arXiv subjects

Jakub Łącki

Publications and source records attributed to Jakub Łącki.

At least 37 records · Page 2Linked to original sources

arXiv2023

Fully Dynamic Consistent $k$-Center Clustering

We study the consistent k-center clustering problem. In this problem, the goal is to maintain a constant factor approximate $k$-center solution during a sequence of $n$ point insertions and deletions while minimizing the recourse, i.e., the number of changes made to the set of centers after each point insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that in the incremental setting, where deletions are not allowed, one can obtain $k \cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and $k$-median, and demonstrated a matching lower bound. However, no algorithm for the fully dynamic setting achieves less than the trivial $O(k)$ changes per update, which can be obtained by simply reclustering the full dataset after every update. In this work, we give the first algorithm for consistent $k$-center clustering for the fully dynamic setting, i.e., when both point insertions and deletions are allowed, and improves upon a trivial $O(k)$ recourse bound. Specifically, our algorithm maintains a constant factor approximate solution while ensuring worst-case constant recourse per update, which is optimal in the fully dynamic setting. Moreover, our algorithm is deterministic and is therefore correct even if an adaptive adversary chooses the insertions and deletions.

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arXiv2023

Adaptive Massively Parallel Connectivity in Optimal Space

We study the problem of finding connected components in the Adaptive Massively Parallel Computation (AMPC) model. We show that when we require the total space to be linear in the size of the input graph the problem can be solved in $O(\log^* n)$ rounds in forests (with high probability) and $2^{O(\log^* n)}$ expected rounds in general graphs. This improves upon an existing $O(\log \log_{m/n} n)$ round algorithm. For the case when the desired number of rounds is constant we show that both problems can be solved using $Θ(m + n \log^{(k)} n)$ total space in expectation (in each round), where $k$ is an arbitrarily large constant and $\log^{(k)}$ is the $k$-th iterate of the $\log_2$ function. This improves upon existing algorithms requiring $Ω(m + n \log n)$ total space.

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arXiv2022

Constant Approximation for Normalized Modularity and Associations Clustering

We study the problem of graph clustering under a broad class of objectives in which the quality of a cluster is defined based on the ratio between the number of edges in the cluster, and the total weight of vertices in the cluster. We show that our definition is closely related to popular clustering measures, namely normalized associations, which is a dual of the normalized cut objective, and normalized modularity. We give a linear time constant-approximate algorithm for our objective, which implies the first constant-factor approximation algorithms for normalized modularity and normalized associations.

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arXiv2022

Near-Optimal Decremental Hopsets with Applications

Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $β$ and stretch $(1+ε)$ is a set of edges such that for any pair of nodes $u, v \in V$, there is a path in $G \cup H$ of at most $β$ hops, whose length is within a $(1+ε)$ factor from the distance between $u$ and $v$ in $G$. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of $2^{\log^{Ω(1)} n}$ [Bernstein, FOCS'09; HKN, FOCS'14; Chechik, FOCS'18]. By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain $(2k-1)(1+ε)$-approximate all-pairs shortest paths (for any constant $k \geq 2)$, in $\tilde{O}(n^{1/k})$ amortized update time and $O(k)$ query time. This improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS'18] that has a query time of $O(\log \log(nW))$, where $W$ is the aspect ratio, and the amortized update time is $n^{1/k}\cdot(\frac{1}ε)^{\tilde{O}(\sqrt{\log n})}$. For sparse graphs our construction nearly matches the best known static running time / query time tradeoff. We also obtain near-optimal bounds for maintaining approximate multi-source shortest paths and distance sketches, and get improved bounds for approximate single-source shortest paths. Our algorithms are randomized and our bounds hold with high probability against an oblivious adversary.

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arXiv2022

Hierarchical Agglomerative Graph Clustering in Poly-Logarithmic Depth

Obtaining scalable algorithms for hierarchical agglomerative clustering (HAC) is of significant interest due to the massive size of real-world datasets. At the same time, efficiently parallelizing HAC is difficult due to the seemingly sequential nature of the algorithm. In this paper, we address this issue and present ParHAC, the first efficient parallel HAC algorithm with sublinear depth for the widely-used average-linkage function. In particular, we provide a $(1+ε)$-approximation algorithm for this problem on $m$ edge graphs using $\tilde{O}(m)$ work and poly-logarithmic depth. Moreover, we show that obtaining similar bounds for exact average-linkage HAC is not possible under standard complexity-theoretic assumptions. We complement our theoretical results with a comprehensive study of the ParHAC algorithm in terms of its scalability, performance, and quality, and compare with several state-of-the-art sequential and parallel baselines. On a broad set of large publicly-available real-world datasets, we find that ParHAC obtains a 50.1x speedup on average over the best sequential baseline, while achieving quality similar to the exact HAC algorithm. We also show that ParHAC can cluster one of the largest publicly available graph datasets with 124 billion edges in a little over three hours using a commodity multicore machine.

