arXiv subjects

Piotr Sankowski

Publications and source records attributed to Piotr Sankowski.

At least 55 records · Page 3Linked to original sources

arXiv2017

Approximate nearest neighbors search without false negatives for $l_2$ for $c>\sqrt{\log\log{n}}$

In this paper, we report progress on answering the open problem presented by Pagh~[14], who considered the nearest neighbor search without false negatives for the Hamming distance. We show new data structures for solving the $c$-approximate nearest neighbors problem without false negatives for Euclidean high dimensional space $\mathcal{R}^d$. These data structures work for any $c = ω(\sqrt{\log{\log{n}}})$, where $n$ is the number of points in the input set, with poly-logarithmic query time and polynomial preprocessing time. This improves over the known algorithms, which require $c$ to be $Ω(\sqrt{d})$. This improvement is obtained by applying a sequence of reductions, which are interesting on their own. First, we reduce the problem to $d$ instances of dimension logarithmic in $n$. Next, these instances are reduced to a number of $c$-approximate nearest neighbor search instances in $\big(\mathbb{R}^k\big)^L$ space equipped with metric $m(x,y) = \max_{1 \le i \le L}(\lVert x_i - y_i\rVert_2)$.

cs.CG↗

arXiv2017

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted graph $G$ embedded on a surface of genus $g$ and a designated face $f$ bounded by a simple cycle of length $k$, uncovers a set $F \subseteq E(G)$ of size polynomial in $g$ and $k$ that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of $f$. We apply this general theorem to prove that: * given an unweighted graph $G$ embedded on a surface of genus $g$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal Steiner tree $T$ for $S$ and that has size polynomial in $g$ and $|E(T)|$; * an analogous result holds for an optimal Steiner forest for a set $S$ of terminal pairs; * given an unweighted planar graph $G$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal (edge) multiway cut $C$ separating $S$ and that has size polynomial in $|C|$. In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution.

cs.DS↗

arXiv2017

Contracting a Planar Graph Efficiently

We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in $O(1)$ time. Moreover, it can report all the arising self-loops and parallel edges. By applying the data structure, we can achieve optimal running times for decremental bridge detection, 2-edge connectivity, maximal 3-edge connected components, and the problem of finding a unique perfect matching for a static planar graph. Furthermore, we improve the running times of algorithms for several planar graph problems, including decremental 2-vertex and 3-edge connectivity, and we show that using our data structure in a black-box manner, one obtains conceptually simple optimal algorithms for computing MST and 5-coloring in planar graphs.

cs.DS↗

arXiv2017

Why Do Cascade Sizes Follow a Power-Law?

We introduce random directed acyclic graph and use it to model the information diffusion network. Subsequently, we analyze the cascade generation model (CGM) introduced by Leskovec et al. [19]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.

cs.SI↗

arXiv2016

RecSys Challenge 2016: job recommendations based on preselection of offers and gradient boosting

We present the Mim-Solution's approach to the RecSys Challenge 2016, which ranked 2nd. The goal of the competition was to prepare job recommendations for the users of the website Xing.com. Our two phase algorithm consists of candidate selection followed by the candidate ranking. We ranked the candidates by the predicted probability that the user will positively interact with the job offer. We have used Gradient Boosting Decision Trees as the regression tool.

cs.AI↗

arXiv2016

Locality-Sensitive Hashing without False Negatives for l_p

In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius $R$ in $d$ dimensional space equipped with $l_p$ norm when $p \in [1,\infty]$. Furthermore, we show how to use these hash functions to solve the $c$-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance $R$, we will certainly report it and points at distance greater than $cR$ will not be reported for $c=Ω(\sqrt{d},d^{1-\frac{1}{p}})$. The constructed algorithms work: - with preprocessing time $\mathcal{O}(n \log(n))$ and sublinear expected query time, - with preprocessing time $\mathcal{O}(\mathrm{poly}(n))$ and expected query time $\mathcal{O}(\log(n))$. Our paper reports progress on answering the open problem presented by Pagh [8] who considered the nearest neighbor search without false negatives for the Hamming distance.

