Discovering similar objects in a social network has many interesting issues. Here, we present ASCOS, an Asymmetric Structure COntext Similarity measure that captures the similarity scores among any pairs of nodes in a network. The definition of ASCOS is similar to that of the well-known SimRank since both define score values recursively. However, we show that ASCOS outputs a more complete similarity score than SimRank because SimRank (and several of its variations, such as PRank and SimFusion) on average ignores half paths between nodes during calculation. To make ASCOS tractable in both computation time and memory usage, we propose two variations of ASCOS: a low rank approximation based approach and an iterative solver Gauss-Seidel for linear equations. When the target network is sparse, the run time and the required computing space of these variations are smaller than computing SimRank and ASCOS directly. In addition, the iterative solver divides the original network into several independent sub-systems so that a multi-core server or a distributed computing environment, such as MapReduce, can efficiently solve the problem. We compare the performance of ASCOS with other global structure based similarity measures, including SimRank, Katz, and LHN. The experimental results based on user evaluation suggest that ASCOS gives better results than other measures. In addition, the asymmetric property has the potential to identify the hierarchical structure of a network. Finally, variations of ASCOS (including one distributed variation) can also reduce computation both in space and time.