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

Ruby DSA Implementation

basic.rb 
def list_string(values) = "[#{values.join(', ')}]"
def heap_insert(heap, value)
  heap << value
  child = heap.length - 1
  while child > 0
    parent = (child - 1) / 2
    break if heap[parent] <= heap[child]
    heap[parent], heap[child] = heap[child], heap[parent]
    child = parent
  end
end
def heap_pop(heap)
  smallest = heap[0]
  heap[0] = heap.pop
  parent = 0
  loop do
    left = parent * 2 + 1
    right = left + 1
    break if left >= heap.length
    child = right < heap.length && heap[right] < heap[left] ? right : left
    break if heap[parent] <= heap[child]
    heap[parent], heap[child] = heap[child], heap[parent]
    parent = child
  end
  smallest
end
heap = [2, 4, 7, 9, 6]
heap_insert(heap, 1)
puts 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

Implementation notes

  • The heap is a Ruby Array laid out as a binary min-heap in level order.
  • heap_insert(heap, value) mutates the passed array; it does not return a new heap.
  • Insert uses heap << value, so the checked value 1 is appended after [2, 4, 7, 9, 6].
  • child = heap.length - 1 starts at the appended slot, and parent = (child - 1) / 2 uses Ruby integer division.
  • The loop runs while child > 0 and stops early with break if heap[parent] <= heap[child], which is the min-heap direction.
  • Swaps use Ruby parallel assignment: heap[parent], heap[child] = heap[child], heap[parent].
  • The trace shows [2, 4, 7, 9, 6, 1], then swaps 1 with 7, then swaps 1 with 2.
  • list_string(heap) formats the final array with join(', '), so puts prints [1, 4, 2, 9, 6, 7].

Output

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