Combine and Conquer

We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We introduce three fully dynamic data structures, called path attribute systems, tree attribute systems, and linear attribute grammars, which extend and generalize the dynamic trees of Sleator and Tarjan. More specifically, we associate values, called attributes, with the nodes and paths of a rooted tree. Path attributes form a path attribute system if they can be maintained in constant time under path concatenation. Node attributes form a tree attribute system if the tree attributes of the tail of a path Π can be determined in constant time from the path attributes of Π. A linear attribute grammar is a tree-based linear expression such that the values of a node μ are calculated from the values at the parent, siblings, and/for children of μ. We provide a framework for maintaining path attribute systems, tree attribute systems, and linear attribute grammars in a fully dynamic environment using linear space and logarithmic time per operation. Also, we demonstrate the applicability of our techniques by showing examples of graph and geometric problems that can be efficiently dynamized, including biconnectivity and triconnectivity queries, planarity testing, drawing trees and series-parallel digraphs, slicing floorplan compaction, point location, and many optimization problems on bounded tree-width graphs. Received May 13, 1994; revised October 12, 1995.