Showing papers on "Disjoint sets published in 2005"

Efficient identification of overlapping communities

Abstract: In this paper, we present an efficient algorithm for finding overlapping communities in social networks. Our algorithm does not rely on the contents of the messages and uses the communication graph only. The knowledge of the structure of the communities is important for the analysis of social behavior and evolution of the society as a whole, as well as its individual members. This knowledge can be helpful in discovering groups of actors that hide their communications, possibly for malicious reasons. Although the idea of using communication graphs for identifying clusters of actors is not new, most of the traditional approaches, with the exception of the work by Baumes et al, produce disjoint clusters of actors, de facto postulating that an actor is allowed to belong to at most one cluster. Our algorithm is significantly more efficient than the previous algorithm by Baumes et al; it also produces clusters of a comparable or better quality.

Logarithmic Lower Bounds in the Cell-Probe Model

Abstract: We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized Omega(lg n) lower bound per operation for several data structural problems on n elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of Omega(lg n / lglg n) for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first Omega(log_B n) lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update.

Hermitean matrix model free energy: Feynman graph technique for all genera

Abstract: We present a diagrammatic technique for calculating the free energy of the Hermitian one-matrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

Quantitative noise sensitivity and exceptional times for percolation

Abstract: One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.

Hardness of the undirected edge-disjoint paths problem

Abstract: We show that there is no log 1 over 3-eM approximation for the undirected Edge-Disjoint Paths problem unless NP ⊆ ZPTIME(npolylog(n), where M is the size of the graph and e is any positive constant. This hardness result also applies to the undirected All-or-Nothing Multicommodity Flow problem and the undirected Node-Disjoint Paths problem.

Multicomplementary operators via finite Fourier transform

Abstract: A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position–momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

Decentralized algorithms using both local and random probes for P2P load balancing

Abstract: We study randomized algorithms for placing a sequence of n nodes on a circle with unit perimeter. Nodes divide the circle into disjoint arcs. We desire that a newly-arrived node (which is oblivious of its index in the sequence) choose its position on the circle by learning the positions of as few existing nodes as possible. At the same time, we desire that that the variation in arc-lengths be small. To this end, we propose a new algorithm that works as follows: The kth node chooses r random points on the circle, inspects the sizes of v arcs in the vicinity of each random point, and places itself at the mid-point of the largest arc encountered. We show that for any combination of r and v satisfying rv iÝ c log k, where c is a small constant, the ratio of the largest to the smallest arc-length is at most eight w.h.p., for an arbitrarily long sequence of n nodes. This strategy of node placement underlies a novel decentralized load-balancing algorithm that we propose for Distributed Hash Tables (DHTs) in peer-to-peer environments.Underlying the analysis of our algorithm is Structured Coupon Collection over n/b disjoint cliques with b nodes per clique, for any n, b ≥ 1. Nodes are initially uncovered. At each step, we choose d nodes independently and uniformly at random. If all the nodes in the corresponding cliques are covered, we do nothing. Otherwise, from among the chosen cliques with at least one uncovered node, we select one at random and cover an uncovered node within that clique. We show that as long as bd ≥ c log n, O(n) steps are sufficient to cover all nodes w.h.p. and each of the first Ω(n) steps succeeds in covering a node w.h.p. These results are then utilized to analyze a stochastic process for growing binary trees that are highly balanced -- the leaves of the tree belong to at most four different levels with high probability.

Combining Nonstably Infinite Theories

Abstract: The Nelson---Oppen combination method combines decision procedures for first-order theories over disjoint signatures into a single decision procedure for the union theory. In order to be correct, the method requires that the component theories be stably infinite. This restriction makes the method inapplicable to many interesting theories such as, for instance, theories having only finite models. In this paper, we describe two extensions of the Nelson---Oppen method that address the problem of combining theories that are not stably infinite. In our extensions, the component decision procedures exchange not only equalities between shared variables but also certain cardinality constraints. Applications of our results include the combination of theories having only finite models, as well as the combination of nonstably infinite theories with the theory of equality, the theories of total and partial orders, and the theory of lattices with maximum and minimum.

