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

PHP DSA Implementation

basic.php 
<?php
function list_string(array $values): string { return "[" . implode(", ", $values) . "]"; }
function heap_insert(array &$heap, int $value): void {
    $heap[] = $value;
    $child = count($heap) - 1;
    while ($child > 0) {
        $parent = intdiv($child - 1, 2);
        if ($heap[$parent] <= $heap[$child]) break;
        [$heap[$parent], $heap[$child]] = [$heap[$child], $heap[$parent]];
        $child = $parent;
    }
}
function heap_pop(array &$heap): int {
    $smallest = $heap[0];
    $heap[0] = array_pop($heap);
    $parent = 0;
    while (true) {
        $left = $parent * 2 + 1; $right = $left + 1;
        if ($left >= count($heap)) break;
        $child = $left;
        if ($right < count($heap) && $heap[$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);
echo list_string($heap) . PHP_EOL;
?>

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 PHP array of integers, starting as [2, 4, 7, 9, 6].
  • heap_insert(array &$heap, int $value): void takes the heap by reference, so appends and swaps mutate the caller's $heap.
  • Insert uses $heap[] = $value; this checked run appends 1, producing [2, 4, 7, 9, 6, 1].
  • $child = count($heap) - 1 starts at the appended slot, and $parent = intdiv($child - 1, 2) finds the parent index.
  • This is a min-heap: if $heap[$parent] <= $heap[$child], the loop stops.
  • Swaps use PHP array destructuring assignment: [$heap[$parent], $heap[$child]] = [$heap[$child], $heap[$parent]].
  • The trace first 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].
  • echo list_string($heap) . PHP_EOL prints [1, 4, 2, 9, 6, 7].

Output

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