X-tree

Abstract: It was recently shown that just five characters (functions on a finite set X) suffice to convexly define a trivalent tree with leaf set X. Here we show that four characters suffice which, since three characters are not enough in general, is the best possible.

Abstract: The intent of this paper was to investigate data movement techniques for some special networks which are derived from the binary tree and the mesh machines. We presented optimal bounds for some problems and close bounds for others. A new lower bound technique which incorporates the entire network topdogy was introduced. We believe that this technique is quite powerful and can be exploited to yield good lower bounds for conservative flow algorithms on other networks. However, it seems to be diacult to generalize it for nonconservative flow algorithms. Though we have obtained close bounds, several problems that remain open are noted.

Abstract: A two-centre binary tree (BT) can be constructed by linking the root nodes of two binary trees. Variations can also be constructed by making additional linkages within the two-centre binary tree. Three of these variations, referred to as the quaternary tree (QT), the X-tree (XT) and the ring tree (RT), are studied. Comparisons using performance measures of traffic flows and path reliabilities are made with respect to the BT and some popular topologies in the literature. Various implementations issues that take advantage of these tree networks are then proposed and discussed. The study indicates that the variants of the two-centre trees are viable topologies for the implementation of a metropolitan area network (MAN).

Abstract: At the conference Dress defined parity split maps by triple point distance and asked for a characterisation of such maps coming from binary phylogenetic X-trees. This article gives an answer to that question. The characterisation for X-trees can be easily described as follows: If all restrictions of a split map to sets of five or fewer elements is a parity split map for an X-tree, then so is the entire map. To ensure that the parity split map comes from an X-tree which is binary and phylogenetic, we add two more technical conditions also based on studying at most five points at a time.