This paper is a study of persistence in data structures. Ordinary data structures are ephemeral in the sense that a change to the structure destroys the old version, leaving only the new version available for use. In contrast, a persistent structure allows access to any version, old or new, at any time. We develop simple, systematic, and effiient techniques for making linked data structures persistent. We use our techniques to devise persistent forms of binary search trees with logarithmic access, insertion, and deletion times and O(1) space bounds for insertion and deletion.

: The aspect of purity versus impurity that we address involves the absence versus presence of mutation: the use of primitives (RPLACA and RPLACD in Lisp, set-car! and set-cdr! in Scheme) that change the state of pairs without creating new pairs. It is well known that cyclic list structures can be created by impure programs, but not by pure ones. In this sense, impure Lisp is "more powerful" than pure Lisp. If the inputs and outputs of programs are restricted to be sequences of atomic symbols, however, this difference in computability disappears. We shall show that if the temporal sequence of input and output operations must be maintained (that is, if computations must be "online "), then a difference in complexity remains: for a pure program to do what an impure program does in n steps, O(n log n) steps are sufficient, and in some cases\Omega\Gamma n log n) steps are necessary. * This research was partially supported by an NSERC Operating Grant. 1. Introduction The programming la...

Abstract. Relaxed balancing means that, in a dictionary stored as a balanced tree, the necessary rebalancing after updates may be delayed. This is in contrast to strict balancing meaning that rebalancing is performed immediately after the update. Relaxed balancing is important for efficiency in highly dynamic applications where updates can occur in bursts. The rebalancing tasks can be performed gradually after all urgent updates, allowing the concurrent use of the dictionary even though the underlying tree structure is not completely in balance. In this paper we propose a new scheme of how to make known rebalancing techniques relaxed in an efficient way. The idea is applied to the red-black trees, but can be applied to any class of balanced trees. The key idea is to accumulate insertions and deletions such that they can be settled in arbitrary order using the same rebalancing operations as for standard balanced search trees. As a result it can be shown that the number of needed rebalancing operations known from the strict balancing scheme carry over to relaxed balancing. 1

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