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Attention operator has been widely used as a basic brick in visual understanding since it provides some flexibility through its adjustable kernels. However, this operator suffers from inherent limitations: (1) the attention kernel is not discriminative enough, resulting in high redundancy, and (2) the complexity in computation and memory is quadratic in the sequence length. In this paper, we propose a novel attention operator, called Lightweight Structure-aware Attention (LiSA), which has a better representation power with log-linear complexity. Our operator transforms the attention kernels to be more discriminative by learning structural patterns. These structural patterns are encoded by exploiting a set of relative position embeddings (RPEs) as multiplicative weights, thereby improving the representation power of the attention kernels. Additionally, the RPEs are approximated to obtain log-linear complexity. Our experiments and analyses demonstrate that the proposed operator outperforms self-attention and other existing operators, achieving state-of-the-art results on ImageNet-1K and other downstream tasks such as video action recognition on Kinetics-400, object detection \& instance segmentation on COCO, and semantic segmentation on ADE-20K.