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:
x
x
in an already sorted array at the backx
down until the array is sortedarray[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;
};
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.
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.
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)
.