$input
$input :
A simple array-backed queue, based off of the classic Okasaki persistent amortized queue. The basic idea is to maintain two stacks: an input stack and an output stack. When the output stack runs out, reverse the input stack and use it as the output stack.
We don't use the SPL implementation because it's only supported on PHP 5.3 and later.
Exercise: Prove that push/pop on this queue take amortized O(1) time.
Exercise: Extend this queue to be a deque, while preserving amortized O(1) time. Some care must be taken on rebalancing to avoid quadratic behaviour caused by repeatedly shuffling data from the input stack to the output stack and back.