5 Conclusion and Open Problems

cost of the overhead is O(lglg n). Hence the implementation cost is O(lg(jA(v)j), which is proportional to the number of comparisons needed by the Koml os algorithm to nd the heaviest edges in 2m half-paths of the tree in the worst case. Summed over all nodes, this comes to O(n log(m+n n)) as Koml os has shown. The only additional costs are in forming the LCA 0 s which take O(m+n) and in processing the tables which takes O(n), and comparing the heaviest edges in each half-path, which takes O(m). Finally, to complete the minimum spanning tree verication algorithm, one compares the weight of each nontree edge to the weight of the heaviest tree edge in the tree path connecting its endpoints. for an additional O(m) cost. We have reduced Koml os's algorithm to the simpler case of the full branching tree. And, we have devised a novel data structure which gives the rst algorithm with linear time overhead for its implementation. It is still an open question as to whether one can nd a linear time algorithm for a pointer machine. Such a result would imply a linear time algorithm for a pointer machine which can compute the lowest common ancestor. None is known for that problem which seems easier. Given a static tree, the [SV] lowest common ancestor algorithm can process on-line query paths in constant time for each. An open problem is to solve the tree path problem in constant time per query path, where the query paths are given on-line. The functions we use are in some sense natural. It is possible that they may be useful for implementing other algorithms which are not known to have linear implementations or whose implementations involve more specialized table lookup functions, as the DRT implementation did. (See [DRT] for references to some of these algorithms.) Finally, any other applications of Theorem 1 would be of interest. of size r=2, looking up the result for the two halves and in constant time, putting the results together to form the entry. For example, for index r , if a table is built for index r=2 , then one can easily construct the table for input strings of size r, in a constant number of operations per entry, as follows: Let I be the rst half of the input and J be its second half. Add weight r=2 (I) to each subword …