Optimum Binary Search Trees on the Hierarchical Memory Model

Abstract

The Hierarchical Memory Model (HMM) of computation is similar to the standard Random Access Machine (RAM) model except that the HMM has a non-uniform memory organized in a hierarchy of levels numbered 1 through h. The cost of accessing a memory location increases with the level number, and accesses to memory locations belonging to the same level cost the same. Formally, the cost of a single access to the memory location at address a is given by m(a), where m: N -> N is the memory cost function, and the h distinct values of m model the different levels of the memory hierarchy. We study the problem of constructing and storing a binary search tree (BST) of minimum cost, over a set of keys, with probabilities for successful and unsuccessful searches, on the HMM with an arbitrary number of memory levels, and for the special case h=2. While the problem of constructing optimum binary search trees has been well studied for the standard RAM model, the additional parameter m for the HMM increases the combinatorial complexity of the problem. We present two dynamic programming algorithms to construct optimum BSTs bottom-up. These algorithms run efficiently under some natural assumptions about the memory hierarchy. We also give an efficient algorithm to construct a BST that is close to optimum, by modifying a well-known linear-time approximation algorithm for the RAM model. We conjecture that the problem of constructing an optimum BST for the HMM with an arbitrary memory cost function m is NP-complete.