PHP と MySQL で再帰を行わずにフラット データを階層的なネストされたメニューに効率的に変換するにはどうすればよいですか?

<code class="php">function build_menu($page_objects) {

  // Create an array to store parent-child relationships
  $parents = array();

  // Loop through the page objects
  foreach ($page_objects as $page) {
    $parents[$page['id']] = $page['parent_id'];
  }

  // Initialize the menu string
  $menu = '';

  // Loop through the page objects again
  foreach ($page_objects as $page) {

    // Check if the current page has no parent (root page)
    if (!isset($parents[$page['id']])) {
      $menu .= "<ul><li>{$page['title']}";
    }

    // Handle child pages
    else {
      $parent_id = $parents[$page['id']];

      // Find the parent 'ul' tag for the child page
      $parent_index = strrpos($menu, "<li>{$page_objects[$parent_id]['title']}");

      // Append the child page to the parent 'ul' tag
      $menu = substr_replace($menu, "<li>{$page['title']}", $parent_index + strlen($page_objects[$parent_id]['title']) + 4, 0);
    }

    // Close the 'li' tag
    $menu .= '</li>';

    // Handle closing the 'ul' tag for root pages
    if (!isset($parents[$page['parent_id']])) {
      $menu .= "</ul>";
    }
  }

  return $menu;
}</code>

<code class="php">$page_objects = array(
    array('id' => 1, 'title' => 'Menu 1', 'parent_id' => null),
    array('id' => 2, 'title' => 'Sub 1.1', 'parent_id' => 1),
    array('id' => 3, 'title' => 'Sub 1.2', 'parent_id' => 1),
    array('id' => 4, 'title' => 'Sub 1.3', 'parent_id' => 1),
    array('id' => 5, 'title' => 'Menu 2', 'parent_id' => null),
    array('id' => 6, 'title' => 'Sub 2.1', 'parent_id' => 5),
    array('id' => 7, 'title' => 'Sub Sub 2.1.1', 'parent_id' => 6),
    array('id' => 8, 'title' => 'Sub 2.2', 'parent_id' => 5),
    array('id' => 9, 'title' => 'Menu 3', 'parent_id' => null)
);</code>

<code class="php">$menu_html = build_menu($page_objects);</code>