08-heaps

Min-Heap Pop (Sift Down)

Remove the minimum value, move the last item to the root, and sift downward.

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 down After removing the root, the last value moves to the root and swaps with the smaller child until order is restored.

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 = [1, 4, 2, 9, 6, 7];
$popped = heap_pop($heap);
echo $popped . " -> " . list_string($heap) . PHP_EOL;
?>

After popping the minimum, the last value moves to the root and sifts down by swapping with the smaller child.

Step 1 - Replace root

The saved minimum is 1; the last value 7 moves to the root before sifting down.

Replacement state [7, 4, 2, 9, 6] with removed value 1.1removed7root4left2smaller9i36i4

Step 2 - Swap with the smaller child

7 swaps with 2, producing the final heap [2, 4, 7, 9, 6].

Final heap after pop and one sift-down swap.2root4i17i29i36i4

Implementation notes

  • The heap is a PHP array of integers, starting as [1, 4, 2, 9, 6, 7].
  • heap_pop(array &$heap): int takes the heap by reference, so replacing the root and later swaps mutate the caller's $heap.
  • $smallest = $heap[0] saves the popped min value before the root changes.
  • $heap[0] = array_pop($heap) removes the last array value, 7, and moves it to the root, giving the trace state [7, 4, 2, 9, 6].
  • Sift-down starts with $parent = 0; child indexes are $left = $parent * 2 + 1 and $right = $left + 1.
  • This is a min-heap, so the code chooses the smaller child with $heap[$right] < $heap[$left] when the right child exists.
  • The loop stops if there is no left child or if $heap[$parent] <= $heap[$child].
  • Swaps use PHP array destructuring assignment: [$heap[$parent], $heap[$child]] = [$heap[$child], $heap[$parent]].
  • The checked trace swaps 7 with smaller child 2, producing [2, 4, 7, 9, 6], then stops.
  • echo $popped . " -> " . list_string($heap) . PHP_EOL prints 1 -> [2, 4, 7, 9, 6].

Output

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