Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $\alpha$ be the cardinality of the largest independent set. Our goal is to estimate $\alpha$ in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating $\beta$, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating $\alpha$ and $\beta$. For unit-length intervals, we obtain a $(2+\epsilon)$-approximation to $\alpha$ and $\beta$ in poly$(\frac{\log(n)}{\epsilon})$ space. We also show a matching lower bound. Combined with the $3/2$-approximation for insertion-only streams by Cabello and Perez-Lanterno [CP15], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a $\left(\frac{8\sqrt{3}}{\pi}\right)$-approximation to $\alpha$ and $\beta$ in poly$(\log(n), \epsilon^{-1})$ space, which is closely related to the hexagonal circle packing constant. We provide algorithms for estimating $\alpha$ for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating $\alpha$ and $\beta$, where the parameter is the ratio of maximum to minimum interval length.Comment: The lower bound for arbitrary length intervals in the previous version contains a bug, we are updating the submission to reflect thi

Finding structure in data streams : correlations, independent sets, and matchings

The streaming model supposes that, rather than being available all at once, the data is received in a piecemeal fashion. In a world of massive data sets, streaming algorithms give a complementary approach to distributed algorithms: with the data all being available in one place but at different times, rather than at the same time in different places. We examine three different single-pass streaming problems where existing results show limited feasibility. We consider realistic relaxations or restrictions of these problems which allow for more efficient algorithms. In the correlation outliers problem, we wish to identify pairs of unusually correlated signals from a streamed matrix of observations. We show that a simple application of existing technique is space-optimal but has slow query time when the outlier threshold is small. We demonstrate how we can achieve faster query times at the cost of storing a larger data summary. In the maximum independent set problem, we wish to find an edge-less induced subgraph of maximum size. For arbitrary graphs, given as a stream of edges, it is known that no space-efficient algorithm exists. We consider a variant streaming model, where the graph is received vertex by vertex. While we show this model still does not admit efficient algorithms for general graphs, we demonstrate efficient approximation algorithms for various special graph classes. In the maximum matching problem, we wish to find a disjoint subset of edges of largest possible size. The greedy algorithm gives us an easy 2-approximation for streams of edges, but the problem becomes infeasible to solve if we allow unlimited edge deletions. We consider a model where, instead, a limited number of deletions are allowed. We describe several new approximation algorithms with complexity parameterised by the number of deletions. We also present new techniques which may lead to the development of corresponding tight lower bounds

Independent Sets in Vertex-Arrival Streams

We consider the maximal and maximum independent set problems in three models of graph streams: - In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set. - In the "explicit" vertex model, the input stream is a sequence of vertices making up the graph. Every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than in edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires Omega({n^2}/{c^6}) bits of space, where n is the number of vertices of the input graph. It is already known that Theta~({n^2}/{c^2}) bits of space are necessary and sufficient in the edge arrival model (Halldórsson et al. 2012), thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping. - In the "implicit" vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Sublinear Algorithm And Lower Bound For Combinatorial Problems

As the scale of the problems we want to solve in real life becomes larger, the input sizes of the problems we want to solve could be much larger than the memory of a single computer. In these cases, the classical algorithms may no longer be feasible options, even when they run in linear time and linear space, as the input size is too large. In this thesis, we study various combinatorial problems in different computation models that process large input sizes using limited resources. In particular, we consider the query model, streaming model, and massively parallel computation model. In addition, we also study the tradeoffs between the adaptivity and performance of algorithms in these models.We first consider two graph problems, vertex coloring problem and metric traveling salesman problem (TSP). The main results are structure results for these problems, which give frameworks for achieving sublinear algorithms of these problems in different models. We also show that the sublinear algorithms for (∆ + 1)-coloring problem are tight. We then consider the graph sparsification problem, which is an important technique for designing sublinear algorithms. We give proof of the existence of a linear size hypergraph cut sparsifier, along with a polynomial algorithm that calculates one. We also consider sublinear algorithms for this problem in the streaming and query models. Finally, we study the round complexity of submodular function minimization (SFM). In particular, we give a polynomial lower bound on the number of rounds we need to compute s − t max flow - a special case of SFM - in the streaming model. We also prove a polynomial lower bound on the number of rounds we need to solve the general SFM problem in polynomial queries

Approximate Computing Survey, Part I: Terminology and Software & Hardware Approximation Techniques

The rapid growth of demanding applications in domains applying multimedia processing and machine learning has marked a new era for edge and cloud computing. These applications involve massive data and compute-intensive tasks, and thus, typical computing paradigms in embedded systems and data centers are stressed to meet the worldwide demand for high performance. Concurrently, the landscape of the semiconductor field in the last 15 years has constituted power as a first-class design concern. As a result, the community of computing systems is forced to find alternative design approaches to facilitate high-performance and/or power-efficient computing. Among the examined solutions, Approximate Computing has attracted an ever-increasing interest, with research works applying approximations across the entire traditional computing stack, i.e., at software, hardware, and architectural levels. Over the last decade, there is a plethora of approximation techniques in software (programs, frameworks, compilers, runtimes, languages), hardware (circuits, accelerators), and architectures (processors, memories). The current article is Part I of our comprehensive survey on Approximate Computing, and it reviews its motivation, terminology and principles, as well it classifies and presents the technical details of the state-of-the-art software and hardware approximation techniques.Comment: Under Review at ACM Computing Survey

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