Showing papers on "Disjoint sets published in 2016"

The second eigenvalue of the fractional p-Laplacian

Abstract: We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \[ \inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}. \] We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.

Bounds on the Parameters of Locally Recoverable Codes

Abstract: A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert–Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets.

Connected Component Model for Multi-Object Tracking

Abstract: In multi-object tracking, it is critical to explore the data associations by exploiting the temporal information from a sequence of frames rather than the information from the adjacent two frames. Since straightforwardly obtaining data associations from multi-frames is an NP-hard multi-dimensional assignment (MDA) problem, most existing methods solve this MDA problem by either developing complicated approximate algorithms, or simplifying MDA as a 2D assignment problem based upon the information extracted only from adjacent frames. In this paper, we show that the relation between associations of two observations is the equivalence relation in the data association problem, based on the spatial–temporal constraint that the trajectories of different objects must be disjoint. Therefore, the MDA problem can be equivalently divided into independent subproblems by equivalence partitioning. In contrast to existing works for solving the MDA problem, we develop a connected component model (CCM) by exploiting the constraints of the data association and the equivalence relation on the constraints. Based upon CCM, we can efficiently obtain the global solution of the MDA problem for multi-object tracking by optimizing a sequence of independent data association subproblems. Experiments on challenging public data sets demonstrate that our algorithm outperforms the state-of-the-art approaches.

Orthopairs and granular computing

Abstract: Pairs of disjoint sets (orthopairs) naturally arise or have points in common with many tools to manage uncertainty: rough sets, shadowed sets, version spaces, three-valued logics, etc. Indeed, they can be used to model partial knowledge, borderline cases, consensus, examples and counter-examples pairs. Moreover, generalized versions of orthopairs are the well known theories of Atanassov intuitionistic fuzzy sets and possibility theory and the newly established three-way decision theory. Thus, it is worth studying them on an abstract level in order to outline general properties that can then be casted to the different paradigms they are in connection with. In this paper, we will review how to define orthopairs and a hierarchy on them in the light of granular computing. Aggregation operators will also be discussed as well as possible generalizations and connections with different paradigms. This will permit us to point out new facets of these paradigms and outline some possible future developments.

Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms

Abstract: Let M=(E, I) be a matroid and let S=lS1, ċ , Str be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y⊆E of size at most q, if there is a set X ∈ S disjoint from Y with X∪ Y ∈ I, then there is a set Xˆ ∈ S disjoint from Y with Xˆ ∪ Y ∈ I. By the classic result of Bollobas, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+qp) sets. In his famous “two families theorem” from 1977, Lovasz proved that the same bound also holds for any matroid representable over a field F. We give an efficient construction of a q-representative family of size at most (p+qp) in time bounded by a polynomial in (p+qp), t, and the time required for field operations.We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include the following:—In the Long Directed Cycle problem, the input is a directed n-vertex graph G and the positive integer k. The task is to find a directed cycle of length at least k in G, if such a cycle exists. As a consequence of our 6.75k+o(k)nO(1) time algorithm, we have that a directed cycle of length at least log n, if such a cycle exists, can be found in polynomial time.—In the Minimum Equivalent Graph (MEG) problem, we are seeking a spanning subdigraph D′ of a given n-vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D.—We provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many “connectivity” problems such as Hamiltonian Cycle or Steiner Tree can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t.For the special case of uniform matroids on n elements, we give a faster algorithm to compute a representative family. We use this algorithm to provide the fastest known deterministic parameterized algorithms for k-Path, k-Tree, and, more generally, k-Subgraph Isomorphism, where the k-vertex pattern graph is of constant treewidth.

LIMITS and Applications of Group Algebras for Parameterized Problems

Abstract: The fastest known randomized algorithms for several parameterized problems use reductions to the k-MlD problem: detection of multilinear monomials of degree k in polynomials presented as circuits The fastest known algorithm for k-MlD is based on 2k evaluations of the circuit over a suitable algebra We use communication complexity to show that it is essentially optimal within this evaluation framework On the positive side, we give additional applications of the method: finding a copy of a given tree on k nodes, a minimum set of nodes that dominate at least t nodes, and an m-dimensional k-matching In each case, we achieve a faster algorithm than what was known before We also apply the algebraic method to problems in exact counting Among other results, we show that a variation of it can break the trivial upper bounds for the disjoint summation problem

Computing inversion-free mappings by simplex assembly

Abstract: We present a novel method, called Simplex Assembly, to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.

