Sensors typically use wireless transmitters to communicate with each other. However, sensors may be located in a way that they cannot even form a connected network (e.g, due to failures of some sensors, or loss of battery power). In this paper we consider the problem of adding the smallest number of additional (relay) nodes so that the induced communication graph is 2-connected 1. The problem is NP-hard. In this paper we develop O(1)-approximation algorithms that find close to optimal solutions in time O((kn) 2) for achieving k-edge connectivity of n nodes. The worst case approximation guarantee is 10, but the algorithm produces solutions that are far better than this bound suggests. We also consider extensions to higher dimensions, and the scheme that we develop for points in the plane, yields a bound of 2dMST where dMST is the maximum degree of a minimum-degree Minimum Spanning Tree in d dimensions using Euclidean metrics. In addition, our methods extend with the same approximation guarantees to a generalization when the locations of relays are required to avoid certain polygonal regions (obstacles). We also prove that if the sensors are uniformly and identically distributed in a unit square, the expected number of relay nodes required goes to zero as the number of sensors goes to infinity.

We propose a new heuristic for deterministic deployment of wireless sensor networks when 1-connectivity and minimum cost are the two competing objectives. Given a set of data sources and a base station, our aim is to introduce the minimum number of relays to the network so that every sensor is connected to the base station via some multihop path. We assume that the data sources and base station lie in a plane, and that every sensor and relay has the same fixed communication radius. Our heuristic is based on the GEOSTEINER algorithms for the Steiner minimal tree problem, and proves to be much more accurate than the current best heuristics for the 1-connected deployment problem, especially in the case of sparse data source distributions.