A hierarchical graph 𝒢 consists of a vertex set V(𝒢) and L pairwise disjoint edge sets E_1, … , E_L. Such a structure naturally defines a hierarchy of L unweighted graphs, where the 𝓁-th graph is G_𝓁 = (V, ⋃_{i=1}^𝓁 E_i). In this paper, we initiate the study of hierarchical spanners, namely subgraphs of 𝒢 that approximately preserve distances among a given set of pairs of vertices in V(𝒢) at every level of the hierarchy. This notion generalizes classical spanners, and thus all known lower bounds extend to this setting; however, it is not clear whether the same size-stretch trade-offs can be achieved. We investigate this question by devising both upper and lower bounds for hierarchical spanners under various types of stretch and pairs of vertices of interest whose approximate (or exact) distances are to be maintained. On the positive side, a trivial adaptation of the greedy construction yields (2k-1)-spanners of size O(n^{1+1/k}), matching the classical bounds. However, the non-hierarchical bounds do not extend to the hierarchical case when additive or nearly-additive spanners are considered. For instance, we prove that any β-additive single-pair hierarchical spanner must have size Ω (n √{n/(β+1)}) in the worst case. This bound is tight, as we provide a matching upper bound for every β ≥ 0, which in turn implies a O(n√n)-size single-pair hierarchical preserver. Finally, we present additional positive results among which a 4-additive all-pairs hierarchical spanner of size Õ(n^{5/3}), an (essentially tight) single-source hierarchical (1+ε)-spanner of size Õ(n/ε), an all-pairs hierarchical spanner of size Õ(n√{n/(ε)}) achieving stretch (1+ε,2), for any constant value of ε > 0, and a subsetwise hierarchical preserver of size O(n √{n|S|}), where S ⊆ V(𝒢).

We consider the task of performing Jaccard similarity queries over a large collection of items that are dynamically updated according to a streaming input model. An item here is a subset of a large universe U of elements. A well-studied approach to address this important problem in data mining is to design fast-similarity data sketches. In this paper, we focus on global solutions for this problem, i.e., a single data structure which is able to answer both Similarity Estimation and All-Candidate Pairs queries, while also dynamically managing an arbitrary, online sequence of element insertions and deletions received in input.

In this work, we propose, analyze and empirically validate a lazy-update approach to maintain accurate approximations of the 2-hop neighborhoods of dynamic graphs resulting from sequences of edge insertions. We first show that under random input sequences, our algorithm exhibits an optimal trade-off between accuracy and insertion cost: it only performs [EQUATION] (amortized) updates per edge insertion, while the estimated size of any vertex's 2-hop neighborhood is at most a factor ε away from its true value in most cases, regardless of the underlying graph topology and for any ε > 0. As a further theoretical contribution, we explore adversarial scenarios that can force our approach into a worst-case behavior at any given time t of interest. We show that while worst-case input sequences do exist, a necessary condition for them to occur is that the girth of the graph released up to time t be at most 4. Finally, we conduct extensive experiments on a collection of real, incremental social networks of different sizes, which typically have low girth. Empirical results are consistent with and typically better than our theoretical analysis anticipates. This further supports the robustness of our theoretical findings: forcing our algorithm into a worst-case behavior not only requires topologies characterized by a low girth, but also carefully crafted input sequences that are unlikely to occur in practice. Combined with standard sketching techniques, our lazy approach proves an effective and efficient tool to support key neighborhood queries on large, incremental graphs, including neighborhood size, Jaccard similarity between neighborhoods and, in general, functions of the union and/or intersection of 2-hop neighborhoods.

A spanner is a sparse subgraph of a given graph $G$ which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t.\ the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

In a temporal forest each edge has an associated set of time labels that specify the time instants in which the edges are available. A temporal path from vertex $u$ to vertex $v$ in the forest is a selection of a label for each edge in the unique path from $u$ to $v$, assuming it exists, such that the labels selected for any two consecutive edges are non-decreasing. We design linear-size data structures that maintain a temporal forest of rooted trees under addition and deletion of both edge labels and singleton vertices, insertion of root-to-node edges, and removal of edges with no labels. Such data structures can answer temporal reachability, earliest arrival, and latest departure queries. All queries and updates are handled in polylogarithmic worst-case time. Our results can be adapted to deal with latencies. More precisely, all the worst-case time bounds are asymptotically unaffected when latencies are uniform. For arbitrary latencies, the update time becomes amortized in the incremental case where only label additions and edge/singleton insertions are allowed as well as in the decremental case in which only label deletions and edge/singleton removals are allowed. To the best of our knowledge, the only previously known data structure supporting temporal reachability queries is due to Brito, Albertini, Casteigts, and Travençolo [Social Network Analysis and Mining, 2021], which can handle general temporal graphs, answers queries in logarithmic time in the worst case, but requires an amortized update time that is quadratic in the number of vertices, up to polylogarithmic factors.