Cannot compile the following code,

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int triangle;
Vector3d vertex;
set<pair<int, Vector3d>> tri_set;
tri_set.insert(make_pair(triangle, vertex));


After insertion, I may need to get the index of the matching pair. This should be performed really fast and I have no clue right now.

CXXFLAGS = -std=c++1y -O3 -ffast-math -fno-cx-limited-range -funroll-loops --param max-unroll-times=2 -march=native
gcc version: 8.3.0 (Debian 8.3.0-6)
eigen version: 3.3.7-1

All the pairs are added once. It is the searching of a pair that is performed millions of times.

A fast searching method is also I am looking forward...

If you can allow some more up-front calculations, then after creating the set:

copy it into a vector
sort it (std::sort)
then lookup using binary search, http://www.cplusplus.com/reference/algorithm/binary_search/