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In this paper, we consider an energy harvesting sensor network where sensors are powered by reusable energy such as solar energy, wind energy, and so on, from their surroundings. We first formulate a novel monitoring quality maximization problem that aims to maximize the quality, rather than the quantity, of collected data, by incorporating spatial data correlation among sensors. An optimization framework consisting of dynamic rate weight assignment, fair data rate allocation, and flow routing for the problem is proposed. To fairly allocate sensors with optimal data rates and efficiently route sensing data to the sink, we then introduce a weighted, fair data rate allocation and flow routing problem, subject to energy budgets of sensors. Unlike the most existing work that formulated the similar problem as a linear programming (LP) and solved the LP, we develop fast approximation algorithms with provable approximation ratios through exploiting the combinatorial property of the problem. A distributed implementation of the proposed algorithm is also developed. The key ingredients in the design of algorithms include a dynamic rate weight assignment and a reduction technique to reduce the problem to a special maximum weighted concurrent flow problem, where all source nodes share the common destination. We finally conduct extensive experiments by simulation to evaluate the performance of the proposed algorithm. The experimental results demonstrate that the proposed algorithm is very promising, and the solution to the weighted, fair data rate allocation and flow routing problem is fractional of the optimum.