General Splay: A Basic Theory and Calculus

The directory problem, that of handling an on-line series of INSERT, DELETE and FIND requests, is for most applications addressed satisfactorily by balanced search trees (BST) (see e.g., [Ov83] for a trace of BST's from [AVL62] and on), which guarantee O(logN) worst time for each of these operations. For applications however in which the access frequencies are highly biased the " overbalancedness " of these trees yields suboptimal performance. This was addressed first, statically, in the 1970's with biased search trees (see [Kn71], [HT71], [BST85]), and later dynamically in the 1980's with splay trees (see [ST85]). Splay trees achieve a logarithmic amortized cost for searching and other operations, and are competitive w.r.t. any other static binary search tree on the same set of elements (see also [Ta85]). However [ST85] left some important issues open, among which the most intriguing is the dynamic optimality conjecture: Are splay trees competitive even when compared to dynamically maintained search trees? To the best of the authors' knowledge this issue remains open, so we still do not know which is the best, at least in this sense, solution to the directory problem. Most probably this can be attributed to the lack of a general self-adjustment theory. In the 1990's Subramanian ([Sub96]) made a step towards this by describing splaying as a set of rules, called templates. Templates are comprised of two trees, the before-and after-schema, each with a special current node. Templates specify a splay-step in which the before-schema is replaced by the after-schema; this is repeated continuously upwards along a path, until the current node reaches the root. Subramanian proved that for such rules to have logarithmic amortized cost, the following conditions are sufficient for binary search trees: (1) " Strict growth " : the set of nodes below the current node is strictly augmented at each stage; (2) " Depth reduction " : the (two) subtrees of current node are linked strictly nearer the root; (3) " Progress " : descendants of template nodes must be moved below the current node within a constant number of steps. In this paper we adopt the template framework of [Sub96] and employ a potential function analysis (i.e., we define for a tree T a potential function-7), and for each operation, q, which transforms T to T' we bound the change in potential-(T)−-(T') ≥ − a(q)+c(q) where c(q) is the actual and a(q) the amortized cost of …