Having fetched all needed data, can update all particles in processors with high efficiency seen in O(N2) problem.
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Namely communicated data is re-used many times as needed by many particles in processor
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In practice have some 10,000 or more particles per processor
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In astrophysics, also need to construct tree and rebalance load at essentially each time step
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There is a straightforward parallel tree building algorithm
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These methods outperform O(N2) when more than a few thousand particles (unless very inhomogeneous)
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