Analytic Variations on the Common Subexpression Problem

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Any tree can be represented in a max/ma//y 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 programminglanguages, as well as in symbolic manipulation and computer a/gebra systems. In compiling, the compaction problem is known as the "common subexpression problem" and it plays a central r61ein 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 n C . - ogWi- ' 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 restmcted to leaves (atoms).

Philippe Flajolet, Paolo Sipala, Jean-Marc Steyaer

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