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.
Hub sets with near to optimal size can be constructed in polynomial time. A greedy setcover-type @math -approximation algorithm (with respect to average size of a hub set) was proposed by @cite_21 . For the case of USP graphs, this approximation ratio was improved by @cite_40 to the logarithm of the graph hop-diameter, i.e., the maximum number of hops of a shortest path in @math .
@cite_40 @cite_21
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There has been significant success in designing highly efficient algorithms for distance and shortest-path queries in recent years; many of the state-of-the-art algorithms use the hub labeling framework. In this paper, we study the approximability of the Hub Labeling problem. We prove a hardness of Ω(log n) for Hub Labeling, matching known approximation guarantees. The hardness result applies to graphs that have multiple shortest paths between some pairs of vertices. No hardness of approximation results were known previously. Then, we focus on graphs that have a unique shortest path between each pair of vertices. This is a very natural family of graphs, and much research on the Hub Labeling problem has studied such graphs. We give an O(log D) approximation algorithm for graphs of shortest-path diameter D with unique shortest paths. In particular, we get an O(log log n) approximation for graphs of polylogarithmic diameter, while previously known algorithms gave an O(log n) approximation. Finally, we present a polynomial-time approximation scheme (PTAS) and quasi-polynomial-time algorithms for Hub Labeling on trees; additionally, we analyze a simple combinatorial heuristic for Hub Labeling on trees, proposed by Peleg in 2000. We show that this heuristic gives an approximation factor of 2. We present a deterministic (1+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o(1))-approximation O(n1 2D1 4+D)-time algorithm of Nanongkai [STOC 2014] by a factor of as large as n1 8, and (ii) the O(є−1logє−1)-approximation factor of Lenzen and Patt-Shamir’s O(n1 2+є+D)-time algorithm [STOC 2013] within the same running time. Our running time matches the known time lower bound of Ω(n1 2 logn + D) [Das STOC 2011] modulo some lower-order terms, thus essentially settling the status of this problem which was raised at least a decade ago [Elkin SIGACT News 2004]. It also implies a (2+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for approximating a network’s weighted diameter which almost matches the lower bound by [PODC 2012]. In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the “hitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (no(1), o(1))-hop set of size O(n1+o(1)). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [ STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+o(1))-approximation O(no(1))-time algorithm on congested cliques, and (ii) a (1+o(1))-approximation O(no(1)logW)-pass O(n1+o(1)logW)-space streaming algorithm, when edge weights are in 1, 2, …, W . The first result answers an open problem in [Nanongkai, STOC 2014]. The second result partially answers an open problem raised by McGregor in 2006 [ sublinear.info , Problem 14]. A 5 4-approximation algorithm is presented for the minimum cardinality 2-edge-connected spanning subgraph problem in undirected graphs. This improves the previous best approximation ratio of 4 3. It is shown that our ratio is tight with respect to current lower bounds, and any further improvement is possible only if new lower bounds are discovered. For a positive parameter @math , the @math -bounded distance between a pair of vertices @math in a weighted undirected graph @math is the length of the shortest @math path in @math with at most @math edges, aka hops . For @math as above and @math , a @math -hopset of @math is a graph @math on the same vertex set, such that all distances in @math are @math -approximated by @math -bounded distances in @math . Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with @math edges, or with a hopbound @math . In this paper we devise a construction of linear-size hopsets with hopbound @math . This improves the previous bound almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm EN16 for computing hopsets with a constant (i.e., independent of @math ) hopbound requires @math time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from EN16 . We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between individual memory requirement @math of vertices throughout preprocessing and routing phases of the algorithm, and stretch @math , along with a near-optimal construction time @math , where @math is the hop-diameter of the input graph.