Density-based cluster algorithms for the identification of core sets

Abstract: The core-set approach is a discretization method for Markov state models of complex molecular dynamics. Core sets are disjoint metastable regions in the conformational space, which need to be known prior to the construction of the core-set model. We propose to use density-based cluster algorithms to identify the cores. We compare three different density-based cluster algorithms: the CNN, the DBSCAN, and the Jarvis-Patrick algorithm. While the core-set models based on the CNN and DBSCAN clustering are well-converged, constructing core-set models based on the Jarvis-Patrick clustering cannot be recommended. In a well-converged core-set model, the number of core sets is up to an order of magnitude smaller than the number of states in a conventional Markov state model with comparable approximation error. Moreover, using the density-based clustering one can extend the core-set method to systems which are not strongly metastable. This is important for the practical application of the core-set method because most biologically interesting systems are only marginally metastable. The key point is to perform a hierarchical density-based clustering while monitoring the structure of the metric matrix which appears in the core-set method. We test this approach on a molecular-dynamics simulation of a highly flexible 14-residue peptide. The resulting core-set models have a high spatial resolution and can distinguish between conformationally similar yet chemically different structures, such as register-shifted hairpin structures.

Quantum corrections to holographic mutual information

Abstract: We compute the leading contribution to the mutual information (MI) of two disjoint spheres in the large distance regime for arbitrary conformal field theories (CFT) in any dimension. This is achieved by refining the operator product expansion method introduced by Cardy [1]. For CFTs with holographic duals the leading contribution to the MI at long distances comes from bulk quantum corrections to the Ryu-Takayanagi area formula. According to the FLM proposal [2] this equals the bulk MI between the two disjoint regions spanned by the boundary spheres and their corresponding minimal area surfaces. We compute this quantum correction and provide in this way a non-trivial check of the FLM proposal.

A survey of Turan problems for expansions

Abstract: The r-expansion G+ of a graph G is the r-uniform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let ex r (n, F) denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of the r-uniform hypergraph F. Many problems in extremal set theory ask for the determination of ex r (n, G+) for various graphs G. We survey these Turan-type problems, focusing on recent developments.

The Calder\'on problem for the fractional Schr\"odinger equation

Abstract: We show global uniqueness in an inverse problem for the fractional Schrodinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderon problem.

Spin structures and entanglement of two disjoint intervals in conformal field theories

Abstract: We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We nd that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal eld theory on the underlying Riemann surface with a given spin structure. The analytical ndings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point.

Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping

Abstract: We study interior regularity issues for systems of elliptic equations of the type − Δ u i = f i , β ( x ) − β ∑ j ≠ i a i j u i | u i | p − 1 | u j | p + 1 set in domains Ω ⊂ R N , for N ⩾ 1 . The paper is devoted to the derivation of C 0 , α estimates that are uniform in the competition parameter β > 0 , as well as to the regularity of the limiting free-boundary problem obtained for β → + ∞ . The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters a i j are only non-negative, and thus may vanish for specific couples ( i , j ) . As a main consequence, in the limit β → + ∞ , densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary p > 0 . In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups. These equations are very common in the study of Bose–Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.

Low-Congestion Shortcuts without Embedding

Abstract: Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. Recent work by [Ghaffari and Hauepler; SODA'16] showed that this phenomenon can be seen as the broad underlying reason for the pervasive $\Omega(\sqrt{n} + D)$ lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, this work also introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast $O(D \log^{O(1)} n)$ distributed algorithms for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner. Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA'16] relies heavily on having access to a genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem - even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining a restricted and more structured form of shortcuts and giving a novel construction algorithm which efficiently finds a shortcut which is, up to a logarithmic factor, as good as the best shortcut that exists for a given network. This new construction algorithm directly leads to an $O(D \log^{O(1)} n)$-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist - without the need to compute any embedding. This includes the first efficient algorithm for bounded genus graphs.

