Closing the Gap Between Directed Hopsets and Shortcut Sets

For an n-vertex directed graph $G = (V,E)$, a $\beta$-\emph{shortcut set} $H$ is a set of additional edges $H \subseteq V \times V$ such that $G \cup H$ has the same transitive closure as $G$, and for every pair $u,v \in V$, there is a $uv$-path in $G \cup H$ with at most $\beta$ edges. A natural generalization of shortcut sets to distances is a $(\beta,\epsilon)$-\emph{hopset} $H \subseteq V \times V$, where the requirement is that $H$ and $G \cup H$ have the same shortest-path distances, and for every $u,v \in V$, there is a $(1+\epsilon)$-approximate shortest path in $G \cup H$ with at most $\beta$ edges. There is a large literature on the tradeoff between the size of a shortcut set / hopset and the value of $\beta$. We highlight the most natural point on this tradeoff: what is the minimum value of $\beta$, such that for any graph $G$, there exists a $\beta$-shortcut set (or a $(\beta,\epsilon)$-hopset) with $O(n)$ edges? Not only is this a natural structural question in its own right, but shortcuts sets / hopsets form the core of many distributed, parallel, and dynamic algorithms for reachability / shortest paths. Until very recently the best known upper bound was a folklore construction showing $\beta = O(n^{1/2})$, but in a breakthrough result Kogan and Parter [SODA 2022] improve this to $\beta = \tilde{O}(n^{1/3})$ for shortcut sets and $\tilde{O}(n^{2/5})$ for hopsets. Our result is to close the gap between shortcut sets and hopsets. That is, we show that for any graph $G$ and any fixed $\epsilon$ there is a $(\tilde{O}(n^{1/3}),\epsilon)$ hopset with $O(n)$ edges. More generally, we achieve a smooth tradeoff between hopset size and $\beta$ which exactly matches the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors). Using a very recent black-box reduction of Kogan and Parter, our new hopset implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ

Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond

For fixed h >= 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest h-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact h-hopset and may be applied in processing distance queries on graph G. In particular, a 2-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3-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 3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle, and also offer a speedup in query time when compared to simple oracles based on a direct application of 2-hopsets. Finally, we consider the problem of computing minimum-size h-hopset (for any h >= 2) for a given graph G, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When h=3, for a given bound on the space used by the distance oracle, we provide a construction of hopset achieving polylog approximation both for space and query time compared to the optimal 3-hopset oracle given the space bound

Folklore Sampling is Optimal for Exact Hopsets: Confirming the n\sqrt{n} Barrier

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. 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 diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where 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 $\widetilde{\Omega}(n^{1/3})$ 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 $p \in [1, n^2]$

Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication

Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+\epsilon)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. We devise a significantly improved algorithm for this problem in the entire range of parameter $r$, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of $r$ in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time $\tilde{O}(|E| \cdot n^{o(1)} + n^{\omega(r)})$, where $n^{\omega(r)}$ is the time required to multiply an $n^r \times n$ matrix by an $n \times n$ one. Our PRAM algorithm has polylogarithmic time $(\log n)^{O(1/\rho)}$, and its work complexity is $\tilde{O}(|E| \cdot n^\rho + n^{\omega(r)})$, for any arbitrarily small constant $\rho >0$. In particular, for $r \le 0.313\ldots$, our centralized algorithm computes $S \times V$ $(1+\epsilon)$-approximate shortest paths in $n^{2 + o(1)}$ time. Our PRAM polylogarithmic-time algorithm has work complexity $O(|E| \cdot n^\rho + n^{2+o(1)})$, for any arbitrarily small constant $\rho >0$. Previously existing solutions either require centralized time/parallel work of $O(|E| \cdot |S|)$ or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for $|S| = n^r$ sources, for $r \le 0.655$, while previous state-of-the-art algorithms did so only for $r \le 1/2$. Moreover, it improves previous bounds for all $r > 1/2$. For unweighted graphs, the running time is improved further to $poly(\log\log n)$

DISTRIBUTED, PARALLEL AND DYNAMIC DISTANCE STRUCTURES

Many fundamental computational tasks can be modeled by distances on a graph. This has inspired studying various structures that preserve approximate distances, but trade off this approximation factor with size, running time, or the number of hops on the approximate shortest paths. Our focus is on three important objects involving preservation of graph distances: hopsets, in which our goal is to ensure that small-hop paths also provide approximate shortest paths; distance oracles, in which we build a small data structure that supports efficient distance queries; and spanners, in which we find a sparse subgraph that approximately preserves all distances. We study efficient constructions and applications of these structures in various models of computation that capture different aspects of computational systems. Specifically, we propose new algorithms for constructing hopsets and distance oracles in two modern distributed models: the Massively Parallel Computation (MPC) and the Congested Clique model. These models have received significant attention recently due to their close connection to present-day big data platforms. In a different direction, we consider a centralized dynamic model in which the input changes over time. We propose new dynamic algorithms for constructing hopsets and distance oracles that lead to state-of-the-art approximate single-source, multi-source and all-pairs shortest path algorithms with respect to update-time. Finally, we study the problem of finding optimal spanners in a different distributed model, the LOCAL model. Unlike our other results, for this problem our goal is to find the best solution for a specific input graph rather than giving a general guarantee that holds for all inputs. One contribution of this work is to emphasize the significance of the tools and the techniques used for these distance problems rather than heavily focusing on a specific model. In other words, we show that our techniques are broad enough that they can be extended to different models

Distributed Exact Shortest Paths in Sublinear Time

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in $O(n)$ time, where $n$ is the number of vertices in the input graph $G$. Peleg and Rubinovich (FOCS'99) showed a lower bound of $\tilde{\Omega}(D + \sqrt{n})$ for this problem, where $D$ is the hop-diameter of $G$. Whether or not this problem can be solved in $o(n)$ time when $D$ is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires $O((n \log n)^{5/6})$ time, for $D = O(\sqrt{n \log n})$, and $O(D^{1/3} \cdot (n \log n)^{2/3})$ time, for larger $D$. This running time is sublinear in $n$ in almost the entire range of parameters, specifically, for $D = o(n/\log^2 n)$. For the all-pairs shortest paths problem, our algorithm requires $O(n^{5/3} \log^{2/3} n)$ time, regardless of the value of $D$. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset $G"$ of a skeleton graph $G'$ of $G$ without first computing $G'$ itself. We then conduct a Bellman-Ford exploration in $G' \cup G"$, while computing the required edges of $G'$ on the fly. As a result, our algorithm computes exactly those edges of $G'$ that it really needs, rather than computing approximately the entire $G'$

(1 + )-Approximate shortest paths in dynamic streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2−1 versus n1+1/, for an integer parameter . (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2−O(1/) passes over the stream (for space O(n1+1/)), and applies only to unweighted graphs. In this paper we show that (1+)-approximate single-source shortest paths can be computed with ˜O (n1+1/) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/). We achieve these results by devising efficient dynamic streaming constructions of (1 + , )-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest. 2012 ACM Subject Classification Theory of computation ! Streaming models; Theory of computation ! Streaming, sublinear and near linear time algorithms; Theory of computation ! Shortest paths; Theory of computation ! Sparsification and spanner

Bridge Girth: A Unifying Notion in Network Design

A classic 1993 paper by Alth\H{o}fer 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: 1) We write the maximum possible size of an $n$-node, $p$-path system with bridge girth $>k$ 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 polynomially 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 $\beta^*(n, p, \infty)$. 2) 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