Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation such as the Congested Clique model and the Massively Parallel Computation (MPC) model, as well as in related big data models such as streaming and semi-streaming. We provide algorithms for constructing the well-known Thorup-Zwick distance oracle (or distance sketches) in these different computational models and discuss the tradeoffs between various complexity measures such as space, time, stretch, and message complexity. One key component we use to construct distance sketches are the hopsets of Elkin and Neiman 2016. In particular, we show that these hopsets can be constructed efficiently in the MPC model, even in the very low memory setting. This result has additional applications such as the first subpolynomial algorithm for approximate single-source shortest paths for weighted graphs in the low memory MPC setting.
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