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Given a set of objects and a query q, a point p is called the reverse k nearest neighbor (RkNN) of q if q is one of the k closest objects of p. In this paper, we introduce the concept of influence zone that is the area such that every point inside this area is the RkNN of q and every point outside this area is not the RkNN. The influence zone has several applications in location-based services, marketing and decision support systems. It can also be used to efficiently process RkNN queries. First, we present efficient algorithm to compute the influence zone. Then, based on the influence zone, we present efficient algorithms to process RkNN queries that significantly outperform existing best-known techniques for both the snapshot and continuous RkNN queries. We also present a detailed theoretical analysis to analyze the area of the influence zone and IO costs of our RkNN processing algorithms. Our experiments demonstrate the accuracy of our theoretical analysis. This paper is an extended version of our previous work (Cheema et al. in Proceedings of ICDE, pp. 577---588, 2011). We make the following new contributions in this extended version: (1) we conduct a rigorous complexity analysis and show that the complexity of one of our proposed algorithms in Cheema et al. (Proceedings of ICDE, pp. 577---588, 2011) can be reduced from O(m 2) to O( km) where m k is the number of objects used to compute the influence zone, (2) we show that our techniques can be applied to dimensionality higher than two, and (3) we present efficient techniques to handle data updates.