You could always just use Data.Sequence.

Alternatively, a well-known implementation of a purely functional queue is to use two lists. One for enqueue and another for dequeue. Enqueue would simply cons with the enqueue list. Dequeue takes the head of the dequeue list. When the dequeue list is shorter than the enqueue list, refill it by reversing the enqueue list. See Chris Okasaki s Purely Functional Datastructures.

Even though this implementation uses reverse, the amortized time cost of this is insignificant asymptotically. It works out so that for every enqueue, you incur a time debt of Θ(1) for the dequeue list refill. The expected time of a dequeue is therefore at most twice that of an enqueue. This is a constant factor, so the worst-case cost of both operations is O(1).

This question appears as the third result on the first page while I google Haskell queue, but the information previously given is misleading. So, I feel there is a need to clarify a few things. (And the first search result is a blog post which contains a careless implementation...)

Everything below is basically from Okasaki s paper, Simple and efficient purely functional queues and deques in 1995 or his book.

Okay, let s begin.

1. A persistent queue implementation with amortised O(1) time complexity is possible. The trick is to reverse the list representing the rear part of a queue as long as the front part is long enough to amortise the cost of reverse operation. So, instead of reversing the rear part when the front part is empty, we reverse it when the front part is shorter than the rear part. The following code is from the appendix of Okasaki s book

   data BQueue a = BQ !Int [a] !Int [a]
   
   check :: Int -> [a] -> Int -> [a] -> BQueue a
   check lenf fs lenr rs =
     if lenr <= lenf 
     then BQ lenf fs lenr rs 
     else BQ (lenr+lenf) (fs ++ reverse rs) 0 [] 
   
   head :: BQueue a -> a
   head (BQ _ []    _ _) = error "empty queue"
   head (BQ _ (x:_) _ _) = x
   
   (|>) :: BQueue a -> a -> BQueue a 
   (BQ lenf fs lenr rs) |> x = check lenf fs (lenr + 1) (x:rs)
   
   tail :: BQueue a -> BQueue a
   tail (BQ lenf (x:fs) lenr rs) = check (lenf-1) fs lenr rs
   

2. And why is this amortised O(1) even used persistently? Haskell is lazy, so reverse rs does not actually happen until it is needed. To force reverse rs, it has to take |fs| steps before reaching the reverse rs. If we repeat tail before reaching the suspension reverse rs, then the result will be memorised so at the second time it takes only O(1). On the other hand, if we use the version before placing the suspension fs ++ reverse rs, then again it has to go through fs steps before reaching reverse rs. A formal proof using (modified) Banker s method is in Okasaki s book.

3. The answer by @Apocalisp

   When the dequeue list is empty, refill it by reversing the enqueue list

   is the implementation in Ch 5 of his book with a warning in the very beginning

   Unfortunately, the simple view of amortization presented in this chapter breaks in the presence of persistence

   Okasaki described his amortised O(1) persistent queue in Ch 6.

4. So far, we have been talking about amortised time complexity only. It is actually possible to eliminate amortisation completely to achieve the worst-case O(1) time complexity for persistent queue. The trick is that reverse has to be forced incrementally every time a de/enqueue is called. The actual implementation is a bit hard to explain here, though.

Again, everything is in his paper already.

Is Data.Dequeue what you are looking for?

(It doesn t have reverse but you can add it pretty easily and send a patch to the author.)

I m not really a Haskell user, but I found a blog post which claims to describe a Haskell queue that can be operated on in amortized constant time. It s based on a design from Chris Okasaki s excellent Purely Functional Data Structures.