In this thesis we study a problem that arises in answering boolean queries submitted to a search engine. Usually a search engine stores the set of IDs of documents containing each word in a pre-computed sorted order and to evaluate a query like "computer AND science" the search engine has to evaluate the union of the sets of documents containing the words "computer" and "science". More complex queries will result in more complex set expressions. In this thesis we consider the problem of evaluation of a set expression with union and intersection as operators and ordered sets as operands. We explore properties of comparison-based algorithms for the problem. A <i>proof of a set expression</i> is the set of comparisons that a comparison-based algorithm performs before it can determine the result of the expression. We discuss the properties of the proofs of set expressions and based on how complex the smallest proofs of a set expression <i>E</i> are, we define a measurement for determining how difficult it is for <i>E</i> to be computed. Then, we design an algorithm that is adaptive to the difficulty of the input expression and we show that the running time of the algorithm is roughly proportional to difficulty of the input expression, where the factor is roughly logarithmic in the number of the operands of the input expression.

In this thesis we study a problem that arises in answering boolean queries submitted to a search engine. Usually a search engine stores the set of IDs of documents containing each word in a pre-computed sorted order and to evaluate a query like "computer AND science" the search engine has to evaluate the union of the sets of documents containing the words "computer" and "science". More complex queries will result in more complex set expressions. In this thesis we consider the problem of evaluation of a set expression with union and intersection as operators and ordered sets as operands. We explore properties of comparison-based algorithms for the problem. A <i>proof of a set expression</i> is the set of comparisons that a comparison-based algorithm performs before it can determine the result of the expression. We discuss the properties of the proofs of set expressions and based on how complex the smallest proofs of a set expression <i>E</i> are, we define a measurement for determining how difficult it is for <i>E</i> to be computed. Then, we design an algorithm that is adaptive to the difficulty of the input expression and we show that the running time of the algorithm is roughly proportional to difficulty of the input expression, where the factor is roughly logarithmic in the number of the operands of the input expression.