New rotation sets in a family of torus homeomorphisms

Abstract: We construct a family $$\{\varPhi _t\}_{t\in [0,1]}$$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $$\rho (\varPhi _t)$$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of [0, 1], consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $$\rho (\varPhi _t)$$ varies continuously with t, the set of extreme points of $$\rho (\varPhi _t)$$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.

Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture

Abstract: A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence a good understanding of these functions has implications in wider generality. Our goal is to study the topological properties of the Julia sets of entire functions of disjoint type. In particular, we give a detailed description of the topology of their connected components. More precisely, consider a "Julia continuum" C of such a function, i.e. the closure in the Riemann sphere of a component of the Julia set. We show that infinity is a terminal point of C, and that C has span zero in the sense of Lelek; under a mild geometric assumption on the function C is arc-like. (Whether every span zero continuum is also arc-like was a famous question in continuum theory, only recently resolved in the negative.) Conversely, we construct a single disjoint-type entire function with the remarkable property that each arc-like continuum with at least one terminal point is realised as a Julia continuum. The class of arc-like continua with terminal points is uncountable. It includes, in particular, the sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc, so these can all occur as Julia continua of a disjoint-type entire function. We give similar descriptions of the possible topology of Julia continua that contain periodic points or points with bounded orbits, and answer a question of Baranski and Karpinska by showing that Julia continua need not contain points that are accessible from the Fatou set. Furthermore, we construct an entire function whose Julia set has connected components on which the iterates tend to infinity pointwise, but not uniformly. This is related to a famous conjecture of Eremenko concerning escaping sets of entire functions.

In situ and in-transit analysis of cosmological simulations

Abstract: Modern cosmological simulations have reached the trillion-element scale, rendering data storage and subsequent analysis formidable tasks. To address this circumstance, we present a new MPI-parallel approach for analysis of simulation data while the simulation runs, as an alternative to the traditional workflow consisting of periodically saving large data sets to disk for subsequent ‘offline’ analysis. We demonstrate this approach in the compressible gasdynamics/N-body code Nyx, a hybrid $\mbox{MPI}+\mbox{OpenMP}$ code based on the BoxLib framework, used for large-scale cosmological simulations. We have enabled on-the-fly workflows in two different ways: one is a straightforward approach consisting of all MPI processes periodically halting the main simulation and analyzing each component of data that they own (‘in situ’). The other consists of partitioning processes into disjoint MPI groups, with one performing the simulation and periodically sending data to the other ‘sidecar’ group, which post-processes it while the simulation continues (‘in-transit’). The two groups execute their tasks asynchronously, stopping only to synchronize when a new set of simulation data needs to be analyzed. For both the in situ and in-transit approaches, we experiment with two different analysis suites with distinct performance behavior: one which finds dark matter halos in the simulation using merge trees to calculate the mass contained within iso-density contours, and another which calculates probability distribution functions and power spectra of various fields in the simulation. Both are common analysis tasks for cosmology, and both result in summary statistics significantly smaller than the original data set. We study the behavior of each type of analysis in each workflow in order to determine the optimal configuration for the different data analysis algorithms.

Disjoint intersection types

Abstract: Dunfield showed that a simply typed core calculus with intersection types and a merge operator is able to capture various programming language features. While his calculus is type-safe, it is not coherent: different derivations for the same expression can elaborate to expressions that evaluate to different values. The lack of coherence is an important disadvantage for adoption of his core calculus in implementations of programming languages, as the semantics of the programming language becomes implementation-dependent. This paper presents λ_i: a coherent and type-safe calculus with a form of intersection types and a merge operator. Coherence is achieved by ensuring that intersection types are disjoint and programs are sufficiently annotated to avoid type ambiguity. We propose a definition of disjointness where two types A and B are disjoint only if certain set of types are common supertypes of A and B. We investigate three different variants of λ_i, with three variants of disjointness. In the simplest variant, which does not allow ⊤ types, two types are disjoint if they do not share any common supertypes at all. The other two variants introduce ⊤ types and refine the notion of disjointness to allow two types to be disjoint when the only the set of common supertypes are top-like. The difference between the two variants with ⊤ types is on the definition of top-like types, which has an impact on which types are allowed on intersections. We present a type system that prevents intersection types that are not disjoint, as well as an algorithmic specifications to determine whether two types are disjoint for all three variants.

