Labeling Schemes for Dynamic Tree Networks

Abstract

The paper concerns distributed distance labeling schemes on dynamic trees. The paper first presents a labeling scheme for distances in the dynamic tree model, with amortized message complexity O(log2 n) per operation, where n is the size of the tree at the time the operation takes place. The protocol maintains O(log2 n) bit labels, where n is the current tree size. This label size is known to be optimal even in the static scenario. A more general labeling scheme is then introduced for the dynamic tree model, based on extending an existing static tree labeling scheme to the dynamic setting. The approach fits a number of natural tree functions, such as distance, separation level and flow. The main resulting scheme incurs an overhead of a O(log n) multiplicative factor in both the label size and amortized message complexity in the case of dynamically growing trees (with no vertex deletions). In the fullydynamic model the scheme incurs also an increased additive overhead in amortized communication, of O(log2 n) messages per operation.