Combining intruder theories

Abstract: Most of the decision procedures for symbolic analysis of protocols are limited to a fixed set of algebraic operators associated with a fixed intruder theory. Examples of such sets of operators comprise XOR, multiplication/exponentiation, abstract encryption/decryption. In this paper we give an algorithm for combining decision procedures for arbitrary intruder theories with disjoint sets of operators, provided that solvability of ordered intruder constraints, a slight generalization of intruder constraints, can be decided in each theory. This is the case for most of the intruder theories for which a decision procedure has been given. In particular our result allows us to decide trace-based security properties of protocols that employ any combination of the above mentioned operators with a bounded number of sessions.

Optimal System of Loops on an Orientable Surface

Abstract: Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.

Building k edge-disjoint spanning trees of minimum total length for isometric data embedding

Abstract: Isometric data embedding requires construction of a neighborhood graph that spans all data points so that geodesic distance between any pair of data points could be estimated by distance along the shortest path between the pair on the graph. This paper presents an approach for constructing k-edge-connected neighborhood graphs. It works by finding k edge-disjoint spanning trees the sum of whose total lengths is a minimum. Experiments show that it outperforms the nearest neighbor approach for geodesic distance estimation.

Series of independent random variables in rearrangement invariant spaces: An operator approach

Abstract: This paper studies series of independent random variables in rearrangement invariant spaces X on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1 ~ p ~ c~ and Lorentz spaces A¢. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumption Lp C X, p < oc is sufficient to guarantee that convergence of such series in X (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence in X of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.

On a question of Erdős and Moser

Abstract: For two finite sets of real numbers A and B , one says that B is sum-free with respect to A if the sum set b + b ' ∣ b , b ' ∈ B , b ≠ b ' is disjoint from A . Forty years ago, Erdőos and Moser posed the following question. Let A be a set of n real numbers. What is the size of the largest subset B of A which is sum-free with respect to A ? In this paper, we show that any set A of n real numbers contains a set B of cardinality at least g ⁡ n ln ⁡ n which is sum-free with respect to A , where g ⁡ n tends to infinity with n . This improves earlier bounds of Klarner, Choi, and Ruzsa and is the first superlogarithmic bound for this problem. Our proof combines tools from graph theory together with several fundamental results in additive number theory such as Freiman's inverse theorem, the Balog-Szemeredi theorem, and Szemeredi's result on long arithmetic progressions. In fact, in order to obtain an explicit bound on g ⁡ n , we use the recent versions of these results, obtained by Chang and by Gowers, where significant quantitative improvements have been achieved.

Hausdorff dimension, its properties, and its surprises

Abstract: We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set $E\subset\C$ of positive planar measure and with dimension 2 such that each point in $E$ can be joined to $\infty$ by one or several curves in $\C$ such that all curves are disjoint from each other and from $E$, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in $\C$ which is not on one of these curves is in $E$. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as $z\mapsto \pi\sin(z)$.

Ramsey Theoretic Consequences of Some New Results About Algebra in the Stone-Čech Compactification

Abstract: Recently [4] we have obtained some new algebraic results about βN, the Stone-Cech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a nonempty alphabet with infinitely many variables adjoined. (The results about βW extend the Graham-Rothschild Parameter Sets Theorem.) In this paper we derive some Ramsey Theoretic consequences of these results. Among these is the following, which extends the Finite Sums Theorem. Theorem. Let N be finitely colored. Then there is a color class D which is central in N and (i) there exists a pairwise disjoint collection {Di,j : i, j ∈ ω} of central subsets of D and for each i ∈ ω there exists a sequence 〈xi,n〉n=i in Di,i such that whenever F is a finite nonempty subset of ω and f : F → {1, 2, . . . ,minF} one has that Σn∈F xf(n),n ∈ Di,j where i = f(minF ) and j = f(maxF ); and (ii) at stage n when one is chosing (x0,n, x1,n, . . . , xn,n), each xi,n may be chosen as an arbitrary element of a certain central subset of Di,i, with the choice of xi,n independent of the choice of xj,n. An analogous extension of the Graham-Rothschild Theorem is established. Also included are new results about image partition regularity and kernel partition regularity of matrices.

Algorithms for Perfectly Contractile Graphs

Abstract: We consider the class ${\cal A}$ of graphs that contain no odd hole, no antihole of length at least $5$, and no prism (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class ${\cal A}'$ of graphs that contain no odd hole, no antihole of length at least $5$, and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. In contrast we prove that determining if a general graph contains a prism (or an even prism, or an odd prism) is NP-complete.

