An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its label size , that is the maximal number of bits stored in a label, taken over all @math -node trees. The design of ancestry labeling schemes finds applications in XML search engines. In the context of these applications, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of an original scheme of size @math bits, a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is @math bits which is still far from the known @math lower bound. Moreover, the hidden constant factor in the additive @math term is large, which makes this term dominate the label size for typical current XML trees. In attempt to provide good performances for real XML data, we rely on the observation that the depth of a typical XML tree is bounded from above by a small constant. Having this in mind, we present an ancestry labeling scheme of size @math , for the family of trees with at most @math nodes and depth at most @math . In addition to our main result, we prove a result that may be of independent interest concerning the existence of a linear universal graph for the family of forests with trees of bounded depth.
Implicit labeling schemes were first introduced in @cite_0 , where an elegant adjacency labeling schemes of size @math is established on @math -node trees. That paper also notices a relation between adjacency labeling schemes and (see also @cite_9 @cite_22 @cite_3 ). Precisely, it is shown that there exists an adjacency labeling scheme with label size @math for a graph family @math if and only if there exists a universal graph for @math with @math nodes.
@cite_0 @cite_9 @cite_22 @cite_3
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Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let ℱ(n) be a family of connected graphs of size at most n and let @math denote the collection of graphs of size at most n, such that each graph in @math is composed of a disjoint union of some graphs in ℱ(n). We first investigate methods for translating f-labeling schemes for ℱ(n) to f-labeling schemes for @math . In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family ℱ(n), one can construct an f-labeling scheme of size g(n)+log log n+O(1) for @math . We also show that in several cases, the above mentioned extra additive term of log log n+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5log n+O(1) and a lower bound of 4 3log n−O(1) for the size of any such labeling scheme. 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 (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. We consider the following problem. Give a rooted tree T, label the nodes of T in the most compact way such that given the labels of two nodes one can determine in constant time, by looking only at the labels, if one node is an ancestor of the other. The best known labeling scheme is rather straightforward and uses labels of size at most 2 log n, where n is the number of vertices In the tree. Our main result in this paper is a labeling scheme with maximum label size close to 3 2 log n. Our motivation for studying this problem is enhancing the performance of Web search engines. In the context of this application each indexed document is a tree and the labels of all trees are maintained in main memory. Therefore even small improvements in the maximum label size are important. There are no lower bounds known for this problem except for an obvious lower bound of log n that follows from the fact that different vertices must have different labels. The question whether one can find even shorter labels remains an intriguing open question. We consider the following problem. Given a rooted tree @math , label the nodes of @math in the most compact way such that, given the labels of two nodes @math and @math , one can determine in constant time, by looking only at the labels, whether @math is ancestor of @math . The best known labeling scheme is rather straightforward and uses labels of length at most @math bits each, where @math is the number of nodes in the tree. Our main result in this paper is a labeling scheme with maximum label length @math . Our motivation for studying this problem is enhancing the performance of web search engines. In the context of this application each indexed document is a tree, and the labels of all trees are maintained in main memory. Therefore even small improvements in the maximum label length are important.