A method to prove query lower bounds

Abstract

The query-model or decision-tree model is a computational model in which the algorithm has to solve a given problem by making a sequence of queries which have 'Yes' or 'No' answers. A large class of algorithms can be described on this model and we can also prove non-trivial lower bounds for many problems on this model. Many lower bounds on the query-model are proved using a technique called adversary argument. In CS courses, a common example used to illustrate the adversary argument is the following problem: Suppose there is an unweighted graph G with $n$ vertices represented by an adjacency matrix. We want to test if the graph is connected. How many entries in the adjacency matrix do we have to probe in order to test if the graph has this property (property being 'connectivity')? Each probe is considered as a query. Since the adjacency matrix has only n^2 entries, O(n^2) queries are sufficient. It is also known that Omega(n^2) queries are necessary. Proving this lower bound is more difficult and is done using the adversary argument. In literature, we find that lower bound proofs of this problem rely too much on 'connectivity' property and do not generalize well. When the property being tested is changed, the proof changes significantly. Our contribution is a method that gives a systematic way of proving lower bounds for problems involving testing of many graph-properties. We did a pilot experiment and found that students were able to understand and apply our method.