Why are 3D vectors in C++ larger in RAM than a 1D vector

Question

I've discovered by accident that an STL vector defined as follows:

vector < float > test;
test.resize(10000 * 10000 * 5);

Uses up significant less space in RAM than the following definition:

std::vector<std::vector<std::vector< float > > > test;

test.resize(10000);
for(int i = 0;i <  10000;i++)
{
test[i].resize(10000);
for(int j = 0;j <  10000;j++)
{
test[i][j].resize(5);
}
}

The linear vector method (top one) uses the correct amount of RAM (2Gb) as would be calculated by hand. So my question is, why does a 3D vector use up way more RAM than a linear one, I found it was significantly more in this example (about 4Gb).

Answer 1

In the former case you have:

sizeof(vector<float>)                  // outermost vector
  + 10000 * 10000 * 5 * sizeof(float)  // xyz space

In the latter you have:

sizeof(vector<vector<vector<float>>>)      // outermost vector
  + 10000 * sizeof(vector<vector<float>>)  // x axis
  + 10000 * 10000 * sizeof(vector<float>)  // xy plane
  + 10000 * 10000 * 5 * sizeof<float>      // xyz space

The typical value for sizeof(vector<T>) for any T is 3 * sizeof(T*), which is also, I believe, the minimal value allowed by the standard—capacity must be distinct from size because reserve() must change the value of capacity() but not of size().