Heap
The heap is a fundamental data structure. The commonly used priority queue is heap-based, and heap sort is also built on the heap. We previously implemented a heap from first principles; now let’s analyze the STL heap implementation. While the STL version is naturally more elaborate, the underlying principle is the same.
Heap Implementation
The STL provides a suite of heap algorithms. Since a heap is stored as an array, how do we check whether a given array is a valid heap? The implementation is straightforward: verify that every parent is greater than or equal to its children (max-heap property). is_heap has other overloads not shown here.
/**
* @brief Determines whether a range is a heap using comparison functor.
* @param __first Start of range.
* @param __last End of range.
* @param __comp Comparison functor to use.
* @return True if range is a heap, false otherwise.
* @ingroup heap_algorithms
*/
template<typename _RandomAccessIterator, typename _Compare>
inline bool is_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _Compare __comp) { // check if the array satisfies the heap property
// concept requirements
__glibcxx_function_requires(_RandomAccessIteratorConcept<_RandomAccessIterator>)
__glibcxx_requires_valid_range(__first, __last);
__glibcxx_requires_irreflexive_pred(__first, __last, __comp);
const auto __dist = std::distance(__first, __last);
typedef __decltype(__comp) _Cmp;
__gnu_cxx::__ops::_Iter_comp_iter<_Cmp> __cmp(_GLIBCXX_MOVE(__comp));
return std::__is_heap_until(__first, __dist, __cmp) == __dist;
}
template<typename _RandomAccessIterator, typename _Distance, typename _Compare>
_Distance __is_heap_until(_RandomAccessIterator __first, _Distance __n, _Compare& __comp) {
_Distance __parent = 0;
for (_Distance __child = 1; __child < __n; ++__child) { // straightforward: for a max-heap, parent >= child
if (__comp(__first + __parent, __first + __child))
return __child;
if ((__child & 1) == 0)
++__parent;
}
return __n;
}If an array does not satisfy the heap property, how do we transform it into a heap? make_heap does this. It starts from the last non-leaf node and sifts down, checking and swapping parent-child relationships all the way to the root.
/**
* @brief Construct a heap over a range.
* @param __first Start of heap.
* @param __last End of heap.
* @ingroup heap_algorithms
*
* This operation makes the elements in [__first,__last) into a heap.
*/
template<typename _RandomAccessIterator>
inline void make_heap(_RandomAccessIterator __first, _RandomAccessIterator __last) {
// concept requirements
__glibcxx_function_requires(_Mutable_RandomAccessIteratorConcept<_RandomAccessIterator>)
__glibcxx_function_requires(_LessThanComparableConcept<typename iterator_traits<_RandomAccessIterator>::value_type>)
__glibcxx_requires_valid_range(__first, __last);
__glibcxx_requires_irreflexive(__first, __last);
__gnu_cxx::__ops::_Iter_less_iter __comp;
std::__make_heap(__first, __last, __comp);
}
template<typename _RandomAccessIterator, typename _Compare>
void __make_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _Compare& __comp) {
typedef typename iterator_traits<_RandomAccessIterator>::value_type _ValueType;
typedef typename iterator_traits<_RandomAccessIterator>::difference_type _DistanceType;
if (__last - __first < 2)
return;
const _DistanceType __len = __last - __first;
_DistanceType __parent = (__len - 2) / 2;
while (true) {
_ValueType __value = _GLIBCXX_MOVE(*(__first + __parent));
std::__adjust_heap(__first, __parent, __len, _GLIBCXX_MOVE(__value), __comp);
if (__parent == 0)
return;
__parent--;
}
}
template<typename _RandomAccessIterator, typename _Distance, typename _Tp, typename _Compare>
void __adjust_heap(_RandomAccessIterator __first, _Distance __holeIndex, _Distance __len, _Tp __value, _Compare __comp) {
const _Distance __topIndex = __holeIndex;
_Distance __secondChild = __holeIndex;
while (__secondChild < (__len - 1) / 2) {
__secondChild = 2 * (__secondChild + 1);
if (__comp(__first + __secondChild, __first + (__secondChild - 1)))
__secondChild--;
*(__first + __holeIndex) = _GLIBCXX_MOVE(*(__first + __secondChild));
__holeIndex = __secondChild;
}
if ((__len & 1) == 0 && __secondChild == (__len - 2) / 2) {
__secondChild = 2 * (__secondChild + 1);
*(__first + __holeIndex) = _GLIBCXX_MOVE(*(__first + (__secondChild - 1)));
__holeIndex = __secondChild - 1;
}
__decltype(__gnu_cxx::__ops::__iter_comp_val(_GLIBCXX_MOVE(__comp)))
__cmp(_GLIBCXX_MOVE(__comp));
std::__push_heap(__first, __holeIndex, __topIndex, _GLIBCXX_MOVE(__value), __cmp);
}
template<typename _RandomAccessIterator, typename _Distance, typename _Tp, typename _Compare>
void __push_heap(_RandomAccessIterator __first, _Distance __holeIndex, _Distance __topIndex, _Tp __value, _Compare& __comp) {
_Distance __parent = (__holeIndex - 1) / 2;
while (__holeIndex > __topIndex && __comp(__first + __parent, __value)) {
*(__first + __holeIndex) = _GLIBCXX_MOVE(*(__first + __parent));
__holeIndex = __parent;
__parent = (__holeIndex - 1) / 2;
}
*(__first + __holeIndex) = _GLIBCXX_MOVE(__value);
}After adding an element to the heap, push_heap restores the heap property by sifting the new element upward until the heap invariant is satisfied.
