Dancing links

My purpose is to discuss an extremely simple technique that deserves to be better known. Suppose x points to an element of a doubly linked list; let L[x] and R[x] point to the predecessor and successor of that element. Then the operations L R[x] ← L[x], R L[x] ← R[x] (1) remove x from the list; every programmer knows this. But comparatively few programmers have realized that the subsequent operations L R[x] ← x, R L[x] ← x (2) will put x back into the list again. This fact is, of course, obvious, once it has been pointed out. Yet I remember feeling a definite sense of " Aha! " when I first realized that (2) would work, because the values of L[x] and R[x] no longer have their former semantic significance after x has been removed from its list. Indeed, a tidy programmer might want to clean up the data structure by setting L[x] and R[x] both equal to x, or to some null value, after x has been deleted. Danger sometimes lurks when objects are allowed to point into a list from the outside; such pointers can, for example, interfere with garbage collection. Why, therefore, am I sufficiently fond of operation (2) that I am motivated to write an entire paper about it? The element denoted by x has been deleted from its list; why would anybody want to put it back again? Well, I admit that updates to a data structure are usually intended to be permanent. But there are also many occasions when they are not. For example, an interactive program may need to revert to a former state when the user wants to undo an operation or a sequence of operations. Another typical application arises in backtrack programs [16], which enumerate all solutions to a given set of constraints. Backtracking, also called depth-first search, will be the focus of the present paper. The idea of (2) was introduced in 1979 by Hitotumatu and Noshita [22], who showed that it makes Dijkstra's well-known program for the N queens problem [6, pages 72–82] run nearly twice as fast without making the program significantly more complicated. Floyd's elegant discussion of the connection between backtracking and nondeterminis-tic algorithms [11] includes a precise method for updating data structures before choosing between alternative lines of computation, and for downdating the data when it is time to explore another line. In general, …