An optimal rebuilding strategy for an incremental tree problem
Résumé
This paper is devoted to the following incremental problem. Initially, a graph and a distinguished subset of vertices, called initial group, are given. This group is connected by an initial tree. The incremental part of the input is given by an on-line sequence of vertices of the graph, not yet in the current group, revealed on-line one after one. The goal is to connect each new member to the current tree, while satisfying a quality constraint: the average distance between members in each constructed tree must be kept in a given range compared to the best possible one.\\ Under this quality constraint, our objectives are to minimize the number of critical stages and the number of elementary changes of the sequence of constructed trees. We call "critical" a stage where the inclusion of a new member implies heavy changes in the current tree. Otherwise, the new member is just added by connecting it with a (well chosen) path to the currenttree. In both cases, updating a tree implies a certain number of elementary changes (that we define).\\ We propose a strategy leading to at most $O(log i)$ critical stages (i is the number of new members) and to at most a constant average number of elementary changes per stage. We also prove that there exists situations where $\Omega(log i)$ critical stages are necessary to any algorithm to maintain the quality constraint. Our strategy is then worst case optimal in order of magnitude for the number of critical stages criterion and induces a constant number of elementary changes in average per stage.