For solving the All Pair Similarity Search (APSS) problem efficiently, this paper provides a maximum dimension partitioning approach to effectively filter non-similar pairs in an early stage. At first, for each data point, the dimension with the maximum value is used to decide the corresponding segment of data partition. An adjusting method is designed to balance the number of elements in each data segment. The similar pairs consist of inter-segment similar pairs and intra-segment similar pairs, where most effort of computing APSS comes from the computation of finding inter-segment similar pairs. For speeding up the computation, a pilot-vector is used to represent each segment for estimating the upper bound of similarity between each segment pair. Only the segment pairs, whose upper bounds of similarity are larger than the given similarity threshold, need to generate the inter-segment data pairs as candidates. Moreover, based on the proposed partitioning method, we designed a MapReduce framework to solve the APSS problem in parallel. The performance evaluation results show the proposed method provides better pruning effectiveness on non-similar data pairs than the related works. Moreover, the proposed partition-based method can properly fit into the MapReduce programming scheme to effectively reduce the response time of solving the APSS problem.