Heapify in logarithmic time using the C++ standard library

Question

I have a heap using std::make_heap:

std::vector v{1,2,3,5,9,20,3};
std::make_heap(v.begin(), v.end());

now I update

Answer 1

If we look closer at your statement:

now I disturb heap by changing one random element of heap.

For heapifying in O(log n) you can only directly "disturb" the back or the front of the vector (which corresponds somehow to inserting or deleting an element). In these cases, (re)heapification can be then achieved by means of the std::push_heap and std::pop_heap algorithms, which take logarithmic running time.

That is, the back:

v.back() = 35;
std::push_heap(v.begin(), v.end()); // heapify in O(log n)

or the front:

v.front() = 35;

// places the front at the back
std::pop_heap(v.begin(), v.end()); // O(log n)
// v.back() is now 35, but it does not belong to the heap anymore

// make the back belong to the heap again
std::push_heap(v.begin(), v.end()); // O(log n)

Otherwise you need to reheapify the whole vector with std::make_heap, which takes linear running time.

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Summary

It's not possible to modify an arbitrary element of the heap and achieve the heapification in logarithmic running time with the standard library (i.e., the function templates std::push_heap and std::pop_heap). However, you can always implement the heap's swim and sink operations by yourself in order to heapify in logarithmic running time.