arXiv subjects

Noy Rotbart

Publications and source records attributed to Noy Rotbart.

5 recordsLinked to original sources

arXiv2026

Adjacency Labeling Schemes of Size $\log n+O(\log\log^* n)$ for Graphs of Bounded Treewidth

Gavoille and Labourel (ESA 2007) gave an adjacency labeling scheme of size $\log n+O(k\log\log n)$ for graphs of treewidth at most $k$, and asked whether $\log n+O(k)$ bits suffice. We give a labeling scheme of size $\log n+O(\log\log^* n)$ for every fixed $k$, the first improvement of the second-order term for any $k\ge 2$. Equivalently, there is an induced-universal graph with $n(\log^* n)^{O_k(1)}$ vertices for this family. We work with laminar families of intervals, in which a vertex finds each earlier neighbor as the owner of an interval of a given nesting depth around it. As long as every subtree occupies consecutive positions, the nesting depth must be logarithmic, and $\log\log n$ bits per neighbor follow. We give up contiguity. Heavy subtrees are moved out, which brings the depth down to $O(\log^* n)$; the intervals broken by the moves are repaired with $2^{O(k^3)}$ additional intervals per vertex; and the bits needed to find these intervals are paid for by the moved subtrees.

cs.DS↗

arXiv2015

A simple and optimal ancestry labeling scheme for trees

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.

cs.DS↗

arXiv2015

Near-optimal adjacency labeling scheme for power-law graphs

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.

cs.DC↗

arXiv2014

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 $\log n + O(\log \log)$ 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 $\log n + Ω(\log \log n)$ 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 $Ω(n)$ 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. 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.

cs.DS↗

arXiv2014

Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree $Δ= O(1)$. 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 $Δ$ with $(e+1)\sqrt{n} < Δ\leq n/5$.

cs.DM↗