Algorithm (Sorted Set)

1. Create a sorted set structure to store the positions of turned on bulbs. We denote this structure as $S$.
2. We insert into $S$ the positions of bulb in increasing order or days. After each insertion at day $d$, we query the higher and lower positions of the inserted element and calculate the corresponding distance $\ell_{\mathrm{lower}}$ and $\ell_{\mathrm{upper}}$; if any of the two is equal to $K+2$, we return $d.$ Otherwise, we return $-1$ as instructed.

Code (Cpp17)

class Solution {
 public:
  int kEmptySlots(vector<int>& bulbs, int K) {
    const auto n = size(bulbs);

    auto higher = [&](const std::set<int> &x, int val) -> optional<int> {
      auto it = x.upper_bound(val);
      if (it == std::end(x))
        return std::nullopt;
      else
        return *it;
    };

    auto lower = [&](const std::set<int> &x, int val) -> optional<int> {
      auto it = x.lower_bound(val);
      if (it == std::begin(x))
        return std::nullopt;
      else
        return *(--it);
    };

    const auto solution = [&] {
      struct state_t { std::set<int> prev; };

      for (auto [i, state] = pair{0, state_t{}}; i < n; ++i) {
        state.prev.insert(bulbs[i]);
        const auto h = higher(state.prev, bulbs[i]);
        const auto l = lower(state.prev, bulbs[i]);
        if (h and std::abs(*h - bulbs[i]) + 1 == K + 2)
          return i + 1;
        else if (l and std::abs(*l - bulbs[i]) + 1 == K + 2)
          return i + 1;
      }
      return -1;
    }();

    return solution;
  }
};