08-heaps

Min-Heap Insert (Sift Up)

Insert one value into a min-heap and restore the parent-child order by sifting upward.

Algorithm

Steps

  1. Store the heap in an array.
  2. Compare parent and child indexes instead of building explicit tree nodes.
  3. Swap only when the heap order is violated.
  4. Print the deterministic final heap state for replay comparison.

Complexity

  • Time: O(log n)
  • Space: O(1) extra
sift up A new value starts at the end of the array and swaps with its parent while it is smaller.

Visual walkthrough

Python DSA Implementation

basic.py 
def list_string(values):
    return "[" + ", ".join(str(v) for v in values) + "]"

def heap_insert(heap, value):
    heap.append(value)
    child = len(heap) - 1
    while child > 0:
        parent = (child - 1) // 2
        if heap[parent] <= heap[child]:
            break
        heap[parent], heap[child] = heap[child], heap[parent]
        child = parent

def heap_pop(heap):
    smallest = heap[0]
    heap[0] = heap.pop()
    parent = 0
    while True:
        left = parent * 2 + 1
        right = left + 1
        if left >= len(heap):
            break
        child = left
        if right < len(heap) and heap[right] < heap[left]:
            child = right
        if heap[parent] <= heap[child]:
            break
        heap[parent], heap[child] = heap[child], heap[parent]
        parent = child
    return smallest
heap = [2, 4, 7, 9, 6]
heap_insert(heap, 1)
print(list_string(heap))

The heap is still an array, but the tree view makes parent-child swaps visible. The labels use the pinned replay states from the lesson.

Step 1 - Append 1

The new value 1 starts at index 5 and compares with its parent value 7.

Array state [2, 4, 7, 9, 6, 1] as a heap-shaped tree.2i04i17parent9i36i41new

Step 2 - Swap with 7

Because 1 is smaller than 7, those array slots swap.

After the first sift-up swap: [2, 4, 1, 9, 6, 7].2parent4i11child9i36i47i5

Step 3 - Swap with 2

1 is also smaller than 2, so it moves to the root and the heap order is restored.

Final heap after insert: [1, 4, 2, 9, 6, 7].1root4i12i29i36i47i5

Output

[1, 4, 2, 9, 6, 7]

Implementation notes

  • Python stores the heap in one mutable list, so heap.append(value) extends the same backing object and places the new value at len(heap) - 1.
  • This is a min-heap implementation: the sift-up loop computes parent = (child - 1) // 2 and stops when heap[parent] <= heap[child] or when the child reaches index 0.
  • Swaps use Python tuple assignment, heap[parent], heap[child] = heap[child], heap[parent], mutating list slots in place without allocating a new heap list.
  • The replay-visible states follow the append and two swaps: [2, 4, 7, 9, 6, 1], [2, 4, 1, 9, 6, 7], then [1, 4, 2, 9, 6, 7]. Python manages the heap list while it is referenced; append may resize its internal storage, and swap temporaries are runtime-managed while they exist.