Combine and Conquer

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We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called {\em attributes}, with the nodes, paths, and subtrees of our trees. Path attributes form a {\em path attribute system}, if they are maintained in constant time under path concatenation. Additionally, attributes form a {\em tree attribute system} if the tree attributes of the tail of a path $\Pi$ are determined in constant time from the path attributes of $\Pi$. We also introduce a new data structure called a {\em linear attribute grammar}. An {\em attribute grammar} is a tree-based expression where the values a node $\mu$ are calculated from the values at the parent, siblings, and/or the children of $\mu$. A linear attribute grammar, is an attribute grammar where all dependencies are linear. Our contributions can be summarized as follows. We provide a framework for maintaining attribute systems on trees in a fully dynamic environment. We show that given a semiring ${\cal S}$, a set of linear expressions with binary and unary operators over ${\cal S}^k$ can be dynamically maintained in a fully dynamic environment using linear space and logarithmic time per operation. We show that a linear attribute grammar can be dynamically maintained in a fully dynamic environment using linear space and logarithmic time per operation. Finally, we present many examples of application of our technique to dynamic graph and geometric problems.