Existing work on placing additional relay nodes in wireless sensor networks to improve network connectivity typically assumes homogeneous wireless sensor nodes with an identical transmission radius. In contrast, this paper addresses the problem of deploying relay nodes to provide fault tolerance with higher network connectivity in heterogeneous wireless sensor networks, where sensor nodes possess different transmission radii. Depending on the level of desired fault tolerance, such problems can be categorized as: 1) full fault-tolerant relay node placement, which aims to deploy a minimum number of relay nodes to establish k(k >= 1) vertex-disjoint paths between every pair of sensor and/or relay nodes and 2) partial fault-tolerant relay node placement, which aims to deploy a minimum number of relay nodes to establish k(k >= 1) vertex-disjoint paths only between every pair of sensor nodes. Due to the different transmission radii of sensor nodes, these problems are further complicated by the existence of two different kinds of communication paths in heterogeneous wireless sensor networks, namely, two-way paths, along which wireless communications exist in both directions; and one-way paths, along which wireless communications exist in only one direction. Assuming that sensor nodes have different transmission radii, while relay nodes use the same transmission radius, this paper comprehensively analyzes the range of problems introduced by the different levels of fault tolerance (full or partial) coupled with the different types of path (one-way or two-way). Since each of these problems is NP-hard, we develop O(sigma k(2))-approximation algorithms for both one-way and two-way partial fault-tolerant relay node placement, as well as O(sigma k(3))-approximation algorithms for both one-way and two-way full fault-tolerant relay node placement (sigma is the best performance ratio of existing approximation algorithms for finding a minimum k-vertex connected spanning graph). To facilitate the applications in higher dimensions, we also extend these algorithms and derive their performance ratios in d-dimensional heterogeneous wireless sensor networks (d >= 3). Finally, heuristic implementations of these algorithms are evaluated via QualNet simulations.