Testing disjointness of private datasets

Abstract: Two parties, say Alice and Bob, possess two sets of elements that belong to a universe of possible values and wish to test whether these sets are disjoint or not. In this paper we consider the above problem in the setting where Alice and Bob wish to disclose no information to each other about their sets beyond the single bit: “whether the intersection is empty or not.” This problem has many applications in commercial settings where two mutually distrustful parties wish to decide with minimum possible disclosure whether there is any overlap between their private datasets. We present three protocols that solve the above problem that meet different efficiency and security objectives and data representation scenarios. Our protocols are based on Homomorphic encryption and in our security analysis, we consider the semi-honest setting as well as the malicious setting. Our most efficient construction for a large universe in terms of overall communication complexity uses a new encryption primitive that we introduce called “superposed encryption.” We formalize this notion and provide a construction that may be of independent interest. For dealing with the malicious adversarial setting we take advantage of recent efficient constructions of Universally-Composable commitments based on verifiable encryption as well as zero-knowledge proofs of language membership.

On A Hypergraph Turán Problem Of Frankl

Abstract: Let $$C^{{{\left( {2k} \right)}}}_{r}$$ be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets Pi∪Pj with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph Kr: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain $$C^{{{\left( {2k} \right)}}}_{3}$$ can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$ denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no $$C^{{{\left( {2k} \right)}}}_{3}$$ has at most as many edges as $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$.Sidorenko has given an upper bound of $$\frac{{r - 2}}{{r - 1}}$$ for the Tur´an density of $$C^{{{\left( {2k} \right)}}}_{r}$$ for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any $$C^{{{\left( 4 \right)}}}_{r}$$-free hypergraph of density $$\frac{{r - 2}}{{r - 1}} - o{\left( 1 \right)}$$ looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turan density of $$C^{{{\left( 4 \right)}}}_{r}$$ to $$\frac{{r - 2}}{{r - 1}} - c{\left( r \right)}$$, where c(r) is a constant depending only on r.The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.

On nonlocal existence results for elliptic equations with convex-concave nonlinearities

Abstract: In this paper we study the family of nonlinear elliptic Dirichlet boundary value problems with p-Laplacian and with concave–convex nonlinearity which depend on real parameter λ . We introduce nonlocal intervals ( λ i , λ i + 1 ) such that the characteristic points λ i , λ i + 1 (a priori bifurcation values) expressed in terms of exact variational principles. In these intervals, new results on the existence of positive solutions, multiple positive solutions and existence of multiple disjoint sets with infinitely many solutions are proved.

On decompositions of Banach spaces of continuous functions on Mrowka's spaces

Abstract: It is well known that if K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space C(K) of continuous functions on K has complemented copies of c 0 , i.e., C(K) ∼ c 0 ○+ X ∼ c 0 ○+ c 0 ○+ X ∼ c 0 ○+ C(K). We address the question if this could be the only type of decompositions of C(K)? c 0 into infinite-dimensional summands for K infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrowka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

Two Edge-Disjoint Hop-Constrained Paths and Polyhedra

Abstract: Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a fixed integer L \geq 2, the two edge-disjoint hop-constrained paths problem is to find a minimum cost subgraph such that between s and t there exist at least two edge-disjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem when L = 2,3. An extension of this result to the more general case where the number of required paths is arbitrary and L = 2,3 is also given. We discuss the associated polytope, P(G,L), for L = 2,3. In particular, we show in this case that the linear relaxation of P(G,L), Q(G,L), given by the trivial, the st-cut, and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2,3. We also give necessary and sufficient conditions for these inequalities to define facets of P(G,L) for L \geq 2 when G is complete. We finally investigate the dominant of P(G,L) and give a complete description of this polyhedron for L \geq 2 when P(G,L) = Q(G,L).

Finding and Maintaining Rigid Components.

Abstract: We give the first complete analysis that the complexity of finding and maintaining rigid components of planar bar-and-joint frameworks and arbitrary d-dimensional body-and-bar frameworks, using a family of algorithms called pebble games, is O(n). To this end, we introduce a new data structure problem called union pairfind, which maintains disjoint edge sets and supports pair-find queries of whether two vertices are spanned by a set. We present solutions that apply to generalizations of the pebble game algorithms, beyond the original rigidity motivation.