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arXiv2022

Hierarchical Clustering in Graph Streams: Single-Pass Algorithms and Space Lower Bounds

The Hierarchical Clustering (HC) problem consists of building a hierarchy of clusters to represent a given dataset. Motivated by the modern large-scale applications, we study the problem in the \streaming model, in which the memory is heavily limited and only a single or very few passes over the input are allowed. Specifically, we investigate whether a good hierarchical clustering can be obtained, or at least whether we can approximately estimate the value of the optimal hierarchy. To measure the quality of a hierarchy, we use the HC minimization objective introduced by Dasgupta. Assuming that the input is an $n$-vertex weighted graph whose edges arrive in a stream, we derive the following results on space-vs-accuracy tradeoffs: * With $O(n\cdot \text{polylog}\,{n})$ space, we develop a single-pass algorithm, whose approximation ratio matches the currently best offline algorithm. * When the space is more limited, namely, $n^{1-o(1)}$, we prove that no algorithm can even estimate the value of optimum HC tree to within an $o(\frac{\log{n}}{\log\log{n}})$ factor, even when allowed $\text{polylog}{\,{n}}$ passes over the input. * In the most stringent setting of $\text{polylog}\,{n}$ space, we rule out algorithms that can even distinguish between "highly"-vs-"poorly" clusterable graphs, namely, graphs that have an $n^{1/2-o(1)}$ factor gap between their HC objective value. * Finally, we prove that any single-pass streaming algorithm that computes an optimal HC tree requires to store almost the entire input even if allowed exponential time. Our algorithmic results establish a general structural result that proves that cut sparsifiers of input graph can preserve cost of "balanced" HC trees to within a constant factor. Our lower bound results include a new streaming lower bound for a novel problem "One-vs-Many-Expanders", which can be of independent interest.

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arXiv2021

Optimal Decremental Connectivity in Non-Sparse Graphs

We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity on a sparse graph. While being Monte-Carlo, our reduction supports a certain final self-check that can be used in Las Vegas algorithms for static problems such as Unique Perfect Matching. For non-sparse graphs with $Ω(n \operatorname{polylog} n)$ edges, our connectivity and $2$-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an $O(m \log (n^2 / m) + n \operatorname{polylog} n)$-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with $Ω(n^2)$ edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an $Ω(\log n/\log\log n)$ worst-case time lower bound in the decremental setting [Alstrup, Thore Husfeldt, FOCS'98] as well as an $Ω(\log n)$ amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

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arXiv2021

Scalable Community Detection via Parallel Correlation Clustering

Graph clustering and community detection are central problems in modern data mining. The increasing need for analyzing billion-scale data calls for faster and more scalable algorithms for these problems. There are certain trade-offs between the quality and speed of such clustering algorithms. In this paper, we design scalable algorithms that achieve high quality when evaluated based on ground truth. We develop a generalized sequential and shared-memory parallel framework based on the LambdaCC objective (introduced by Veldt et al.), which encompasses modularity and correlation clustering. Our framework consists of highly-optimized implementations that scale to large data sets of billions of edges and that obtain high-quality clusters compared to ground-truth data, on both unweighted and weighted graphs. Our empirical evaluation shows that this framework improves the state-of-the-art trade-offs between speed and quality of scalable community detection. For example, on a 30-core machine with two-way hyper-threading, our implementations achieve orders of magnitude speedups over other correlation clustering baselines, and up to 28.44x speedups over our own sequential baselines while maintaining or improving quality.

