Ralf Hinze and Ross Paterson, Journal of Functional Programming 16(2):197–217, 2006. doi:10.1017/S0956796805005769
We present 2-3 finger trees, a functional representation of persistent sequences supporting access to the ends in amortized constant time, and concatenation and splitting in time logarithmic in the size of the smaller piece. Representations achieving these bounds have appeared previously, but 2-3 finger trees are much simpler, as are the operations on them. Further, by defining the split operation in a general form, we obtain a general purpose data structure that can serve as a sequence, priority queue, search tree, priority search queue and more.
The basic structure is expressed as a non-regular (or nested) type:
data FingerTree a = Empty | Single a | Deep (Digit a) (FingerTree (Node a)) (Digit a) data Digit a = One a | Two a a | Three a a a | Four a a a a data Node a = Node2 a a | Node3 a a aThis produces trees of 2-3 trees, with favoured access (fingers) at the ends, like
(more examples) and also supports efficient concatenation. To support splitting and searching, we annotate the internal nodes of the tree with values drawn from an application-specific monoid.
data ImplicitDeque a = Empty | Single a | Deep (Digit a) (ImplicitDeque (a, a)) (Digit a) data Digit a = One a | Two a a | Three a a aFinger trees result from two extensions of this structure: