kd-tree  has long been deemed unsuitable for exact nearest-neighbor search in high dimensional data. The theoretical guarantees and the empirical performance of kd-tree do not show significant improvements over brute-force nearest-neighbor search in moderate to high dimensions. kd-tree has been used relatively more successfully for approximate search  but lack theoretical guarantees. In the article, we build upon randomized-partition trees  to propose kd-tree based approximate search schemes with O(d log d + log n) query time for data sets with n points in d dimensions and rigorous theoretical guarantees on the search accuracy. We empirically validate the search accuracy and the query time guarantees of our proposed schemes, demonstrating the significantly improved scaling for same level of accuracy.