Sparse Prefix Sums.

The prefix sum approach is a powerful technique to answer range-sum queries over multi-dimensional arrays in constant time by requiring only a few look-ups in an array of precomputed prefix sums. In this paper, we propose the sparse prefix sum approach that is based on relative prefix sums and exploits sparsity in the data to vastly reduce the storage costs for the prefix sums. The proposed approach has desirable theoretical properties and works well in practice. It is the first approach achieving constant query time with sub-linear update costs and storage costs for range-sum queries over sparse low-dimensional arrays. Experiments on real-world data sets show that the approach reduces storage costs by an order of magnitude with only a small overhead in query time, thus preserving microsecond-fast query answering.