arXiv subjects

Piotr Sankowski

Publications and source records attributed to Piotr Sankowski.

At least 37 records · Page 2Linked to original sources

arXiv2024

Modeling Online Paging in Multi-Core Systems

Web requests are growing exponentially since the 90s due to the rapid development of the Internet. This process was further accelerated by the introduction of cloud services. It has been observed statistically that memory or web requests generally follow power-law distribution, Breslau et al. INFOCOM'99. That is, the $i^{\text{th}}$ most popular web page is requested with a probability proportional to $1 / i^α$ ($α> 0$ is a constant). Furthermore, this study, which was performed more than 20 years ago, indicated Zipf-like behavior, i.e., that $α\le 1$. Surprisingly, the memory access traces coming from petabyte-size modern cloud systems not only show that $α$ can be bigger than one but also illustrate a shifted power-law distribution -- called Pareto type II or Lomax. These previously not reported phenomenon calls for statistical explanation. Our first contribution is a new statistical {\it multi-core power-law} model indicating that double-power law can be attributed to the presence of multiple cores running many virtual machines in parallel on such systems. We verify experimentally the applicability of this model using the Kolmogorov-Smirnov test (K-S test). The second contribution of this paper is a theoretical analysis indicating why LRU and LFU-based algorithms perform well in practice on data satisfying power-law or multi-core assumptions. We provide an explanation by studying the online paging problem in the stochastic input model, i.e., the input is a random sequence with each request independently drawn from a page set according to a distribution $π$. We derive formulas (as a function of the page probabilities in $π$) to upper bound their ratio-of-expectations, which help in establishing O(1) performance ratio given the random sequence following power-law and multi-core power-law distributions.

cs.DS↗

arXiv2023

Compositional Search of Stable Crystalline Structures in Multi-Component Alloys Using Generative Diffusion Models

Exploring the vast composition space of multi-component alloys presents a challenging task for both \textit{ab initio} (first principles) and experimental methods due to the time-consuming procedures involved. This ultimately impedes the discovery of novel, stable materials that may display exceptional properties. Here, the Crystal Diffusion Variational Autoencoder (CDVAE) model is adapted to characterize the stable compositions of a well studied multi-component alloy, NiFeCr, with two distinct crystalline phases known to be stable across its compositional space. To this end, novel extensions to CDVAE were proposed, enhancing the model's ability to reconstruct configurations from their latent space within the test set by approximately 30\% . A fact that increases a model's probability of discovering new materials when dealing with various crystalline structures. Afterwards, the new model is applied for materials generation, demonstrating excellent agreement in identifying stable configurations within the ternary phase space when compared to first principles data. Finally, a computationally efficient framework for inverse design is proposed, employing Molecular Dynamics (MD) simulations of multi-component alloys with reliable interatomic potentials, enabling the optimization of materials property across the phase space.

cond-mat.mtrl-sci↗

arXiv2023

Contrastive News and Social Media Linking using BERT for Articles and Tweets across Dual Platforms

X (formerly Twitter) has evolved into a contemporary agora, offering a platform for individuals to express opinions and viewpoints on current events. The majority of the topics discussed on Twitter are directly related to ongoing events, making it an important source for monitoring public discourse. However, linking tweets to specific news presents a significant challenge due to their concise and informal nature. Previous approaches, including topic models, graph-based models, and supervised classifiers, have fallen short in effectively capturing the unique characteristics of tweets and articles. Inspired by the success of the CLIP model in computer vision, which employs contrastive learning to model similarities between images and captions, this paper introduces a contrastive learning approach for training a representation space where linked articles and tweets exhibit proximity. We present our contrastive learning approach, CATBERT (Contrastive Articles Tweets BERT), leveraging pre-trained BERT models. The model is trained and tested on a dataset containing manually labeled English and Polish tweets and articles related to the Russian-Ukrainian war. We evaluate CATBERT's performance against traditional approaches like LDA, and the novel method based on OpenAI embeddings, which has not been previously applied to this task. Our findings indicate that CATBERT demonstrates superior performance in associating tweets with relevant news articles. Furthermore, we demonstrate the performance of the models when applied to finding the main topic -- represented by an article -- of the whole cascade of tweets. In this new task, we report the performance of the different models in dependence on the cascade size.

cs.CL↗

arXiv2023

Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form

Algebraic techniques have had an important impact on graph algorithms so far. Porting them, e.g., the matrix inverse, into the dynamic regime improved best-known bounds for various dynamic graph problems. In this paper, we develop new algorithms for another cornerstone algebraic primitive, the Frobenius normal form (FNF). We apply our developments to dynamic and fault-tolerant exact distance oracle problems on directed graphs. For generic matrices $A$ over a finite field accompanied by an FNF, we show (1) an efficient data structure for querying submatrices of the first $k\geq 1$ powers of $A$, and (2) a near-optimal algorithm updating the FNF explicitly under rank-1 updates. By representing an unweighted digraph using a generic matrix over a sufficiently large field (obtained by random sampling) and leveraging the developed FNF toolbox, we obtain: (a) a conditionally optimal distance sensitivity oracle (DSO) in the case of single-edge or single-vertex failures, providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a multiple-failures DSO improving upon the state of the art (vd. Brand and Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved dynamic distance oracles in the case of single-edge updates, and (d) a dynamic distance oracle supporting vertex updates, i.e., changing all edges incident to a single vertex, in $\tilde{O}(n^2)$ worst-case time and distance queries in $\tilde{O}(n)$ time.