/**
* @brief Push an element onto a heap.
* @param __first Start of heap.
* @param __last End of heap + element.
* @ingroup heap_algorithms
*
* This operation pushes the element at last-1 onto the valid heap
* over the range [__first,__last-1). After completion,
* [__first,__last) is a valid heap.
*/
template<typename _RandomAccessIterator>
inline void push_heap(_RandomAccessIterator __first, _RandomAccessIterator __last) {
typedef typename iterator_traits<_RandomAccessIterator>::value_type _ValueType;
typedef typename iterator_traits<_RandomAccessIterator>::difference_type _DistanceType;
// concept requirements
__glibcxx_function_requires(_Mutable_RandomAccessIteratorConcept<_RandomAccessIterator>)
__glibcxx_function_requires(_LessThanComparableConcept<_ValueType>)
__glibcxx_requires_valid_range(__first, __last);
__glibcxx_requires_irreflexive(__first, __last);
__glibcxx_requires_heap(__first, __last - 1);
__gnu_cxx::__ops::_Iter_less_val __comp;
_ValueType __value = _GLIBCXX_MOVE(*(__last - 1));
std::__push_heap(__first, _DistanceType((__last - __first) - 1), _DistanceType(0), _GLIBCXX_MOVE(__value), __comp);
}Popping the top element moves it to the end of the container, then sifts the replacement root downward to restore the heap property.
/**
* @brief Pop an element off a heap using comparison functor.
* @param __first Start of heap.
* @param __last End of heap.
* @param __comp Comparison functor to use.
* @ingroup heap_algorithms
*
* This operation pops the top of the heap. The elements __first
* and __last-1 are swapped and [__first,__last-1) is made into a
* heap. Comparisons are made using comp.
*/
template<typename _RandomAccessIterator, typename _Compare>
inline void pop_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _Compare __comp) {
// concept requirements
__glibcxx_function_requires(_Mutable_RandomAccessIteratorConcept<_RandomAccessIterator>)
__glibcxx_requires_valid_range(__first, __last);
__glibcxx_requires_irreflexive_pred(__first, __last, __comp);
__glibcxx_requires_non_empty_range(__first, __last);
__glibcxx_requires_heap_pred(__first, __last, __comp);
if (__last - __first > 1) {
typedef __decltype(__comp) _Cmp;
__gnu_cxx::__ops::_Iter_comp_iter<_Cmp> __cmp(_GLIBCXX_MOVE(__comp));
--__last;
std::__pop_heap(__first, __last, __last, __cmp);
}
}
template<typename _RandomAccessIterator, typename _Compare>
inline void __pop_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _RandomAccessIterator __result, _Compare& __comp) {
typedef typename iterator_traits<_RandomAccessIterator>::value_type _ValueType;
typedef typename iterator_traits<_RandomAccessIterator>::difference_type _DistanceType;
_ValueType __value = _GLIBCXX_MOVE(*__result);
*__result = _GLIBCXX_MOVE(*__first);
std::__adjust_heap(__first, _DistanceType(0), _DistanceType(__last - __first), _GLIBCXX_MOVE(__value), __comp);
}The STL also provides a heap sort implementation:
/**
* @brief Sort a heap using comparison functor.
* @param __first Start of heap.
* @param __last End of heap.
* @param __comp Comparison functor to use.
* @ingroup heap_algorithms
*
* This operation sorts the valid heap in the range [__first,__last).
* Comparisons are made using __comp.
*/
template<typename _RandomAccessIterator, typename _Compare>
inline void sort_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _Compare __comp) {
// concept requirements
__glibcxx_function_requires(_Mutable_RandomAccessIteratorConcept<_RandomAccessIterator>)
__glibcxx_requires_valid_range(__first, __last);
__glibcxx_requires_irreflexive_pred(__first, __last, __comp);
__glibcxx_requires_heap_pred(__first, __last, __comp);
typedef __decltype(__comp) _Cmp;
__gnu_cxx::__ops::_Iter_comp_iter<_Cmp> __cmp(_GLIBCXX_MOVE(__comp));
std::__sort_heap(__first, __last, __cmp);
}
template<typename _RandomAccessIterator, typename _Compare>
void __sort_heap(_RandomAccessIterator __first, _RandomAccessIterator __last, _Compare& __comp) {
while (__last - __first > 1) {
--__last;
std::__pop_heap(__first, __last, __last, __comp);
}
}