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arXiv2021

Hierarchical Agglomerative Graph Clustering in Nearly-Linear Time

We study the widely used hierarchical agglomerative clustering (HAC) algorithm on edge-weighted graphs. We define an algorithmic framework for hierarchical agglomerative graph clustering that provides the first efficient $\tilde{O}(m)$ time exact algorithms for classic linkage measures, such as complete- and WPGMA-linkage, as well as other measures. Furthermore, for average-linkage, arguably the most popular variant of HAC, we provide an algorithm that runs in $\tilde{O}(n\sqrt{m})$ time. For this variant, this is the first exact algorithm that runs in subquadratic time, as long as $m=n^{2-ε}$ for some constant $ε> 0$. We complement this result with a simple $ε$-close approximation algorithm for average-linkage in our framework that runs in $\tilde{O}(m)$ time. As an application of our algorithms, we consider clustering points in a metric space by first using $k$-NN to generate a graph from the point set, and then running our algorithms on the resulting weighted graph. We validate the performance of our algorithms on publicly available datasets, and show that our approach can speed up clustering of point datasets by a factor of 20.7--76.5x.

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arXiv2020

Faster DBSCAN via subsampled similarity queries

DBSCAN is a popular density-based clustering algorithm. It computes the $ε$-neighborhood graph of a dataset and uses the connected components of the high-degree nodes to decide the clusters. However, the full neighborhood graph may be too costly to compute with a worst-case complexity of $O(n^2)$. In this paper, we propose a simple variant called SNG-DBSCAN, which clusters based on a subsampled $ε$-neighborhood graph, only requires access to similarity queries for pairs of points and in particular avoids any complex data structures which need the embeddings of the data points themselves. The runtime of the procedure is $O(sn^2)$, where $s$ is the sampling rate. We show under some natural theoretical assumptions that $s \approx \log n/n$ is sufficient for statistical cluster recovery guarantees leading to an $O(n\log n)$ complexity. We provide an extensive experimental analysis showing that on large datasets, one can subsample as little as $0.1\%$ of the neighborhood graph, leading to as much as over 200x speedup and 250x reduction in RAM consumption compared to scikit-learn's implementation of DBSCAN, while still maintaining competitive clustering performance.

cs.LG↗

arXiv2020

Parallel Graph Algorithms in Constant Adaptive Rounds: Theory meets Practice

We study fundamental graph problems such as graph connectivity, minimum spanning forest (MSF), and approximate maximum (weight) matching in a distributed setting. In particular, we focus on the Adaptive Massively Parallel Computation (AMPC) model, which is a theoretical model that captures MapReduce-like computation augmented with a distributed hash table. We show the first AMPC algorithms for all of the studied problems that run in a constant number of rounds and use only $O(n^ε)$ space per machine, where $0 < ε< 1$. Our results improve both upon the previous results in the AMPC model, as well as the best-known results in the MPC model, which is the theoretical model underpinning many popular distributed computation frameworks, such as MapReduce, Hadoop, Beam, Pregel and Giraph. Finally, we provide an empirical comparison of the algorithms in the MPC and AMPC models in a fault-tolerant distriubted computation environment. We empirically evaluate our algorithms on a set of large real-world graphs and show that our AMPC algorithms can achieve improvements in both running time and round-complexity over optimized MPC baselines.

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arXiv2020

Near-Optimal Massively Parallel Graph Connectivity

Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is $n^δ$ for some desirably small constant $δ\in (0, 1)$. We present an algorithm that for graphs with diameter $D$ in the wide range $[\log^ε n, n]$, takes $O(\log D)$ rounds to identify the connected components and takes $O(\log \log n)$ rounds for all other graphs. The algorithm is randomized, succeeds with high probability, does not require prior knowledge of $D$, and uses an optimal total space of $O(m)$. We complement this by showing a conditional lower-bound based on the widely believed TwoCycle conjecture that $Ω(\log D)$ rounds are indeed necessary in this setting. Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni et al. [FOCS 2018] who presented an algorithm with $O(\log D \cdot \log \log n)$ round-complexity. Our algorithm improves this result for the whole range of values of $D$ and almost settles the problem due to the conditional lower-bound. Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of the (CRCW) PRAM in asymptotically the same number of rounds.

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arXiv2019

Walking Randomly, Massively, and Efficiently

We introduce a set of techniques that allow for efficiently generating many independent random walks in the Massive Parallel Computation (MPC) model with space per machine strongly sublinear in the number of vertices. In this space-per-machine regime, many natural approaches to graph problems struggle to overcome the $Θ(\log n)$ MPC round complexity barrier. Our techniques enable breaking this barrier for PageRank---one of the most important applications of random walks---even in more challenging directed graphs, and for approximate bipartiteness and expansion testing. In the undirected case, we start our random walks from the stationary distribution, which implies that we approximately know the empirical distribution of their next steps. This allows for preparing continuations of random walks in advance and applying a doubling approach. As a result we can generate multiple random walks of length $l$ in $Θ(\log l)$ rounds on MPC. Moreover, we show that under the popular 1-vs.-2-Cycles conjecture, this round complexity is asymptotically tight. For directed graphs, our approach stems from our treatment of the PageRank Markov chain. We first compute the PageRank for the undirected version of the input graph and then slowly transition towards the directed case, considering convex combinations of the transition matrices in the process. For PageRank, we achieve the following round complexities for damping factor equal to $1 - ε$: * in $O(\log \log n + \log 1 / ε)$ rounds for undirected graphs (with $\tilde O(m / ε^2)$ total space), * in $\tilde O(\log^2 \log n + \log^2 1/ε)$ rounds for directed graphs (with $\tilde O((m+n^{1+o(1)}) / poly\, ε)$ total space).