cs.DS↗

arXiv2016

There is Something Beyond the Twitter Network

How information spreads through a social network? Can we assume, that the information is spread only through a given social network graph? What is the correct way to compare the models of information flow? These are the basic questions we address in this work. We focus on meticulous comparison of various, well-known models of rumor propagation in the social network. We introduce the model incorporating mass media and effects of absent nodes. In this model the information appears spontaneously in the graph. Using the most conservative metric, we showed that the distribution of cascades sizes generated by this model fits the real data much better than the previously considered models.

cs.SI↗

arXiv2016

Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in $\tilde{O}(m^{10/7} \log W)$ Time

In this paper, we study a set of combinatorial optimization problems on weighted graphs: the shortest path problem with negative weights, the weighted perfect bipartite matching problem, the unit-capacity minimum-cost maximum flow problem and the weighted perfect bipartite $b$-matching problem under the assumption that $\Vert b\Vert_1=O(m)$. We show that each one of these four problems can be solved in $\tilde{O}(m^{10/7}\log W)$ time, where $W$ is the absolute maximum weight of an edge in the graph, which gives the first in over 25 years polynomial improvement in their sparse-graph time complexity. At a high level, our algorithms build on the interior-point method-based framework developed by Madry (FOCS 2013) for solving unit-capacity maximum flow problem. We develop a refined way to analyze this framework, as well as provide new variants of the underlying preconditioning and perturbation techniques. Consequently, we are able to extend the whole interior-point method-based approach to make it applicable in the weighted graph regime.

cs.DS↗

arXiv2016

The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree

In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an $n$-vertex graph $G=(V,E,w)$ with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set $S \subseteq V$, which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear $o(n)$ time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a $(6+\varepsilon)$-approximate Steiner tree of a general graph in $\tilde{O}(\sqrt{n} \log D)$ time per terminal addition or removal. Here, $D$ denotes the stretch of the metric induced by $G$. For planar graphs we achieve the same running time and the approximation ratio of $(2+\varepsilon)$. Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time $(4+\varepsilon)$-approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.

cs.DS↗

arXiv2016

Optimal Dynamic Strings

In this paper we study the fundamental problem of maintaining a dynamic collection of strings under the following operations: concat - concatenates two strings, split - splits a string into two at a given position, compare - finds the lexicographical order (less, equal, greater) between two strings, LCP - calculates the longest common prefix of two strings. We present an efficient data structure for this problem, where an update requires only $O(\log n)$ worst-case time with high probability, with $n$ being the total length of all strings in the collection, and a query takes constant worst-case time. On the lower bound side, we prove that even if the only possible query is checking equality of two strings, either updates or queries take amortized $Ω(\log n)$ time; hence our implementation is optimal. Such operations can be used as a basic building block to solve other string problems. We provide two examples. First, we can augment our data structure to provide pattern matching queries that may locate occurrences of a specified pattern $p$ in the strings in our collection in optimal $O(|p|)$ time, at the expense of increasing update time to $O(\log^2 n)$. Second, we show how to maintain a history of an edited text, processing updates in $O(\log t \log \log t)$ time, where $t$ is the number of edits, and how to support pattern matching queries against the whole history in $O(|p| \log t \log \log t)$ time. Finally, we note that our data structure can be applied to test dynamic tree isomorphism and to compare strings generated by dynamic straight-line grammars.

cs.DS↗

arXiv2015

Algorithmic Complexity of Power Law Networks

It was experimentally observed that the majority of real-world networks follow power law degree distribution. The aim of this paper is to study the algorithmic complexity of such "typical" networks. The contribution of this work is twofold. First, we define a deterministic condition for checking whether a graph has a power law degree distribution and experimentally validate it on real-world networks. This definition allows us to derive interesting properties of power law networks. We observe that for exponents of the degree distribution in the range $[1,2]$ such networks exhibit double power law phenomenon that was observed for several real-world networks. Our observation indicates that this phenomenon could be explained by just pure graph theoretical properties. The second aim of our work is to give a novel theoretical explanation why many algorithms run faster on real-world data than what is predicted by algorithmic worst-case analysis. We show how to exploit the power law degree distribution to design faster algorithms for a number of classical P-time problems including transitive closure, maximum matching, determinant, PageRank and matrix inverse. Moreover, we deal with the problems of counting triangles and finding maximum clique. Previously, it has been only shown that these problems can be solved very efficiently on power law graphs when these graphs are random, e.g., drawn at random from some distribution. However, it is unclear how to relate such a theoretical analysis to real-world graphs, which are fixed. Instead of that, we show that the randomness assumption can be replaced with a simple condition on the degrees of adjacent vertices, which can be used to obtain similar results. As a result, in some range of power law exponents, we are able to solve the maximum clique problem in polynomial time, although in general power law networks the problem is NP-complete.

