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 fully-dynamic model the scheme incurs also an increased additive overhead in amortized communication, of O(log2 n) messages per operation.