28652/2006 † Facultad de Inform´atica, Universidad Polit´ ecnica de Madrid, co-supported by Project Consolider Ingenio 2010 i-MATH C3-0159 and MICINN Project MTM2008-05043 mabellanas@fi.upm.es ‡ Departamento de Matem´ atica & CEOC, Universidade de Aveiro, supported by CEOC through Programa POCTI, FCT, co-financed by EC fund FEDER {leslie,ipmatos}@ua.pt