Design Most Recently Used Queue — C++

// Time:  ctor:  O(n + m), m is the max number of calls
// Space: fetch: O(log(n + m))

class MRUQueue {
public:
    MRUQueue(int n)
     : bit_{n}
     , curr_{n} {
        for (int i = 0; i < n; ++i) {
            lookup_[i] = i + 1;
        }
    }
    
    int fetch(int k) {
        int pos  = bit_.binary_lift(k);
        int val = lookup_[pos];
        lookup_.erase(pos);
        bit_.add(pos, -1);
        bit_.add(curr_, 1);
        lookup_[curr_++] = val;
        return val;
    }

private:
    class BIT {
    public:
        static const int MAX_CALLS = 2000;

        BIT(int n) : bit_(n + MAX_CALLS + 1) {  // 0-indexed
            for (int i = 1; i < size(bit_); ++i) {
                bit_[i] = ((i - 1 < n) ? 1 : 0) + bit_[i - 1];
            }
            for (int i = size(bit_) - 1; i >= 1; --i) {
                int last_i = i - lower_bit(i);
                bit_[i] -= bit_[last_i];
            }
        }
        
        void add(int i, int val) {
            ++i;
            for (; i < size(bit_); i += lower_bit(i)) {
                bit_[i] += val;
            }
        }

        int query(int i) const {
            ++i;
            int total = 0;
            for (; i > 0; i -= lower_bit(i)) {
                total += bit_[i];
            }
            return total;
        }
        
        int binary_lift(int k) const {
            int total = 0;
            int pos = 0;
            for (int i = floor_log2_x(size(bit_) - 1); i >= 0; --i) {
                if (pos + (1 << i) < size(bit_) && !(total + bit_[pos + (1 << i)] >= k)) {
                    total += bit_[pos + (1 << i)];
                    pos += (1 << i);
                }
            }
            return (pos + 1) - 1;
        }
    
    private:
        int lower_bit(int i) const {
            return i & -i;
        }
        
        int floor_log2_x(int x) const {
            return 8 * sizeof(int) - __builtin_clz(x) - 1;
        };
        
        vector<int> bit_;
    };
    
    BIT bit_;
    unordered_map<int, int> lookup_;
    int curr_;
};

// Time:  ctor:  O(n)
// Space: fetch: O(sqrt(n))
// sqrt decomposition solution
class MRUQueue2 {
public:
    MRUQueue2(int n)
     : buckets_(ceil(sqrt(n))) {
        for (int i = 0; i < n; ++i) {
            buckets_[i / size(buckets_)].emplace_back(i + 1);
        }
    }
    
    int fetch(int k) {
        --k;
        int left = k / size(buckets_);
        int idx = k % size(buckets_);
        auto cit = cbegin(buckets_[left]);
        advance(cit, idx);
        int val = *cit;
        buckets_[left].erase(cit);
        buckets_.back().emplace_back(val);
        for (int i = size(buckets_) - 2; i >= left; --i) {
            int x = buckets_[i + 1].front();
            buckets_[i + 1].pop_front();
            buckets_[i].emplace_back(x);
        }
        return val;
    }
    
private:
    vector<list<int>> buckets_;
};