How to compose generators with STL algorithms

Here's how I would write the function in c++20, using range views and algorithms so that there isn't a separate container that stores the intermediate results:

double get_minimum_cost(auto const & inp)
{
  namespace rs = std::ranges;
  namespace rv = std::ranges::views;

  // for each i compute the minimum cost for all j's  
  auto min_cost_from_i = [&](auto i) 
  {

    auto costs_from_i = rv::iota(i + 1, inp.size())
                      | rv::transform([&](auto j) 
                        { 
                          return cost(inp[i], inp[j]); 
                        });

    return *rs::min_element(costs_from_i);
  };

  // compute min costs for all i's
  auto all_costs = rv::iota(0u, inp.size())
                 | rv::transform(min_cost_from_i);

  return *rs::min_element(all_costs);
}

Here's a demo.

Note that the solution doesn't compare the cost between same elements, since the cost function example you showed would have a trivial result of 0. For a cost function that doesn't return 0, you can adapt the solution to generate a range from i instead of i + 1. Also, if the cost function is not symmetric, make that range start from 0 instead of i.

Also, this function has UB if you call it with an empty range, so you should check for that as well.