Dynamic and Multi-Functional Labeling Schemes

We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of logn + O(log log) labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC ’10]. Several multi-functional labeling schemes also enjoy lower or upper bounds of logn+Ω(log logn) or logn + O(log log n) respectively. Notably an upper bound of logn + 5 log logn for adjacency+siblings and a lower bound of logn + log logn for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA ’03]. We improve the constants hidden in the O-notation. In particular we show a logn + 2 log logn lower bound for connectivity+ancestry and connectivity+siblings, as well as an upper bound of logn+ 3 log logn+O(log log logn) for connectivity+adjacency+siblings by altering existing methods. In the context of dynamic labeling schemes it is known that ancestry requires Ω(n) bits [Cohen, et al. PODS ’02]. In contrast, we show upper and lower bounds on the label size for adjacency, siblings, and connectivity of 2 logn bits, and 3 logn to support all three functions. There exist efficient adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and interval graphs. In a dynamic setting, we show a lower bound of Ω(n) for each of those families.