For fixed @math , we consider the task of adding to a graph @math a set of weighted shortcut edges on the same vertex set, such that the length of a shortest @math -hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in @math . A set of shortcut edges with this property is called an exact @math -hopset and may be applied in processing distance queries on graph @math . In particular, a @math -hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on @math -hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that @math -hopsets require exponentially fewer shortcuts per node than any previously described distance oracle while incurring only a quadratic increase in the query decoding time, and actually offer a speedup when compared to simple oracles based on a direct application of @math -hopsets. Finally, we consider the problem of computing minimum-size @math -hopset (for any @math ) for a given graph @math , showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When @math , for a given bound on the space used by the distance oracle, we provide a construction of hopsets achieving polylog approximation both for space and query time compared to the optimal @math -hopset oracle given the space bound.
The notion of 2-hopset studied in this work coincides with the special case of two-hop distance labeling (also called ), where labels are constructed from hub sets: in hub-labeling, a small hub set @math is assigned to each node of a graph @math such that or any pair @math of nodes, the intersection of hub sets @math contains a node on a shortest @math path. Such a construction is proposed by @cite_5 and applies to graphs of treewidth @math with labels of @math size and allows to answer distance queries in @math time; the hub sets have a hierarchical structure, which allows for an improvement of query time to @math time by a binary search over levels. Hub labelings are the best currently known distance labelings for sparse graphs, achieving sublinear node label size @cite_6 @cite_17 , and may also be used to provide a 2-additive-approximation for distance labeling in general graphs using sublinear-space labels @cite_17 .
@cite_5 @cite_6 @cite_17
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We consider the problem of labeling the nodes of a graph in a way that will allow one to compute the distance between any two nodes directly from their labels (without using any additional information). Our main interest is in the minimal length of labels needed in different cases. We obtain upper and lower bounds for several interesting families of graphs. In particular, our main results are the following. For general graphs, we show that the length needed is Θ(n). For trees, we show that the length needed is Θ(log2 n). For planar graphs, we show an upper bound of O(√nlogn) and a lower bound of Ω(n1 3). For bounded degree graphs, we show a lower bound of Ω(√n). The upper bounds for planar graphs and for trees follow by a more general upper bound for graphs with a r(n)-separator. The two lower bounds, however, are obtained by two different arguments that may be interesting in their own right. We also show some lower bounds on the length of the labels, even if it is only required that distances be approximated to a multiplicative factor s. For example, we show that for general graphs the required length is Ω(n) for every s < 3. We also consider the problem of the time complexity of the distance function once the labels are computed. We show that there are graphs with optimal labels of length 3 log n, such that if we use any labels with fewer than n bits per label, computing the distance function requires exponential time. A similar result is obtained for planar and bounded degree graphs. A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node u stores its distance to all nodes from an appropriately chosen set of hubs (S(u) V ). For a queried pair of nodes (u, v), the length of a shortest (u !- !v )-path passing through a hub node from (S(u) S(v) ) is then used as an upper bound on the distance between u and v.