What data structure, exactly, are deques in C++?

Question

Is there a specific data structure that a deque in the C++ STL is supposed to implement, or is a deque just this vague notion of an array growable from both the front and the back, to be implemented however the implementation chooses?

I used to always assume a deque was a circular buffer, but I was recently reading a C++ reference here, and it sounds like a deque is some kind of array of arrays. It doesn't seem like it's a plain old circular buffer. Is it a gap buffer, then, or some other variant of growable array, or is it just implementation-dependent?

UPDATE AND SUMMARY OF ANSWERS:

It seems the general consensus is that a deque is a data structure such that:

• the time to insert or remove an element should be constant at beginning or end of the list and at most linear elsewhere. If we interpret this to mean true constant time and not amortized constant time, as someone comments, this seems challenging. Some have argued that we should not interpret this to mean non-amortized constant time.
• "A deque requires that any insertion shall keep any reference to a member element valid. It's OK for iterators to be invalidated, but the members themselves must stay in the same place in memory." As someone comments: This is easy enough by just copying the members to somewhere on the heap and storing T* in the data structure under the hood.
• "Inserting a single element either at the beginning or end of a deque always takes constant time and causes a single call to a constructor of T." The single constructor of T will also be achieved if the data structure stores T* under the hood.
• The data structure must have random access.

It seems no one knows how to get a combination of the 1st and 4th conditions if we take the first condition to be "non-amortized constant time". A linked list achieves 1) but not 4), whereas a typical circular buffer achieves 4) but not 1). I think I have an implementation that fulfills both below. Comments?

We start with an implementation someone else suggested: we allocate an array and start placing elements from the middle, leaving space in both the front and back. In this implementation, we keep track of how many elements there are from the center in both the front and back directions, call those values F and B. Then, let's augment this data structure with an auxiliary array that is twice the size of the original array (so now we're wasting a ton of space, but no change in asymptotic complexity). We will also fill this auxiliary array from its middle and give it similar values F' and B'. The strategy is this: every time we add one element to the primary array in a given direction, if F > F' or B > B' (depending on the direction), up to two values are copied from the primary array to the auxiliary array until F' catches up with F (or B' with B). So an insert operation involves putting 1 element into the primary array and copying up to 2 from the primary to the auxiliary, but it's still O(1). When the primary array becomes full, we free the primary array, make the auxiliary array the primary array, and make another auxiliary array that's yet 2 times bigger. This new auxiliary array starts out with F' = B' = 0 and having nothing copied to it (so the resize op is O(1) if a heap allocation is O(1) complexity). Since the auxiliary copies 2 elements for every element added to the primary and the primary starts out at most half-full, it is impossible for the auxiliary to not have caught up with the primary by the time the primary runs out of space again. Deletions likewise just need to remove 1 element from the primary and either 0 or 1 from the auxiliary. So, assuming heap allocations are O(1), this implementation fulfills condition 1). We make the array be of T* and use new whenever inserting to fulfill conditions 2) and 3). Finally, 4) is fulfilled because we are using an array structure and can easily implement O(1) access.

Answer 1

It's implementation specific. All a deque requires is constant time insertion/deletion at the start/end, and at most linear elsewhere. Elements are not required to be contiguous.

Most implementations use what can be described as an unrolled list. Fixed-sized arrays get allocated on the heap and pointers to these arrays are stored in a dynamically sized array belonging to the deque.