Removal and Stability for Erdös--Ko--Rado

Abstract: A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdos--Ko--Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $\gamma n \le k \le (\tfrac12 - \gamma)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal lemma for large families, showing that families of size close to $\ell \binom{n-1}{k-1}$ with relatively few disjoint pairs must be close to a union of $\ell$ stars. Our lemma holds for a wide range of uniformities; in particular, when $\ell = 1$, the result holds for all $2 \le k < \frac{n}{2}$ and provides sharp quantitative estimates. We use this removal lemma to answer a question of Bollobas, Naraya...

Succinct Choice Dictionaries.

Abstract: The choice dictionary is introduced as a data structure that can be initialized with a parameter $n\in\mathbb{N}=\{1,2,\ldots\}$ and subsequently maintains an initially empty subset $S$ of $\{1,\ldots,n\}$ under insertion, deletion, membership queries and an operation choice that returns an arbitrary element of $S$. The choice dictionary appears to be fundamental in space-efficient computing. We show that there is a choice dictionary that can be initialized with $n$ and an additional parameter $t\in\mathbb{N}$ and subsequently occupies $n+O(n(t/w)^t+\log n)$ bits of memory and executes each of the four operations insert, delete, contains (i.e., a membership query) and choice in $O(t)$ time on a word RAM with a word length of $w=\Omega(\log n)$ bits. In particular, with $w=\Theta(\log n)$, we can support insert, delete, contains and choice in constant time using $n+O(n/(\log n)^t)$ bits for arbitrary fixed $t$. We extend our results to maintaining several pairwise disjoint subsets of $\{1,\ldots,n\}$. We study additional space-efficient data structures for subsets $S$ of $\{1,\ldots,n\}$, including one that supports only insertion and an operation extract-choice that returns and deletes an arbitrary element of $S$. All our main data structures can be initialized in constant time and support efficient iteration over the set $S$, and we can allow changes to $S$ while an iteration over $S$ is in progress. We use these abilities crucially in designing the most space-efficient algorithms known for solving a number of graph and other combinatorial problems in linear time. In particular, given an undirected graph $G$ with $n$ vertices and $m$ edges, we can output a spanning forest of $G$ in $O(n+m)$ time with at most $(1+\epsilon)n$ bits of working memory for arbitrary fixed $\epsilon>0$.

M\"obius disjointness for non-uniquely ergodic skew products

Abstract: For $\tau>2$, let $T$ be a $C^\tau$ skew product map of the form $(x+\alpha,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the Mobius sequence. This in particular implies that any non-uniquely ergodic $C^\tau$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the Mobius sequence.

Separating Regular Languages with First-Order Logic

Abstract: Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem: given two regular input languages of finite words, decide whether there exists a first-order definable separator. We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation. This yields an EXPTIME algorithm for checking first-order separability. Moreover, the correctness proof of this algorithm yields a stronger result, namely a description of a possible separator. Finally, we generalize this technique to answer the same question for regular languages of infinite words.

ddNF: An Efficient Data Structure for Header Spaces

Abstract: Network Verification is emerging as a critical enabler to manage large complex networks. In order to scale to data-center networks found in Microsoft Azure we developed a new data structure called ddNF, disjoint difference Normal Form, that serves as an efficient container for a small set of equivalence classes over header spaces. Our experiments show that ddNFs outperform representations proposed in previous work, in particular representations based on BDDs, and is especially suited for incremental verification. The advantage is observed empirically; in the worst case ddNFs are exponentially inferior than using BDDs to represent equivalence classes. We analyze main characteristics of ddNFs to explain the advantages we are observing.

Truncated linear statistics associated with the top eigenvalues of random matrices

Abstract: Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(\lambda_1,\ldots,\lambda_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=\sum_{i=1}^Nf(\lambda_i)$, where $f(\lambda)$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1