Adaptive Algorithm for Stochastic Connected Dominating Set

The problem of finding a minimum cardinality/weight connected dominating set (CDS) of a given graph has been studied extensively, because of its wide applications in wireless sensor networks (WSNs). Existing studies typically assume that the underlying network structure is fixed and preknown. However, given the inherent instability of mobile wireless devices, the network structure may be considered a random variable. Furthermore, determining the state of a node (active or inactive) often requires probing its local neighborhood. This motivates us to study the stochastic connected dominating set problem whose goal is to identify a connected dominating set within the graph comprised of active nodes while minimizing the probing cost. In this paper, we study the unweighted stochastic CDS problem, and present an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left ({{\frac {1}{\delta }(H (\Delta -1)+1)+1}}\right)$</tex-math> </inline-formula>-approximation algorithm in expectation, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H(\gamma)=\sum _{i=1}^{\gamma }1/i$</tex-math> </inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma $</tex-math> </inline-formula>th Harmonic number, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta $</tex-math> </inline-formula> is the maximum degree of the graph, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $</tex-math> </inline-formula> is the minimum probability that a node is active.