. In this paper we consider an on--line problem related to minimizing the diameter of a dynamic tree T . A new edge f is added, and our task is to delete the edge e of the induced cycle so as to minimize the diameter of the resulting tree T [ffgnfeg. Starting with a tree with n nodes, we show how each such best swap can be found in worst--case O(log 2 n) time. The problem was raised by Italiano and Ramaswami at ICALP'94 together with a related problem for edge deletions. Italiano and Ramaswami solved both problems in O(n) time per operation. 1 Introduction The diameter of a tree is the length of a longest simple path in the tree and such a path is called a diameter path. The unique midpoint on all diameter paths is called the center, hence the center is the point whose maximal distance to any node is as small as possible. In 1973 Handler [4] showed how one in linear time can compute the diameter (and center) of a tree. However, as pointed out by Rauch [8], too little work has been...

In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameter-constrained minimum spanning tree (DCMST) of a given undirected, edge-weighted graph, G, is the smallest-weight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NP-complete for all values of k; 4 k (n -- 2), except when all edge-weights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree data-structure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomial-time algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter-4 tree is also used for evaluating the quality of o...