cs.DS↗

arXiv2014

Approximation Algorithms for Steiner Tree Problems Based on Universal Solution Frameworks

This paper summarizes the work on implementing few solutions for the Steiner Tree problem which we undertook in the PAAL project. The main focus of the project is the development of generic implementations of approximation algorithms together with universal solution frameworks. In particular, we have implemented Zelikovsky 11/6-approximation using local search framework, and 1.39-approximation by Byrka et al. using iterative rounding framework. These two algorithms are experimentally compared with greedy 2-approximation, with exact but exponential time Dreyfus-Wagner algorithm, as well as with results given by a state-of-the-art local search techniques by Uchoa and Werneck. The results of this paper are twofold. On one hand, we demonstrate that high level algorithmic concepts can be designed and efficiently used in C++. On the other hand, we show that the above algorithms with good theoretical guarantees, give decent results in practice, but are inferior to state-of-the-art heuristical approaches.

cs.DS↗

arXiv2014

Optimal decremental connectivity in planar graphs

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in $O(n)$ time. This results improves over previously known $O(n \log n)$ time algorithm.

cs.DS↗

arXiv2014

Efficiency of Truthful and Symmetric Mechanisms in One-sided Matching

We study the efficiency (in terms of social welfare) of truthful and symmetric mechanisms in one-sided matching problems with {\em dichotomous preferences} and {\em normalized von Neumann-Morgenstern preferences}. We are particularly interested in the well-known {\em Random Serial Dictatorship} mechanism. For dichotomous preferences, we first show that truthful, symmetric and optimal mechanisms exist if intractable mechanisms are allowed. We then provide a connection to online bipartite matching. Using this connection, it is possible to design truthful, symmetric and tractable mechanisms that extract 0.69 of the maximum social welfare, which works under assumption that agents are not adversarial. Without this assumption, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least a third of the maximum social welfare. For normalized von Neumann-Morgenstern preferences, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least $\frac{1}{e}\frac{ν(\opt)^2}{n}$, where $ν(\opt)$ is the maximum social welfare and $n$ is the number of both agents and items. On the hardness side, we show that no truthful mechanism can achieve a social welfare better than $\frac{ν(\opt)^2}{n}$.

cs.GT↗

arXiv2013

Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected $n$-vertex planar graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a min $st$-cut in $G$? We show how to answer such queries in constant time with $O(n\log^4n)$ preprocessing time and $O(n\log n)$ space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in $O(n\log^4n)$ time and $O(n\log n)$ space. Additionally, an explicit representation can be obtained in $O(C)$ time and space where $C$ is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead, or deterministically with an additional $\log^2 n$ factor in the preprocessing times.

cs.DM↗

arXiv2013

Algebraic Algorithms for b-Matching, Shortest Undirected Paths, and f-Factors

Let G=(V,E) be a graph with f:V\to Z_+ a function assigning degree bounds to vertices. We present the first efficient algebraic algorithm to find an f-factor. The time is \tilde{O}(f(V)^ω). More generally for graphs with integral edge weights of maximum absolute value W we find a maximum weight f-factor in time \tilde{O}(Wf(V)^ω). (The algorithms are randomized, correct with high probability and Las Vegas; the time bound is worst-case.) We also present three specializations of these algorithms: For maximum weight perfect f-matching the algorithm is considerably simpler (and almost identical to its special case of ordinary weighted matching). For the single-source shortest-path problem in undirected graphs with conservative edge weights, we present a generalization of the shortest-path tree, and we compute it in \tilde{O(Wn^ω) time. For bipartite graphs, we improve the known complexity bounds for vertex capacitated max-flow and min-cost max-flow on a subclass of graphs.

cs.DS↗

arXiv2012

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.

cs.DM↗

arXiv2012

Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^ω) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^ω) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^ω) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^ω) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.

cs.DS↗