class Solution {
 public:
  double findMaxAverage(vector<int>& nums, int k) {
    constexpr double kErr = 1e-5;
    double l = *min_element(begin(nums), end(nums));
    double r = *max_element(begin(nums), end(nums));

    while (r - l > kErr) {
      const double m = (l + r) / 2;
      if (check(nums, k, m))
        l = m;
      else
        r = m;
    }

    return l;
  }

 private:
  // true if there's a subarray with length >= k and average sum >= m
  bool check(const vector<int>& nums, int k, double m) {
    double sum = 0;
    double prevSum = 0;
    double minPrevSum = 0;

    for (int i = 0; i < nums.size(); ++i) {
      // trick: -m for each num so that we can check if the sum of the
      // subarray >= 0
      sum += nums[i] - m;
      if (i >= k) {
        prevSum += nums[i - k] - m;
        minPrevSum = min(minPrevSum, prevSum);
      }
      // if sum - minPrevSum >= 0,
      // we know there's a subarray with length >= k and average sum >= m
      if (i + 1 >= k && sum >= minPrevSum)
        return true;
    }

    return false;
  };
};

class Solution {
  public double findMaxAverage(int[] nums, int k) {
    final double kErr = 1e-5;
    double l = (double) Arrays.stream(nums).min().getAsInt();
    double r = (double) Arrays.stream(nums).max().getAsInt();

    while (r - l > kErr) {
      final double m = (l + r) / 2;
      if (check(nums, k, m))
        l = m;
      else
        r = m;
    }

    return l;
  }

  // true if there's a subarray with length >= k and average sum >= m
  private boolean check(int[] nums, int k, double m) {
    double sum = 0;
    double prevSum = 0;
    double minPrevSum = 0;

    for (int i = 0; i < nums.length; ++i) {
      // trick: -m for each num so that we can check if the sum of the
      // subarray >= 0
      sum += nums[i] - m;
      if (i >= k) {
        prevSum += nums[i - k] - m;
        minPrevSum = Math.min(minPrevSum, prevSum);
      }
      // if sum - minPrevSum >= 0,
      // we know there's a subarray with length >= k and average sum >= m
      if (i + 1 >= k && sum >= minPrevSum)
        return true;
    }

    return false;
  }
}

class Solution:
  def findMaxAverage(self, nums: List[int], k: int) -> float:
    kErr = 1e-5
    l = min(nums)
    r = max(nums)

    # True if there's a subarray with length >= k and average sum >= m
    def check(m: float) -> bool:
      sum = 0
      prevSum = 0
      minPrevSum = 0

      for i, num in enumerate(nums):
        # trick: -m for each num so that we can check if the sum of the
        # subarray >= 0
        sum += num - m
        if i >= k:
          prevSum += nums[i - k] - m
          minPrevSum = min(minPrevSum, prevSum)
        # if sum - minPrevSum >= 0,
        # we know there's a subarray with length >= k and average sum >= m
        if i + 1 >= k and sum >= minPrevSum:
          return True

      return False

    while r - l > kErr:
      m = (l + r) / 2
      if check(m):
        l = m
      else:
        r = m

    return l