Real-time deques, multihead Turing machines, and purely functional programming

We answer the following question: Can a deque (double-ended queue) be implemented in a purely functional language such that each push or pop operation on either end of a queue is accomplished in O(1) time in the worst case? The answer is yes, thus solving a problem posted by Gajewska and Tarjan and by Ponder, McGeer, and Ng, and refining results of Sarnak and Hoogerwoord. We term such a deque real-time, since its constant worst-case behavior might be useful in real time programs (assuming real-time garbage collection, etc.) Furthermore, we show that no restriction of the functional language is necessary, and that push and pop operations on previous versions of a deque can also be achieved in constant time. We present a purely functional implementation of real-time deques and its complexity analysis. We then show that the implementation has some interesting implications, and can be used to give a real-time simulation of a multihead Turing machine in a purely functional language.