cs.DS↗

arXiv2023

Improved Distance Queries and Cycle Counting by Frobenius Normal Form

Consider an unweighted, directed graph $G$ with the diameter $D$. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time $\widetilde{O}(n^ω)$. The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.

cs.DS↗

arXiv2022

Improving Ads-Profitability Using Traffic-Fingerprints

This paper introduces the concept of traffic-fingerprints, i.e., normalized 24-dimensional vectors representing a distribution of daily traffic on a web page. Using k-means clustering we show that similarity of traffic-fingerprints is related to the similarity of profitability time patterns for ads shown on these pages. In other words, these fingerprints are correlated with the conversions rates, thus allowing us to argue about conversion rates on pages with negligible traffic. By blocking or unblocking whole clusters of pages we were able to increase the revenue of online campaigns by more than 50%.

cs.IR↗

arXiv2022

Subquadratic Dynamic Path Reporting in Directed Graphs Against an Adaptive Adversary

We study reachability and shortest paths problems in dynamic directed graphs. Whereas algebraic dynamic data structures supporting edge updates and reachability/distance queries have been known for quite a long time, they do not, in general, allow reporting the underlying paths within the same time bounds, especially against an adaptive adversary. In this paper we develop the first known fully dynamic reachability data structures working against an adaptive adversary and supporting edge updates and path queries for two natural variants: (1) point-to-point path reporting, and (2) single-source reachability tree reporting. For point-to-point queries in DAGs, we achieve $O(n^{1.529})$ worst-case update and query bounds, whereas for tree reporting in DAGs, the worst-case bounds are $O(n^{1.765})$. More importantly, we show how to lift these algorithms to work on general graphs at the cost of increasing the bounds to $n^{1+5/6+o(1)}$ and making the update times amortized. On the way to accomplishing that, we obtain two interesting subresults. We give subquadratic fully dynamic algorithms for topological order (in a DAG), and strongly connected components. To the best of our knowledge, such algorithms have not been described before. Additionally, we provide deterministic incremental data structures for reachability and shortest paths that handle edge insertions and report the respective paths within subquadratic worst-case time bounds. For reachability and $(1+ε)$-approximate shortest paths in weighted digraphs, these bounds match the best known dynamic matrix inverse-based randomized bounds for fully dynamic reachability [v.d.Brand, Nanongkai and Saranurak, FOCS'19]. For exact shortest paths in unweighted graphs, the obtained bounds in the incremental setting polynomially improve upon the respective best known randomized update/distance query bounds in the fully dynamic setting.

cs.DS↗

arXiv2021

Improved Feature Importance Computations for Tree Models: Shapley vs. Banzhaf

Shapley values are one of the main tools used to explain predictions of tree ensemble models. The main alternative to Shapley values are Banzhaf values that have not been understood equally well. In this paper we make a step towards filling this gap, providing both experimental and theoretical comparison of these model explanation methods. Surprisingly, we show that Banzhaf values offer several advantages over Shapley values while providing essentially the same explanations. We verify that Banzhaf values: (1) have a more intuitive interpretation, (2) allow for more efficient algorithms, and (3) are much more numerically robust. We provide an experimental evaluation of these theses. In particular, we show that on real world instances. Additionally, from a theoretical perspective we provide new and improved algorithm computing the same Shapley value based explanations as the algorithm of Lundberg et al. [Nat. Mach. Intell. 2020]. Our algorithm runs in $O(TLD+n)$ time, whereas the previous algorithm had $O(TLD^2+n)$ running time bound. Here, $T$ is the number of trees, $L$ is the maximum number of leaves in a tree, and $D$ denotes the maximum depth of a tree in the ensemble. Using the computational techniques developed for Shapley values we deliver an optimal $O(TL+n)$ time algorithm for computing Banzhaf values based explanations. In our experiments these algorithms give running times smaller even by an order of magnitude.

cs.LG↗

arXiv2021

Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for $d=2$. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the $A^*$ search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra's algorithm. Specifically, if the $n$ corresponding points of a weighted unit-disk graph $G$ are picked from a unit square uniformly at random, and the connectivity radius is $r\in (0,1)$, $A^*$ finds a shortest path in $G$ in $O(n)$ expected time when $r=Ω(\sqrt{\log n/n})$, even though $G$ has $Θ((nr)^2)$ edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to $A^*$. Specifically, we show that, if we are able to report the set of all $k$ points of $G$ from an arbitrary rectangular region of the plane in $O(k + t(n))$ time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in $O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all $r=Ω(\sqrt{\log n/n})$ and $O(\sqrt{n})$ expected bound for $r=Ω(n^{-1/4})$ after only near-linear preprocessing of the point set.

