Luciano Gualà’s research while affiliated with University of Rome Tor Vergata and other places

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 Traven\c{c}olo [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.

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. We introduce and provide an in-depth analysis of a dynamic, buffered version of the well-known k-MinHash sketch. This buffered version better manages critical update operations thus significantly reducing the number of times the sketch needs to be rebuilt from scratch using expensive recovery queries. We prove that the buffered k-MinHash uses $O(k \log |U|)$ memory words per subset and that its amortized update time per insertion/deletion is $O(k \log |U|)$ with high probability. Moreover, our data structure can return the k-MinHash signature of any subset in O(k) time, and this signature is exactly the same signature that would be computed from scratch (and thus the quality of the signature is the same as the one guaranteed by the static k-MinHash). Analytical and experimental comparisons with the other, state-of-the-art global solutions for this problem given in [Bury et al.,WSDM'18] show that the buffered k-MinHash turns out to be competitive in a wide and relevant range of the online input parameters.

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 the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices as input. The tree is embedded in an unknown metric space and we have unlimited access to an oracle that, given two distinct vertices u and v of T, can answer queries reporting the cost of the edge (u, v) in constant time. We want to augment T with k shortcuts in order to minimize the diameter of the resulting graph.For k=1, $O(n \log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bilò, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than 10/9-approximate solution for trees for $k\ge 3$. For any constant $\varepsilon > 0$, we instead design a linear-time $(1+\varepsilon )$-approximation algorithm for paths and $k = o(\sqrt{\log n})$, thus establishing a dichotomy between paths and trees for $k\ge 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time 4-approximation algorithm for trees, and to compute the diameter of graphs with $n \,+\, k\, -\, 1$ edges in time $O(n k \log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.KeywordsTree diameter augmentationFast diameter computationApproximation algorithmsTime-efficient algorithms

In the \emph{k-Diameter-Optimally Augmenting Tree Problem} we are given a tree T of n vertices as input. The tree is embedded in an unknown \emph{metric} space and we have unlimited access to an oracle that, given two distinct vertices u and v of T, can answer queries reporting the cost of the edge (u,v) in constant time. We want to augment T with k shortcuts in order to minimize the diameter of the resulting graph. For k=1, $O(n \log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bil\`o, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than 10/9-approximate solution for trees for $k\geq 3$. For any constant $\varepsilon > 0$, we instead design a linear-time $(1+\varepsilon)$-approximation algorithm for paths and $k = o(\sqrt{\log n})$, thus establishing a dichotomy between paths and trees for $k\geq 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time 4-approximation algorithm for trees, and to compute the diameter of graphs with $n + k - 1$ edges in time $O(n k \log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.

In this paper we introduce the notions of blackout-tolerant temporal $\alpha$-spanner of a temporal graph G which is a subgraph of G that preserves the distances between pairs of vertices of interest in G up to a multiplicative factor of $\alpha$, even when the graph edges at a single time-instant become unavailable. In particular, we consider the single-source, single-pair, and all-pairs cases and, for each case we look at three quality requirements: exact distances (i.e., $\alpha =1$), almost-exact distances (i.e., $\alpha = 1+\varepsilon$ for an arbitrarily small constant $\varepsilon >0$), and connectivity (i.e., unbounded $\alpha$). For each combination we provide tight bounds, up to polylogarithmic factors, on the size, which is measured as the number of edges, of the corresponding blackout-tolerant $\alpha$-spanner for both general temporal graphs and for temporal cliques. Our result show that such spanners are either very sparse (i.e., they have $\widetilde{O}(n)$ edges) or they must have size $\varOmega (n^2)$ in the worst case, where n is the number of vertices of G. To complete the picture, we also investigate the case of multiple blackouts.KeywordsTemporal graphsTemporal spannersFault-tolerance

We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC’04] in which up to $\delta$ δ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in $O(\delta )$ O ( δ ) worst-case time per operation.

We study the heterogeneous weighted delivery (HWD) problem introduced in [Bärtschi et al., STACS'17] where k heterogeneous mobile agents (e.g., robots, vehicles, etc.), initially positioned on vertices of an n-vertex edge-weighted graph G, have to deliver m messages. Each message is initially placed on a source vertex of G and needs to be delivered to a target vertex of G. Each agent can move along the edges of G and carry at most one message at any time. Each agent has a rate of energy consumption per unit of traveled distance and the goal is that of delivering all messages using the minimum overall amount of energy. This problem has been shown to be NP-hard even when k=1, and is 4ρ-approximable where ρ is the ratio between the maximum and minimum energy consumption of the agents. In this paper, we provide approximation algorithms with approximation ratios independent of the energy consumption rates. First, we design a polynomial-time 8-approximation algorithm for k=O(log⁡n), closing a problem left open in [Bärtschi et al., ATMOS'17]. This algorithm can be turned into an O(k)-approximation algorithm that always runs in polynomial-time, regardless of the values of k. Then, we show that HWD problem is 36-approximable in polynomial-time when each agent has one of two possible consumption rates. Finally, we design a polynomial-time O˜(log3⁡n)-approximation algorithm for the general case.

A temporal graph is an undirected graph G=(V,E) along with a function that assigns a time-label to each edge in E. A path in G with non-decreasing time-labels is called temporal path and the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal $\alpha$-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of $\alpha$. The size of H is the number of its edges. In this work we study the size-stretch trade-offs of temporal spanners. We show that temporal cliques always admit a temporal $(2k-1)-$spanner with $\tilde{O}(kn^{1+\frac{1}{k}})$ edges, where $k>1$ is an integer parameter of choice. Choosing $k=\lfloor\log n\rfloor$, we obtain a temporal $O(\log n)$-spanner with $\tilde{O}(n)$ edges that has almost the same size (up to logarithmic factors) as the temporal spanner in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then consider general temporal graphs. Since $\Omega(n^2)$ edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that $\tilde{O}(n/\log(1+\varepsilon))$ edges suffice to obtain a stretch of $(1+\varepsilon)$, for any small $\varepsilon>0$. This result is essentially tight since there are temporal graphs for which any temporal subgraph preserving exact distances from a single-source must use $\Omega(n^2)$ edges. We extend our analysis to prove an upper bound of $\tilde{O}(n^2/\beta)$ on the size of any temporal $\beta$-additive spanner, which is tight up to polylogarithmic factors. Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.