Does std::vector *have* to move objects when growing capacity? Or, can allocators “reallocate”?

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When std::vector<T> runs out of capacity it has to allocate a new block. You have correctly covered the reasons.

IMO it would make sense to augment the allocator interface. Two of us tried to for C++11 and we were unable to gain support for it: [1] [2]

I became convinced that in order to make this work, an additional C-level API would be needed. I failed in gaining support for that as well: [3]

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