cs.CG↗

arXiv2021

Budget Feasible Mechanisms on Matroids

Motivated by many practical applications, in this paper we study {\em budget feasible mechanisms} where the goal is to procure independent sets from matroids. More specifically, we are given a matroid $\mathcal{M}=(E,\mathcal{I})$ where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements $E$. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with $4$-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns $α-$approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with $(3α+1)-$approximation to the optimal common independent set.

cs.GT↗

arXiv2021

Decomposable Submodular Function Minimization via Maximum Flow

This paper bridges discrete and continuous optimization approaches for decomposable submodular function minimization, in both the standard and parametric settings. We provide improved running times for this problem by reducing it to a number of calls to a maximum flow oracle. When each function in the decomposition acts on $O(1)$ elements of the ground set $V$ and is polynomially bounded, our running time is up to polylogarithmic factors equal to that of solving maximum flow in a sparse graph with $O(\vert V \vert)$ vertices and polynomial integral capacities. We achieve this by providing a simple iterative method which can optimize to high precision any convex function defined on the submodular base polytope, provided we can efficiently minimize it on the base polytope corresponding to the cut function of a certain graph that we construct. We solve this minimization problem by lifting the solutions of a parametric cut problem, which we obtain via a new efficient combinatorial reduction to maximum flow. This reduction is of independent interest and implies some previously unknown bounds for the parametric minimum $s,t$-cut problem in multiple settings.

cs.DS↗

arXiv2021

A Deterministic Parallel APSP Algorithm and its Applications

In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $\tilde{O}(nm+(n/d)^3)$ work and $\tilde{O}(d)$ depth for any depth parameter $d\in [1,n]$. To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP'17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has $\tilde{O}(nm+n^3/d^2)$ work and $\tilde{O}(d)$ depth [Ullman and Yannakakis, SIAM J. Comput. '91]. Our APSP algorithm turns out to be a powerful tool for designing efficient planar graph algorithms in both parallel and sequential regimes. One notable ingredient of our parallel APSP algorithm is a simple deterministic $\tilde{O}(nm)$-work $\tilde{O}(d)$-depth procedure for computing $\tilde{O}(n/d)$-size hitting sets of shortest $d$-hop paths between all pairs of vertices of a real-weighted digraph. Such hitting sets have also been called $d$-hub sets. Hub sets have previously proved especially useful in designing parallel or dynamic shortest paths algorithms and are typically obtained via random sampling. Our procedure implies, for example, an $\tilde{O}(nm)$-time deterministic algorithm for finding a shortest negative cycle of a real-weighted digraph. Such a near-optimal bound for this problem has been so far only achieved using a randomized algorithm [Orlin et al., Discret. Appl. Math. '18].

cs.DS↗

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).

cs.DS↗

arXiv2019

Min-Cost Flow in Unit-Capacity Planar Graphs

In this paper we give an $\widetilde{O}((nm)^{2/3}\log C)$ time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by $C$. For planar multigraphs, this improves upon the best known algorithms for general graphs: the $\widetilde{O}(m^{10/7}\log C)$ time algorithm of Cohen et al. [SODA 2017], the $O(m^{3/2}\log(nC))$ time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the $\widetilde{O}(\sqrt{n}m \log C)$ time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the $\widetilde{O}(m^{3/2})$ time barrier for min-cost flow problem in planar graphs. To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in $\widetilde{O}(m^{3/2}\log{C})$ time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: $r$-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs.

cs.DS↗

arXiv2018

Online Facility Location with Deletions

In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client's location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log n_act / log log n_act)-competitive algorithm, where n_act is the number of active clients at the end of the input sequence. Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log n/ log log n), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log m + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where m is number of points in the input metric and c is the capacity of any open facility.

cs.DS↗

arXiv2018

NC Algorithms for Weighted Planar Perfect Matching and Related Problems

Consider a planar graph $G=(V,E)$ with polynomially bounded edge weight function $w:E\to [0, poly(n)]$. The main results of this paper are NC algorithms for the following problems: - minimum weight perfect matching in $G$, - maximum cardinality and maximum weight matching in $G$ when $G$ is bipartite, - maximum multiple-source multiple-sink flow in $G$ where $c:E\to [1, poly(n)]$ is a polynomially bounded edge capacity function, - minimum weight $f$-factor in $G$ where $f:V\to [1, poly(n)]$, - min-cost flow in $G$ where $c:E\to [1, poly(n)]$ is a polynomially bounded edge capacity function and $b:V\to [1, poly(n)]$ is a polynomially bounded vertex demand function. There have been no known NC algorithms for any of these problems previously (Before this and independent paper by Anari and Vazirani). In order to solve these problems we develop a new relatively simple but versatile framework that is combinatorial in spirit. It handles the combinatorial structure of matchings directly and needs to only know weights of appropriately defined matchings from algebraic subroutines.

cs.DS↗

arXiv2018

Round Compression for Parallel Matching Algorithms

For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+Ω(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+ε)$-approximation to maximum matching, for any fixed constant $ε>0$, in $O((\log \log n)^2)$ rounds.

cs.DS↗

arXiv2017

A Tight Bound for Shortest Augmenting Paths on Trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].

cs.DM↗