In the semi-streaming model, an algorithm must process any \(n\)-vertex graph by making one or few passes over a stream of its edges, use \(\tilde{O}(n) := O(n \cdot \mathrm{polylog}(n))\) words of space, and at the end of the last pass, output a solution to the problem at hand. Approximating (single-source) shortest paths on undirected graphs is a longstanding open question in this model. In this work, we make progress on this question from both upper and lower bound fronts: 1) We present a simple randomized algorithm that for any \(\varepsilon \gt 0\), with high probability computes \((1+\varepsilon)\)-approximate shortest paths from a given source vertex in \(O\left( \frac{1}{\varepsilon} \cdot n \log^3 n \right)\) space and \(O\left( \frac{1}{\varepsilon} \cdot \left(\frac{\log n}{\log\log n}\right)^{2} \right)\) passes. The algorithm can also be derandomized and made to work on dynamic streams at a cost of some extra \(\mathrm{poly}(\log n,1/\varepsilon)\) factors only in the space. Previously, the best known algorithms for this problem required \(1/\varepsilon \cdot \log^{c}n\) passes, for an unspecified large constant \(c\). 2) We prove that any semi-streaming algorithm that with large constant probability outputs any constant approximation to shortest paths from a given source vertex (even to a single fixed target vertex and only the distance, not necessarily the path) requires \(\Omega\left( \frac{\log n}{\log\log n} \right)\) passes. We emphasize that our lower bound holds for any constant-factor approximation of shortest paths. Previously, only constant-pass lower bounds were known and only for small approximation ratios below two. Our results collectively reduce the gap in the pass complexity of approximating single-source shortest paths in the semi-streaming model from \(\mathrm{polylog}(n)\) vs. \(\omega(1)\) to only a quadratic gap.

We aim to revive Thorup’s conjecture [Thorup, WG’92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph G=(V,E) with m edges, we can add m1+o(1) “shortcut” edges E+ from the transitive closure E* of G so that G+(u,v) ≤ mo(1) for all (u,v)∈ E*, where G+=(V,E∪ E+). The conjecture was refuted by Hesse [Hesse, SODA’03], followed by significant efforts in the last few years to optimize the lower bounds.

A k-fault-tolerant connectivity preserver of a directed n-vertex graph G is a subgraph H such that, for any edge set F ⊆ E(G) of size |F| ≤ k, the strongly connected components of G−F and H −F are the same. While some graphs require a preserver with Ω(2kn) edges [1], the best-known upper bound is $\tilde O\left( {k{2^k}{n^{2 - /k}}} \right)$ edges [2], leaving a significant gap of Ω(n1−1/k). In contrast, there is no gap in undirected graphs; the optimal bound of Θ(kn) has been well-established since the 90s [3].We nearly close the gap for directed graphs; we prove that there exists a k-fault-tolerant connectivity preserver with O(k4knlogn) edges, and we can construct one with O(8knlog5/2n) edges in poly(2kn) time.Our results also improve the state-of-the-art for a closely related object; a k-connectivity preserver of G is a subgraph H where, for all i ≤ k, the strongly i-connected components of G and H agree. By a known reduction, we obtain a k-connectivity preserver with O(k4knlogn) edges, improving the previous best bound of $\tilde O\left( {k{2^k}{n^{2 - 1/(k - 1)}}} \right)$ [2]. Therefore, for any constant k, our results are optimal to a logn factor for both problems.Lastly, we show that the exponential dependency on k is not inherent for k-connectivity preservers by presenting another construction with $O\left( {n{\text{ }}\sqrt {kn} } \right)$ edges.

We study distance preservers, hopsets, and shortcut sets in n-node, m-edge directed graphs, and show improved bounds and new reductions for various settings for these problems.

Peer Reviewed

The hereditary discrepancy of a set system is a certain quantitative measure of the pseudorandom properties of the system. Roughly, hereditary discrepancy measures how well one can $2$-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has well-studied applications e.g. in communication complexity and derandomization. More recently, the hereditary discrepancy of set systems of shortest paths has found applications in differential privacy [Chen et al.~SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy $\widetilde{O}(n^{1/4})$, and we construct lower bound examples demonstrating that this bound is tight up to hidden $\text{polylog } n$ factors. Our lower bounds apply even in the planar and bipartite settings, and they improve on a previous lower bound of $Ω(n^{1/6})$ obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from $Ω(n^{1/6})$ [Chen et al.~SODA 23] to $Ω(n^{1/4})$, and we improve the lower bound on additive error for the differentially-private all sets range queries problem to $Ω(n^{1/4})$, which is tight up to hidden $\text{polylog } n$ factors [Deng et al.~WADS 23].

We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+Ω(n^{3/17})$, improving on the previous best lower bound of $Ω(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for $O(n)$-size additive emulators to $+Ω(n^{1/14})$.

