Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?

On Constructing Spanners from Random Gaussian Projections

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work

Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners

We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost ($\textsf{LIC}$) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that $\Omega(\textsf{LIC}_\gamma(P)/ \log\tau \log n)$ bits are required for solving a graph problem $P$ with a $\tau$-round algorithm that errs with probability at most $\gamma$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a $(2t-1)$-spanner that consists of at most $O(n^{1+1/t + \epsilon})$ edges, where $\epsilon = \Theta(1/t^2)$. Our main result is that any $O(\textsf{poly}(n))$-time algorithm must send at least $\tilde\Omega((1/t^2) n^{1+1/2t})$ bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for this problem. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with $\tilde O(n)$ communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth

Online Directed Spanners and Steiner Forests

We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $\tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $\tilde{O}(n^{2/3+\epsilon})$-competitive algorithm, and an efficient deterministic $\tilde{O}(k^{1/2+\epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $\tilde{O}(n^{3/5 + \epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $\tilde{O}(n^{2/3 + \epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $\tilde{O}(k^{1/2 + \epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $\tilde{O}(n^{26/45 + \epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion

(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

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph $G=(V, E)$ in the streaming model of computation. The goal is to find a partitioning $\mathcal{C}$ of vertices $V$ such that the subgraphs of $G$ induced by the clusters $C \in \mathcal{C}$ are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of $V$) to within a $(\delta, \epsilon)$-multiplicative/additive error with high probability. The power cut sparsifier uses $\tilde{O}(n/\epsilon\delta)$ space and edges, which we show is asymptotically tight up to polylogarithmic factors in $n$ for constant $\delta$.Comment: 31 pages, 0 figures, to appear in ITCS 202

Streaming Algorithms for Connectivity Augmentation

We study the $k$-connectivity augmentation problem ($k$-CAP) in the single-pass streaming model. Given a $(k-1)$-edge connected graph $G=(V,E)$ that is stored in memory, and a stream of weighted edges $L$ with weights in $\{0,1,\dots,W\}$, the goal is to choose a minimum weight subset $L'\subseteq L$ such that $G'=(V,E\cup L')$ is $k$-edge connected. We give a $(2+\epsilon)$-approximation algorithm for this problem which requires to store $O(\epsilon^{-1} n\log n)$ words. Moreover, we show our result is tight: Any algorithm with better than $2$-approximation for the problem requires $\Omega(n^2)$ bits of space even when $k=2$. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for $k$-CAP. We further consider a natural generalization to the fully streaming model where both $E$ and $L$ arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a $(2t-1+\epsilon)$-approximate weighted spanner of size at most $O(\epsilon^{-1} n^{1+1/t}\log n)$ for integer $t$, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on $\log W$. Using our spanner result, we provide an optimal $O(t)$-approximation for $k$-CAP in the fully streaming model with $O(nk + n^{1+1/t})$ words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), $k$-edge connected spanning subgraph ($k$-ECSS), and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass $O(t\log k)$-approximation for SNDP using $O(kn^{1+1/t})$ words of space, where $k$ is the maximum connectivity requirement

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

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum