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

Perl DSA Implementation

basic.pl 
use strict;
use warnings;
sub list_string { return "[" . join(", ", @_) . "]"; }
sub heap_insert {
    my ($heap, $value) = @_;
    push @$heap, $value;
    my $child = scalar(@$heap) - 1;
    while ($child > 0) {
        my $parent = int(($child - 1) / 2);
        last if $heap->[$parent] <= $heap->[$child];
        ($heap->[$parent], $heap->[$child]) = ($heap->[$child], $heap->[$parent]);
        $child = $parent;
    }
}
sub heap_pop {
    my ($heap) = @_;
    my $smallest = $heap->[0];
    $heap->[0] = pop @$heap;
    my $parent = 0;
    while (1) {
        my $left = $parent * 2 + 1; my $right = $left + 1;
        last if $left >= scalar @$heap;
        my $child = $left;
        $child = $right if $right < scalar @$heap && $heap->[$right] < $heap->[$left];
        last if $heap->[$parent] <= $heap->[$child];
        ($heap->[$parent], $heap->[$child]) = ($heap->[$child], $heap->[$parent]);
        $parent = $child;
    }
    return $smallest;
}
my @heap = (2, 4, 7, 9, 6);
heap_insert(\@heap, 1);
print list_string(@heap) . "\n";

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

  • my @heap = (2, 4, 7, 9, 6) stores the min-heap in a Perl array with the @ sigil.
  • heap_insert(\@heap, 1) passes an array reference, so the helper mutates the original heap.
  • Inside heap_insert, my ($heap, $value) = @_ stores that array reference in the scalar $heap.
  • push @$heap, $value appends the new value to the end of the referenced array.
  • Perl heap indexing here is zero-based: after append, my $child = scalar(@$heap) - 1 points at the inserted slot.
  • Parent index math is int(($child - 1) / 2).
  • The loop runs while $child > 0, so sift-up stops at the root.
  • This is a min-heap: last if $heap->[$parent] <= $heap->[$child] stops when parent order is valid.
  • Swaps use Perl list assignment: ($heap->[$parent], $heap->[$child]) = ($heap->[$child], $heap->[$parent]).
  • The trace starts from [2, 4, 7, 9, 6], appends 1, and shows [2, 4, 7, 9, 6, 1].
  • Sift-up swaps 1 with parent 7, giving [2, 4, 1, 9, 6, 7], then swaps 1 with parent 2, giving [1, 4, 2, 9, 6, 7].
  • list_string(@heap) joins the final array with ", ", and print ... . "\n" outputs [1, 4, 2, 9, 6, 7].

Output

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