Optimal Search Trees with 2-Way Comparisons

We study the following problem: given a set of keys and access probabilities, find a minimum-cost binary search tree that uses only 2-way comparisons (=, <,≤) at each node. We give the first polynomialtime algorithm when both successful and unsuccessful queries are allowed, settling a long-standing open question. Our algorithm relies on a new structural theorem that proves a conjecture of Spuler [13,14]. We also give a fast approximation algorithm, improved algorithms for special cases, and a counter-example revealing an error in previous works. 1 Background and statement of results We study the problem of finding optimal binary search trees using 2-way comparisons. Given a set K of keys and a query set Q from an ordered universe, and a set C ⊆ {=, <,≤} of binary operators, a (2-way comparison) binary search tree (2wcst) is a rooted binary tree where each internal node has a key in K and an associated operator in C. Given a query value v ∈ Q, a search for v compares v to the key at the root using the root’s comparison operator. If the outcome is yes, the search recurses into the left subtree, otherwise into the right subtree. The search halts when it reaches a leaf. For any v ∈ Q, the outcomes of the comparisons made during the search must suffice to determine the relation of v to every key in K. Fig. 1 gives two examples. v = H? v < O? v < W? yes