Smallest Number of Sensors for k-Covering

This paper presents some theoretical results on the smaller number N-k (a, b) of sensors to achieve k coverage for the rectangular area [0, a] x [0, b]. The first properties show the numbers N-k(a,b) are sub-additive and increasing on each variable. Based on these results, some lower and upper bounds for N-k(a,b) are introduced. The main result of the article proves that the minimal density of sensors to achieve k-coverage is lambda(k) <= k/2 improving a previous result of Ammari and Das [2]. Finally, the numbers N-1 (a, b) are tabled for some small values of a, b.