Heapbuilding

Heapbuilding is the process of transforming an unordered collection into a heap data structure, typically a binary heap, so that the resulting structure satisfies the heap property: in a max-heap, every node is greater than or equal to its children; in a min-heap, every node is less than or equal to its children. Heapbuilding is a common step when preparing data for priority queues, heapsort, or related algorithms that require efficient access to the extreme value.

In an array-based binary heap, the structure is a complete binary tree stored in an array. For 0-based indexing, the left child of index i is at 2i + 1, the right child at 2i + 2, and the parent at floor((i - 1) / 2). The heap property ensures that the root contains the maximum (for a max-heap) or minimum (for a min-heap) value, with each subtree also satisfying the property.

The standard method to build a heap efficiently is the bottom-up approach known as heapify or Floyd’s algorithm. Starting from the last non-leaf node, floor(n / 2) - 1, and moving upward to the root, each node is sifted down to its proper position. This process runs in linear time, O(n), and can be done in place with O(1) extra space. By contrast, inserting elements one by one into a heap generally requires O(n log n) time.

Applications of heapbuilding include constructing priority queues, performing heapsort, and supporting various graph and scheduling algorithms. Variants such as d-ary heaps or Fibonacci heaps offer different performance trade-offs, but the core construction principle—reorganizing elements to satisfy the heap property—remains central.