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arXiv2019

Fully Dynamic Matching: Beating 2-Approximation in $Δ^ε$ Update Time

In fully dynamic graphs, we know how to maintain a 2-approximation of maximum matching extremely fast, that is, in polylogarithmic update time or better. In a sharp contrast and despite extensive studies, all known algorithms that maintain a $2-Ω(1)$ approximate matching are much slower. Understanding this gap and, in particular, determining the best possible update time for algorithms providing a better-than-2 approximate matching is a major open question. In this paper, we show that for any constant $ε> 0$, there is a randomized algorithm that with high probability maintains a $2-Ω(1)$ approximate maximum matching of a fully-dynamic general graph in worst-case update time $O(Δ^ε+\text{polylog } n)$, where $Δ$ is the maximum degree. Previously, the fastest fully dynamic matching algorithm providing a better-than-2 approximation had $O(m^{1/4})$ update-time [Bernstein and Stein, SODA 2016]. A faster algorithm with update-time $O(n^ε)$ was known, but worked only for maintaining the size (and not the edges) of the matching in bipartite graphs [Bhattacharya, Henzinger, and Nanongkai, STOC 2016].

cs.DS↗

arXiv2019

Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs

We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in $\widetilde{O}(mn^{4/3}\log{W}/ε)$ total time (where the edge weights are from $[1,W]$) and explicitly maintains a $(1+ε)$-approximate distance matrix. For a fixed $ε>0$, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is $o(n^2)$ regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC'02, Bernstein STOC'13] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the $\widetilde{O}(\cdot)$ notation). Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of $\widetilde{O}(n/d)$ vertices which hit a shortest path between each pair of vertices, provided it has hop-length $Ω(d)$. We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions.

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arXiv2019

Massively Parallel Computation via Remote Memory Access

We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms using MapReduce and a distributed hash table service. This extension allows us to give new graph algorithms with much lower round complexities compared to the best known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in $O(1)$ rounds and connectivity/minimum spanning tree in $O(\log\log_{m/n} n)$ rounds both using $O(n^δ)$ space per machine for constant $δ< 1$. In the same memory regime for MPC, the best known algorithms for these problems require polylog $n$ rounds. Our results imply that the 2-Cycle conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.

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arXiv2018

Connected Components at Scale via Local Contractions

As a fundamental tool in hierarchical graph clustering, computing connected components has been a central problem in large-scale data mining. While many known algorithms have been developed for this problem, they are either not scalable in practice or lack strong theoretical guarantees on the parallel running time, that is, the number of communication rounds. So far, the best proven guarantee is $\Oh(\log n)$, which matches the running time in the PRAM model. In this paper, we aim to design a distributed algorithm for this problem that works well in theory and practice. In particular, we present a simple algorithm based on contractions and provide a scalable implementation of it in MapReduce. On the theoretical side, in addition to showing $\Oh(\log n)$ convergence for all graphs, we prove an $\Oh(\log \log n)$ parallel running time with high probability for a certain class of random graphs. We work in the MPC model that captures popular parallel computing frameworks, such as MapReduce, Hadoop or Spark. On the practical side, we show that our algorithm outperforms the state-of-the-art MapReduce algorithms. To confirm its scalability, we report empirical results on graphs with several trillions of edges.

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arXiv2018

Decremental SPQR-trees for Planar Graphs

We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over $Ω(n)$ operations, is $O(\log^2 n)$. Via SPQR-trees, we give a decremental data structure for maintaining $3$-vertex connectivity in planar graphs. It answers queries in $O(1)$ time and processes edge deletions and contractions in $O(\log^2 n)$ amortized time. This is an exponential improvement over the previous best bound of $O(\sqrt{n}\,)$ that has stood for over 20 years. In addition, the previous data structures only supported edge deletions.

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