114 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Zooming in on the Universe: In Search of Quantum Spacetime

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    This thesis investigates low-dimensional models of nonperturbative quantum gravity, with a special focus on Causal Dynamical Triangulations (CDT). We define the so-called curvature profile, a new quantum gravitational observable based on the quantum Ricci curvature. We subsequently study its coarse-graining capabilities on a class of regular, two-dimensional polygons with isolated curvature singularities, and we determine the curvature profile of (1+1)-dimensional CDT with toroidal topology. Next, we focus on CDT in 2+1 dimensions, intvestigating the behavior of the two-dimensional spatial slice geometries. We then turn our attention to matrix models, exploring a differential reformulation of the integrals over one- and two-matrix ensembles. Finally, we provide a hands-on introduction to computer simulations of CDT quantum gravity.Comment: Ph.D. thesi

    On learning the structure of clusters in graphs

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    Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs. The first part of the thesis studies the classical spectral clustering algorithm, and presents a tighter analysis on its performance. This result explains why it works under a much weaker and more natural condition than the ones studied in the literature, and helps to close the gap between the theoretical guarantees of the spectral clustering algorithm and its excellent empirical performance. The second part of the thesis builds on the theoretical guarantees of the previous part and shows that, when the clusters of the underlying graph have certain structures, spectral clustering with fewer than k eigenvectors is able to produce better output than classical spectral clustering in which k eigenvectors are employed, where k is the number of clusters. This presents the first work that discusses and analyses the performance of spectral clustering with fewer than k eigenvectors, and shows that general structures of clusters can be learned with spectral methods. The third part of the thesis considers efficient learning of the structure of clusters with local algorithms, whose runtime depends only on the size of the target clusters and is independent of the underlying input graph. While the objective of classical local clustering algorithms is to find a cluster which is sparsely connected to the rest of the graph, this part of the thesis presents a local algorithm that finds a pair of clusters which are densely connected to each other. This result demonstrates that certain structures of clusters can be learned efficiently in the local setting, even in the massive graphs which are ubiquitous in real-world applications. The final part of the thesis studies the problem of learning densely connected clusters in hypergraphs. The developed algorithm is based on a new heat diffusion process, whose analysis extends a sequence of recent work on the spectral theory of hypergraphs. It allows the structure of clusters to be learned in datasets modelling higher-order relations of objects and can be applied to efficiently analyse many complex datasets occurring in practice. All of the presented theoretical results are further extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs

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    We consider the massively parallel computation (MPC) model, which is a theoretical abstraction of large-scale parallel processing models such as MapReduce. In this model, assuming the widely believed 1-vs-2-cycles conjecture, solving many basic graph problems in O(1)O(1) rounds with a strongly sublinear memory size per machine is impossible. We improve on the recent work of Holm and T\v{e}tek [SODA 2023] that bypass this barrier for problems when a planar embedding of the graph is given. In the previous work, on graphs of size nn with O(n/S)O(n/\mathcal{S}) machines, the memory size per machine needs to be at least S=n2/3+Ω(1)\mathcal{S} = n^{2/3+\Omega(1)}, whereas we extend their work to the fully scalable regime, where the memory size per machine can be S=nδ\mathcal{S} = n^{\delta} for any constant 0<δ<10< \delta < 1. We give the first constant round fully scalable algorithms for embedded planar graphs for the problems of (i) connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the ε\varepsilon-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be incorporated into our recursive framework to obtain constant-round (1+ε)(1+\varepsilon)-approximation algorithms for the problems of computing (iii) single source shortest path (SSSP), (iv) global min-cut, and (v) stst-max flow. All previous results on cuts and flows required linear memory in the MPC model. Furthermore, our results give new algorithms for problems that implicitly involve embedded planar graphs. We give as corollaries constant round fully scalable algorithms for (vi) 2D Euclidean MST using O(n)O(n) total memory and (vii) (1+ε)(1+\varepsilon)-approximate weighted edit distance using O~(n2δ)\widetilde{O}(n^{2-\delta}) memory. Our main technique is a recursive framework combined with novel graph drawing algorithms to compute smaller embedded planar graphs in constant rounds in the fully scalable setting.Comment: To appear in SODA24. 55 pages, 9 figures, 1 table. Added section on weighted edit distance and shortened abstrac

    Cover and Hitting Times of Hyperbolic Random Graphs

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    We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range (2,3)(2,3). In particular, we focus on the expected times for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that up to multiplicative constants: the cover time is n(logn)2n(\log n)^2, the maximum hitting time is nlognn\log n, and the average hitting time is nn. The first two results hold in expectation and a.a.s. and the last in expectation (with respect to the HRG). We prove these results by determining the effective resistance either between an average vertex and the well-connected "center" of HRGs or between an appropriately chosen collection of extremal vertices. We bound the effective resistance by the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane on which we overlay a forest-like structure.Comment: 34 pages, 2 figures. To appear at the conference RANDOM 202

    FKT is not universal — A planar Holant dichotomy for symmetric constraints

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    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    Testing, Learning, Sampling, Sketching

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    We study several problems about sublinear algorithms, presented in two parts. Part I: Property testing and learning. There are two main goals of research in property testing and learning theory. The first is to understand the relationship between testing and learning, and the second is to develop efficient testing and learning algorithms. We present results towards both goals. - An oft-repeated motivation for property testing algorithms is to help with model selection in learning: to efficiently check whether the chosen hypothesis class (i.e. learning model) will successfully learn the target function. We present in this thesis a proof that, for many of the most useful and natural hypothesis classes (including halfspaces, polynomial threshold functions, intersections of halfspaces, etc.), the sample complexity of testing in the distribution-free model is nearly equal to that of learning. This shows that testing does not give a significant advantage in model selection in this setting. - We present a simple and general technique for transforming testing and learning algorithms designed for the uniform distribution over {0, 1}^d or [n]^d into algorithms that work for arbitrary product distributions over R d . This leads to an improvement and simplification of state-of-the-art results for testing monotonicity, learning intersections of halfspaces, learning polynomial threshold functions, and others. Part II. Adjacency and distance sketching for graphs. We initiate the thorough study of adjacency and distance sketching for classes of graphs. Two open problems in sublinear algorithms are: 1) to understand the power of randomization in communication; and 2) to characterize the sketchable distance metrics. We observe that constant-cost randomized communication is equivalent to adjacency sketching in a hereditary graph class, which in turn implies the existence of an efficient adjacency labeling scheme, the subject of a major open problem in structural graph theory. Therefore characterizing the adjacency sketchable graph classes (i.e. the constant-cost communication problems) is the probabilistic equivalent of this open problem, and an essential step towards understanding the power of randomization in communication. This thesis gives the first results towards a combined theory of these problems and uses this connection to obtain optimal adjacency labels for subgraphs of Cartesian products, resolving some questions from the literature. More generally, we begin to develop a theory of graph sketching for problems that generalize adjacency, including different notions of distance sketching. This connects the well-studied areas of distance sketching in sublinear algorithms, and distance labeling in structural graph theory
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