The Coast of Searching in General Trees versus Complete Binary Trees

We consider the problem of searching in natural families of trees. A search in a tree T is a sequence of queries used to locate a pre-designated node called the \buggy" node. A query is made to any subtree in T. If the queried subtree contains the buggy node, then we restrict the search to that subtree, otherwise the search is restricted to the complement of that subtree in T. Two problems are considered, estimating the cost of an optimal search and devising fast algorithms that compute reasonably good search strategies. It follows that for general trees the cost is (log d+1 d n), where d is the maximal degree and n is the size of the tree. For the restricted family of complete binary trees, the cost of the optimal search is reduced to O(log n+log n). We also describe a fast algorithm that computes a search strategy with cost of O(log n + log log n) for complete binary trees. Finally, we give a simple fast algorithm that computes an optimal search strategy for the special case of top-down search. Computing this strategy can be done in O(n) steps, where n is the size of the underlying tree. Search in trees is often encountered in program testing or debugging. The bug should be found by searching the syntax tree of the program using a minimal set of queries. Another area where search problems appear is searching in classiied large tree-like data bases (e.g. the Yahoo's data base of the Internet).