Analytic Variations on the Common Subexpression Problem

Any tree can be represented in a maximally compact form as a directed acyclic graph where common subtrees are factored and shared, being represented only once. Such a compaction can be effected in linear time. It is used to save storage in implementations of functional programming languages, as well as in symbolic manipulation and computer algebra systems. In compiling, the compaction problem is known as the “common subexpression problem” and it plays a central role in register allocation, code generation and optimisation. We establish here that, under a variety of probabilistic models, a tree of size n has a compacted form of expected size asymptotically $$C\frac{n}{{\sqrt {\log n} }},$$ where the constant C is explicitly related to the type of trees to be compacted and to the statistical model reflecting tree usage. In particular the savings in storage approach 100% on average for large structures, which overperforms the commonly used form of sharing that is restricted to leaves (atoms).