Fast algorithm for repeated calculation of percentile?

Question

In an algorithm I have to calculate the 75th percentile of a data set whenever I add a value. Right now I am doing this:

1. Get value x
2. Insert x in an already sorted array at the back
3. swap x down until the array is sorted
4. Read the element at position array[array.size * 3/4]

Point 3 is O(n), and the rest is O(1), but this is still quite slow, especially if the array gets larger. Is there any way to optimize this?

UPDATE

Thanks Nikita! Since I am using C++ this is the solution easiest to implement. Here is the code:

template<class T>
class IterativePercentile {
public:
  /// Percentile has to be in range [0, 1(
  IterativePercentile(double percentile)
    : _percentile(percentile)
  { }

  // Adds a number in O(log(n))
  void add(const T& x) {
    if (_lower.empty() || x <= _lower.front()) {
      _lower.push_back(x);
      std::push_heap(_lower.begin(), _lower.end(), std::less<T>());
    } else {
      _upper.push_back(x);
      std::push_heap(_upper.begin(), _upper.end(), std::greater<T>());
    }

    unsigned size_lower = (unsigned)((_lower.size() + _upper.size()) * _percentile) + 1;
    if (_lower.size() > size_lower) {
      // lower to upper
      std::pop_heap(_lower.begin(), _lower.end(), std::less<T>());
      _upper.push_back(_lower.back());
      std::push_heap(_upper.begin(), _upper.end(), std::greater<T>());
      _lower.pop_back();
    } else if (_lower.size() < size_lower) {
      // upper to lower
      std::pop_heap(_upper.begin(), _upper.end(), std::greater<T>());
      _lower.push_back(_upper.back());
      std::push_heap(_lower.begin(), _lower.end(), std::less<T>());
      _upper.pop_back();
    }            
  }

  /// Access the percentile in O(1)
  const T& get() const {
    return _lower.front();
  }

  void clear() {
    _lower.clear();
    _upper.clear();
  }

private:
  double _percentile;
  std::vector<T> _lower;
  std::vector<T> _upper;
};

Answer 1

You can use binary search to do find the correct position in O(log n). However, shifting the array up is still O(n).

Answer 2

You can do it with two heaps. Not sure if there's a less 'contrived' solution, but this one provides O(logn) time complexity and heaps are also included in standard libraries of most programming languages.

First heap (heap A) contains smallest 75% elements, another heap (heap B) - the rest (biggest 25%). First one has biggest element on the top, second one - smallest.

1. Adding element.

See if new element x is <= max(A). If it is, add it to heap A, otherwise - to heap B.
Now, if we added x to heap A and it became too big (holds more than 75% of elements), we need to remove biggest element from A (O(logn)) and add it to heap B (also O(logn)).
Similar if heap B became too big.

1. Finding "0.75 median"

Just take the largest element from A (or smallest from B). Requires O(logn) or O(1) time, depending on heap implementation.

edit
As Dolphin noted, we need to specify precisely how big each heap should be for every n (if we want precise answer). For example, if size(A) = floor(n * 0.75) and size(B) is the rest, then, for every n > 0, array[array.size * 3/4] = min(B).

Answer 3

A simple Order Statistics Tree is enough for this.

A balanced version of this tree supports O(logn) time insert/delete and access by Rank. So you not only get the 75% percentile, but also the 66% or 50% or whatever you need without having to change your code.

If you access the 75% percentile frequently, but only insert less frequently, you can always cache the 75% percentile element during an insert/delete operation.

Most standard implementations (like Java's TreeMap) are order statistic trees.