Catenable double-ended queues

Abstract

Catenabledouble-ended queues are double-ended queues (deques) thatsupportcatenation(i.e., append)efficiently without sacrificing the efficiency of other operations. We present a purely functional implementationof catenable deques for which every operation, including catenation,takea0(1) amortizedtime. Kaplan and Tarjan haveindependentlydeveloped a much more complicated implementationof crdenabledeques thatachieves similar worst-casebounds. The two designs are superficially similar, but differ in the underlying mechanismused to achieve efficiency in a persistentsetting(i.e., whenused in a non-single-threadedfashion). Their implementation uses a techniquecalled recursiveslowdown, while ours relies on the simpler mechanism of lazy evaluation.Besides lazy evaluation, our implementation rdso exemplifies theuse of two additional language features: polymorphic recursion and views. Neitheris indispensable, but both significantly simplify the presentation.

Purely functional, real-time deques with catenation

“…Our definition of purely functional is extremely strict; we do not, for example, allow techniques such as memoization. This contrasts our work with, for example, that ofOkasaki [1995a; 1995b;1997;]. For more discussion of this issue, see Sections 2 and 7.…”

Generics for the masses

“…If a data type has more than two alternatives, or if a constructor has more than two arguments, we have to nest the binary type constructors Plus and Pair accordingly. Actually, we are more flexible than this: we can map the new type to any other type as long as the target type is an instance of Rep. For instance, we could map elements of some efficient sequence type (Okasaki, 1997) to lists. This flexibility is even necessary if the structure of a type is not available as in the case of abstract types.…”

Finger trees: a simple general-purpose data structure

“…Okasaki also presents a wealth of functional implementations of sequences and other data types. Implementations of functional deques with constant time concatenation are known (Kaplan & Tarjan, 1995;Okasaki, 1997), but these structures do not admit efficient insertion, deletion or splitting. Kaplan and Tarjan (1996) defined two functional data structures with the same time bounds as here, except that their bounds are worst case, and do not require laziness.…”