Distance labeling schemes for trees

We consider distance labeling schemes for trees: given a tree with $n$ nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg (J. Graph Theory 2000) establish that labels must use $\Theta(\log^2 n)$ bits\footnote{Throughout this paper we use $\log$ for $\log_2$.}. Gavoille et. al. (ESA 2001) show that for very small approximate stretch, labels use $\Theta(\log n \log \log n)$ bits. Several other papers investigate various variants such as, for example, small distances in trees (Alstrup et. al., SODA'03). We improve the known upper and lower bounds of exact distance labeling by showing that $\frac{1}{4} \log^2 n$ bits are needed and that $\frac{1}{2} \log^2 n$ bits are sufficient. We also give ($1+\epsilon$)-stretch labeling schemes using $\Theta(\log n)$ bits for constant $\epsilon>0$. ($1+\epsilon$)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar (STOC 2004). In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size $2\log n - \Theta(\log\log n)$. For simple paths with $k$ nodes and edge weights in $[1,n]$, we show that labels must have size $\frac{k-1}{k}\log n+\Theta(\log k)$

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

On the Succinct Representation of Equivalence Classes

Given a set of n elements that are partitioned into equivalence classes, we study the problem of assigning unique labels to these elements in order to support the query that asks whether the elements corresponding to two given labels belong to the same equivalence class. This problem has been studied by Katz et al., Alstrup et al., and Lewenstein et al.. Lewenstein et al. showed that with no auxiliary data structure, a label space of size nlg(n) is necessary and sufficient to represent the equivalence relation. They also showed that if the labels were to be assigned from the set [n], a data structure of square root of n bits is necessary and sufficient to represent the equivalence relation and to answer the equivalence query in O(lg(n)) time. In this thesis, we give an improved data structure that uses O(square root of n) bits and can answer queries in constant time, when the label space is of size n. Moreover, we study the case where we allow the label space to be of size cn for any constant c > 1. We show that with such a label space, a data structure of O(lg(n)) bits is necessary and sufficient to represent the equivalence relation and to answer the equivalence query in constant time. We believe that our work can trigger further work on tradeoffs between label space and auxiliary data structure space for other labeling problems