A labeling scheme is a method of distributing the information about the structure of a graph among its vertices by assigning short labels, such that a selected function on pairs of vertices can be computed using only their labels. A labeling scheme consists of an encoder that has access to the entire graph and assigns labels to vertices, and a decoder that has access to only the labels of a smaller set of vertices (typically a pair) and returns information about this subset (e.g., whether two vertices are adjacent, or the distance between them in the graph). The main objective is to minimize the maximum label size: the maximum number of bits used in a label of any vertex. Among the applications of labeling schemes are XML search engines, mapping services, and internet routing.

We present a $$\lg n + 2 \lg \lg n+3$$ ancestry labeling scheme for trees. The problem was first presented by Kannan et al. [STOC 88’] along with a simple $$2 \lg n$$ solution. Motivated by applications to XML files, the label size was improved incrementally over the course of more than 20 years by a series of papers. The last, due to Fraigniaud and Korman [STOC 10’], presented an asymptotically optimal $$\lg n + 4 \lg \lg n+O(1)$$ labeling scheme using non-trivial tree-decomposition techniques. By providing a framework generalizing interval based labeling schemes, we obtain a simple, yet asymptotically optimal solution to the problem. Furthermore, our labeling scheme is attained by a small modification of the original $$2 \lg n$$ solution.

An adjacency labeling scheme is a method that assigns labels to the vertices of a graph such that adjacency between vertices can be inferred directly from the assigned label, without using a centralized data structure. We devise adjacency labeling schemes for the family of power-law graphs. This family that has been used to model many types of networks, e.g. the Internet AS-level graph. Furthermore, we prove an almost matching lower bound for this family. We also provide an asymptotically near- optimal labeling scheme for sparse graphs. Finally, we validate the efficiency of our labeling scheme by an experimental evaluation using both synthetic data and real-world networks of up to hundreds of thousands of vertices.

We consider ancestry labeling schemes: Given a rooted tree T, assign a binary string (label) to each node, such that given the labels of any two nodes, one can determine whether the first is an ancestor of the second in T. Recently, Fraigniaud and Korman [STOC'10] showed that such labels can be assigned using log n + O(loglog n) bits per label, solving a long standing open question and matching a lower bound of log n + Omega(loglog n) due to Alstrup et al. [SODA'03]. In this paper we present an alternative ancestry labeling scheme using log n + 2loglog n + O(1) bits. Similar to the previously known schemes, our scheme relies on intervals and can encode any tree in O(n) time. Rather than using complicated decompositions, our scheme uses approximate interval sizes, which gives a clean algorithm that is simple enough to be taught and implemented.

We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree $\Delta$ with $(e+1)\sqrt{n} < \Delta \leq n/5$.

We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of $\log n + O(\log \log)$ 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 $\log n + \Omega(\log \log n)$ or $\log n + O(\log \log n)$ respectively. Notably an upper bound of $\log n + 5\log \log n$ 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. In particular we show a $\log n + 2\log \log n$ lower bound for connectivity+ancestry and connectivity+siblings, as well as an upper bound of $\log n + 3\log \log n + O(\log \log \log n)$ for connectivity+adjacency+siblings by altering existing methods. In the context of dynamic labeling schemes it is known that ancestry requires $\Omega(n)$ 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. 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 $\Omega(n)$ for each of those families.

We present an implementation and evaluation based on simulation of dynamic labeling schemes for tree networks. Two algorithms are studied: a general scheme that converts static labeling schemes to dynamic, and a specialized dynamic distance labeling scheme. Our study shows that theoretical bounds only partially portray the performance of such dynamic labeling schemes in practice. First, we observe order-of-magnitude differences between the gains in label size when compared to the number of messages passed. Second, we demonstrate a significant bottleneck in the tree network, suggesting that the current practice of counting total messages passed in the whole network is insufficient to properly characterize performance of these distributed algorithms. Finally, our experiments provide intuition on the worst case scenarios for the stated algorithms, in particular path tree networks and fully dynamic schemes permitting both node additions and deletions.