O(1) Reversible Tree Navigation Without Cycles

Imperative programmers often use cyclically linked trees in order to achieve O(1) navigation time to neighbours. Some logic programmers believe that cyclic terms are necessary to achieve the same in logic-based languages. An old but little-known technique provides O(1) time and space navigation without cyclic links, in the form of reversible predicates. A small modification provides O(1) amortised time and space editing. 1 The problem When imperative programmers think of lists, they commonly choose doubly linked lists, instead of the singly linked lists that logic and functional programmers use. In the same way, it is extremely common for trees to be given parent links, whether they are really needed or not. A typical C example might be typedef int Datum; typedef struct TreeRec *TreePtr; struct TreeRec { TreePtr left, right, up, down; Datum datum; }; where ‘down’ points to the first child of a node, ‘up’ to its parents, and the children of a node form a doubly linked list with ‘left’ and ‘right’ pointers. Essentially this representation is required by the Document Object Model (W3C, 1998; W3C, 2000), for example. Cyclically linked trees in imperative languages such as Java provide constant time navigation in any of the four directions (up, down, left, right) and also constant time and constant space editing (insert, delete, replace). They do so at a price: each element is rigidly locked into place, so that any kind of space sharing (such as hash consing) is made impossible. Some logic programming languages have been designed to support cyclically linked terms. That does provide constant time navigation, but not editing. The locking into place that is a nuisance in imperative languages is a very serious difficulty