We consider wireless sensor networks secured by the heterogeneous random key predistribution scheme under an on off channel model. The heterogeneous random key predistribution scheme considers the case when the network includes sensor nodes with varying levels of resources, features, or connectivity requirements; e.g., regular nodes vs. cluster heads, but does not incorporate the fact that wireless channel are unreliable. To capture the unreliability of the wireless medium, we use an on off channel model; wherein, each wireless channel is either on (with probability @math ) or off (with probability @math ) independently. We present conditions (in the form of zero-one laws) on how to scale the parameters of the network model so that with high probability the network is @math -connected, i.e., the network remains connected even if any @math nodes fail or leave the network. We also present numerical results to support these conditions in the finite-node regime.
Our paper completes the analysis that we started in @cite_24 concerning the minimum node degree of the intersection model @math . There, we presented conditions on how to scale the parameters of the intersection model @math so that the minimum node degree is no less than @math , with high probability when the number of nodes @math gets large. It is clear that a graph can not be @math -connected if its minimum node degree is less than @math . Thus, we readily obtain the zero-law of Theorem by virtue of the zero-law of the minimum node degree being no less than @math [Theorem 3.1] EletrebyISIT . Our paper completes the analysis of @cite_24 by means of establishing the one-law of @math -connectivity; thus, obtaining a fuller understanding of the properties of the intersection model @math . In particular, it was conjectured in [Conjecture 3.2] EletrebyISIT , that under some additional conditions, the zero-one laws for @math -connectivity would resemble those for the minimum node degree being no less than @math . In our paper, we prove that this conjecture is correct and provide the extra conditions needed for it to hold.
@cite_24
2005079131 2161190897 2079430060 2174245573
In this paper, we consider a wireless network with unreliable links and investigate its minimum node degree and k- connectivity. In such network, n nodes are uniformly distributed in a region, and edges are established for any two nodes within a certain distance and with a probabilistically active link in between. On a torus topology, we present a zero-one law for the property of minimum degree being at least k, leading to a zero-law for k-connectivity and an intermediate result towards a one-law. On a square with boundary effect, we establish a one- law for minimum degree. Our results are derived from rigorous analysis and also confirmed by the simulation, and they provide guidelines for the design of wireless networks. Index Terms—Connectivity, Erdos-Renyi graph, geometric graph, graph intersection, minimum node degree, random graph, unreliable link, wireless network, zero-one law. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and @math -out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node @math in the solution is at most @math , where @math is the degree upper bound on @math . For undirected graphs and element-connectivity requirements with maximum connectivity requirement @math , our algorithm computes a solution that is a @math -approximation on the cost, and the degree of each node @math in the solution is at most @math . These ratios improve the previous @math -approximation on the cost and @math -approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k...