Space Time Tradeoffs for Graph Properties

Given a graph property P, we study the tradeoff between the pre-processing space and the query time in the following scenario. We are given a graph G, a boolean function family F, and a parameter s which indicates the amount of space available for storing a data structure D that contains information about G. The data structure D satisfies the constraint that the value of each cell c in D corresponds to an application of some function drawn from F. Our queries are of the form: "Does the subgraph Gx induced by the vertex set X satisfy property P?" For various settings of F and s, this model unifies many well-studied problems. At one extreme, when the space s is unrestricted, we study the generalized decision tree complexity of evaluating P on the entire graph G itself; each tree node stores a function in F applied to some subset of edges. A special case of this model is the famous AKR conjecture (where .F, contains merely the identity function g(x) = x) which states that any non-trivial monotone graph property is evasive. At the other extreme, when the function family F is unrestricted, our problem is an example of the classical static data structure problem and we examine the cell probe complexity of our problem. We study graph properties across this broad spectrum of computational frameworks. A central thesis of our work is that "polynomial preprocessing space yields only a negligible (poly-logarithmic) speedup". While proving such a result for an unrestricted F is unlikely, we provide formal evidence towards this thesis by establishing near-quadratic (optimal in many cases) lower bounds under a variety of natural restrictions. Our results are built upon a diverse range of techniques drawn from communication complexity, the probabilistic method and algebraic representations of boolean functions. We also study the problem from an algorithmic viewpoint and develop a framework for designing algorithms that efficiently answer queries using bounded space. We conclude with a study of space-time tradeoffs in an abstract setting of general interest that highlights certain structural issues underlying our problem. for their valuable comments and suggestions. Special thanks to my family for their continuing support.