For a graph G, a D-diameter-reducing exact hopset is a small set of additional edges H that, when added to G, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most D edges. A shortcut set is the analogous concept for reachability rather than distances. These objects have been studied since the early ’90s, due to applications in parallel, distributed, dynamic, and streaming graph algorithms.For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA ’22] and Bernstein and Wein [SODA ’23] have finally improved over the folklore algorithm for shortcut sets and for $(1+\varepsilon)$-approximate hopsets. For either object, it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1 / 3})$, improving over the folklore diameter bound of $\widetilde{O}(n^{1 / 2})$. The only setting in which folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued?We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1 / 2})$. This improves on the previous lower bound of $\Omega(n^{1 / 3})$ by Kogan and Parter [FOCS ’22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1 / 4})$. This improves on the previous lower bound of $\Omega(n^{1 / 6})$ by Huang and Pettie [SIAM J. Disc. Math. ’18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of p; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of parameters $p \in[1, n^{2}]$.

A classic 1993 paper by Althöfer et al. proved a tight reduction from spanners, emulators, and distance oracles to the extremal function $\gamma$ of high-girth graphs. This paper initiated a large body of work in network design, in which problems are attacked by reduction to $\gamma$ or the analogous extremal function for other girth concepts. In this paper, we introduce and study a new girth concept that we call the bridge girth of path systems, and we show that it can be used to significantly expand and improve this web of connections between girth problems and network design. We prove two kinds of results:•We write the maximum possible size of an n-node, p-path system with bridge girth $\gt k$ as $\beta(n, p, k)$, and we write a certain variant for “ordered” path systems as $\beta^{*}(n, p, k)$. We identify several arguments in the literature that implicitly show upper or lower bounds on $\beta, \beta^{*}$, and we provide some polynomial improvements to these bounds. In particular, we construct a tight lower bound for $\beta(n, p, 2)$, and we polynomially improve the upper bounds for $\beta(n, p, 4)$ and $\beta^{*}(n, p, \infty)$.•We show that many state-of-the-art results in network design can be recovered or improved via black-box reductions to $\beta$ or $\beta^{*}$. Examples include bounds for distance/reachability preservers, exact hopsets, shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an integrality gap for directed Steiner forest.We believe that the concept of bridge girth can lead to a stronger and more organized map of the research area. Towards this, we leave many open problems related to both bridge girth reductions and extremal bounds on the size of path systems with high bridge girth.

For an input graph G, an additive spanner is a sparse subgraph H whose shortest paths match those of G up to small additive error. We prove two new lower bounds in the area of additive spanners:•We construct n-node graphs G for which any spanner on $O(n)$ edges must increase a pairwise distance by $+\Omega(n^{1/7})$. This improves on a recent lower bound of $+\Omega(n^{1/10.5})$ by Lu, Wein, Vassilevska Williams, and Xu [SODA 22].•A classic result by Coppersmith and Elkin [SODA 05] proves that for any n-node graph G and set of $p=O(n^{1/2})$ demand pairs, one can exactly preserve all pairwise distances among demand pairs using a spanner on $O(n)$ edges. They also provided a lower bound construction, establishing that that this range $p=O(n^{1/2})$ cannot be improved. We strengthen this lower bound by proving that, for any constant k, this range of p is still unimprovable even if the spanner is allowed $+k$ additive error among the demand pairs. This negatively resolves an open question asked by Coppersmith and Elkin [SODA 05] and again by Cygan, Grandoni, and Kavitha [STACS 13] and Abboud and Bodwin [SODA 16].At a technical level, our lower bounds are obtained by an improvement to the entire obstacle product framework used to compose “inner” and outer” graphs into lower bound instances. In particular, we develop a new strategy for analysis that allows certain non-layered graphs to be used in the product, and we use this freedom to design better inner and outer graphs that lead to our new lower bounds.

We present the first fine-grained complexity results on two classic problems on strings. The first one is the k-Median-Edit-Distance problem, where the input is a collection of k strings, each of length at most n, and the task is to find a new string that minimizes the sum of the edit distances from itself to all other strings in the input. Arising frequently in computational biology, this problem provides an important generalization of edit distance to multiple strings and is similar to the multiple sequence alignment problem in bioinformatics. We demonstrate that for any ε > 0 and k ≥ 2, an O(n^{k-ε}) time solution for the k-Median-Edit-Distance problem over an alphabet of size O(k) refutes the Strong Exponential Time Hypothesis (SETH). This provides the first matching conditional lower bound for the O(n^k) time algorithm established in 1975 by Sankoff. The second problem we study is the k-Center-Edit-Distance problem. Here also, the input is a collection of k strings, each of length at most n. The task is to find a new string that minimizes the maximum edit distance from itself to any other string in the input. We prove that the same conditional lower bound as before holds. Our results also imply new conditional lower bounds for the k-Tree-Alignment and the k-Bottleneck-Tree-Alignment problems studied in phylogenetics.