Dynamic and Multi-Functional Labeling Schemes (original) (raw)

2014, Lecture Notes in Computer Science

We investigate labeling schemes supporting adjacency, ancestry, sibling,and connectivity queries in forests. In the course of more than 20 years, the existence of log n + O(log log n) 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 log n + Ω(log log n) or log n + O(log log n) respectively. Notably an upper bound of log n + 2 log log n for adjacency+siblings and a lower bound of log n + log log n for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We improve the constants hidden in the O-notation, where our main technical contribution is a log n + 2 log log n lower bound for connectivity+ancestry and connectivity+siblings. 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 log n bits, and 3 log n to support all three functions. We also show that there exist no efficient dynamic adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and bounded degree graphs.