Generalising tree traversals and tree transformations to DAGs: Exploiting sharing without the pain

Abstract

We present a recursion scheme based on attribute grammars that can be transparently applied to trees and acyclic graphs. Our recursion scheme allows the programmer to implement a tree traversal or a tree transformation and then apply it to compact graph representations of trees instead. The resulting graph traversal or graph transformation avoids recomputation of intermediate results for shared nodes – even if intermediate results are used in different contexts. Consequently, this approach leads to asymptotic speedup proportional to the compression provided by the graph representation. In general, however, this sharing of intermediate results is not sound. Therefore, we complement our implementation of the recursion scheme with a number of correspondence theorems that ensure soundness for various classes of traversals. We illustrate the practical applicability of the implementation as well as the complementing theory with a number of examples. • We give a method for defining traversals that run seamlessly on trees and DAGs. • We extend the method to sharing-preserving transformations. • Correspondence theorems characterise the semantics of traversals on trees vs. DAGs. • The method is based on attribute grammars, implemented as a Haskell library. • We demonstrate the technique on a number of example algorithms.