Labeling Schemes with Queries

We study the question of ``how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels''. Let $f$ be a function on pairs of vertices. An $f$-labeling scheme for a family of graphs \cF labels the vertices of all graphs in \cF such that for every graph G\in\cF and every two vertices $u,v\in G$, the value $f(u,v)$ can be inferred by merely inspecting the labels of $u$ and $v$. This paper introduces a natural generalization: the notion of $f$-labeling schemes with queries, in which the value $f(u,v)$ can be inferred by inspecting not only the labels of $u$ and $v$ but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one {\em query}) enables us to reduce the label size of many labeling schemes significantly

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 $\log n + \Omega(\log \log n)$ 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.Comment: 17 pages, 5 figure

A Bi-Labeling Based XPath Processing System

We present BLAS, a Bi-LAbeling based XPath processing System. BLAS uses two labeling schemes to speed up query processing: P-labeling for processing consecutive child (or parent) axis traversals, and D-labeling for processing descendant (or ancestor) axis traversals. XML data are stored in labeled form and indexed. Algorithms are presented for translating XPath queries to SQL expressions. BLAS reduces the number of joins in the SQL query translated from a given XPath query and reduces the number of disk accesses required to execute the SQL query compared with the traditional XPath processing using D-labeling alone. We also propose an approximate P-labeling scheme and the corresponding query translation algorithm to handle XML data trees that contain a large number of distinct tag names, and/or are very deep. This extension captures a spectrum of XPath-to-SQL query translation schemes, ranging from existing schemes that do not use P-labels to the one that uses exact P-labels. Experimental results demonstrate the efficiency of the BLAS system

Connectivity Labeling for Multiple Vertex Failures

We present an efficient labeling scheme for answering connectivity queries in graphs subject to a specified number of vertex failures. Our first result is a randomized construction of a labeling function that assigns vertices $O(f^3\log^5 n)$-bit labels, such that given the labels of $F\cup \{s,t\}$ where $|F|\leq f$, we can correctly report, with probability $1-1/\mathrm{poly}(n)$, whether $s$ and $t$ are connected in $G-F$. However, it is possible that over all $n^{O(f)}$ distinct queries, some are answered incorrectly. Our second result is a deterministic labeling function that produces $O(f^7 \log^{13} n)$-bit labels such that all connectivity queries are answered correctly. Both upper bounds are polynomially off from an $\Omega(f)$-bit lower bound. Our labeling schemes are based on a new low degree decomposition that improves the Duan-Pettie decomposition, and facilitates its distributed representation. We make heavy use of randomization to construct hitting sets, fault-tolerant graph sparsifiers, and in constructing linear sketches. Our derandomized labeling scheme combines a variety of techniques: the method of conditional expectations, hit-miss hash families, and $\epsilon$-nets for axis-aligned rectangles. The prior labeling scheme of Parter and Petruschka shows that $f=1$ and $f=2$ vertex faults can be handled with $O(\log n)$- and $O(\log^3 n)$-bit labels, respectively, and for $f>2$ vertex faults, $\tilde{O}(n^{1-1/2^{f-2}})$-bit labels suffice

Simpler, faster and shorter labels for distances in graphs

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n/2. We also give results for bipartite graphs and for exact and 1-additive distance oracles