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.