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

Fortran DSA Implementation

basic.f90 
program main
  implicit none
  integer :: heap(16), n, popped, top(16)
  heap = 0; heap(1:5) = [2, 4, 7, 9, 6]; n = 5
  call heap_insert(heap, n, 1)
  call print_list(heap, n)
contains
  subroutine print_list(values, count)
    integer, intent(in) :: values(16), count
    integer :: i
    write(*, '(A)', advance='no') '['
    do i = 1, count
      if (i > 1) write(*, '(A)', advance='no') ', '
      write(*, '(I0)', advance='no') values(i)
    end do
    print '(A)', ']'
  end subroutine print_list
  subroutine swap(a, b)
    integer, intent(inout) :: a, b
    integer :: tmp
    tmp = a; a = b; b = tmp
  end subroutine swap
  subroutine heap_insert(heap, n, value)
    integer, intent(inout) :: heap(16), n
    integer, intent(in) :: value
    integer :: child, parent
    n = n + 1; heap(n) = value; child = n
    do while (child > 1)
      parent = child / 2
      if (heap(parent) <= heap(child)) exit
      call swap(heap(parent), heap(child))
      child = parent
    end do
  end subroutine heap_insert
  subroutine heap_pop(heap, n, popped)
    integer, intent(inout) :: heap(16), n
    integer, intent(out) :: popped
    integer :: parent, left, right, child
    popped = heap(1); heap(1) = heap(n); n = n - 1; parent = 1
    do
      left = parent * 2; right = left + 1
      if (left > n) exit
      child = left
      if (right <= n .and. heap(right) < heap(left)) child = right
      if (heap(parent) <= heap(child)) exit
      call swap(heap(parent), heap(child))
      parent = child
    end do
  end subroutine heap_pop
  subroutine keep_top(heap, n, value, k)
    integer, intent(inout) :: heap(16), n
    integer, intent(in) :: value, k
    integer :: removed
    call heap_insert(heap, n, value)
    if (n > k) call heap_pop(heap, n, removed)
  end subroutine keep_top
end program main

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]