Exact algorithms for the joint object placement and request routing problem in content distribution networks

This paper describes two exact algorithms for the joint problem of object placement and request routing in a content distribution network (CDN). A CDN is a technology used to efficiently distribute electronic content throughout an existing Internet Protocol network. The problem consists of replicating content on the proxy servers and routing the requests for the content to a suitable proxy server in a CDN such that the total cost of distribution is minimized. An upper bound on end-to-end object transfer time is also taken into account. The problem is formulated as a nonlinear integer programming formulation which is linearized in three different ways. Two algorithms, one based on Benders decomposition and the other based on Lagrangean relaxation and decomposition, are described for the solution of the problem. Computational experiments are conducted by comparing the proposed linearizations and the two algorithms on randomly generated Internet topologies.     

A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

Inclusion-exclusion for k-CNF formulas

We show that the number of satisfying assignments of a k-CNF formula is determined uniquely from the numbers of unsatisfying assignments for clause-sets of size up to ⌊logk⌋+2. This amount of information is also shown to be necessary.     

Efficient algorithms for the longest common subsequence in k-length substrings

Finding the longest common subsequence in k-length substrings (LCSk) is a recently proposed problem motivated by computational biology. This is a generalization of the well-known LCS problem in which matching symbols from two sequences A and B are replaced with matching non-overlapping substrings of length k from A and B. We propose several algorithms for LCSk, being non-trivial incarnations of the major concepts known from LCS research (dynamic programming, sparse dynamic programming, tabulation). Our algorithms make use of a linear-time and linear-space preprocessing finding the occurrences of all the substrings of length k from one sequence in the other sequence.     

Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

The rotation graph of k-ary trees is Hamiltonian

In this paper we show that the graph of k-ary trees, connected by rotations, contains a Hamilton cycle. Our proof is constructive and thus provides a cyclic Gray code for k-ary trees. Furthermore, we identify a basic building block of this graph as the 1-skeleton of the polytopal complex dual to the lower faces of a certain cyclic polytope.     

Almost k-wise independence versus k-wise independence

We say that a distribution over {0,1}ⁿ is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε,k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (ε,k)-wise independent distributions whose statistical distance is at least n^O(k)·ε from any k-wise independent distribution. In addition, we show that for any (ε,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n^O(k)·ε.     

Upper bounds on the queuenumber of k-ary n-cubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927-958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n−1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398-412] showed that the ternary n-cube can be laid out using at most 2n−2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41-44] improved the upper bound on queuenumber to n−2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

(1)

if n?3.

(2)

if n?2 and 4?k?8.

(3)

if n?1 and k?9.

     

On structural properties of eta-expansions of identity

We investigate some structural properties of a particular class of λ-terms, the so-called η-expansions of identity , considered inside the β-calculus. This class of terms can be directly identified with a class of (unlabeled) trees, plane (or ordered) trees, which correspond to the so-called parentheses systems. By means of this identification, the class of plane trees can be equipped with an application operation as well as with a composition operation. We show that composition corresponds to a union operation on trees, while application can be characterized in terms of composition and a natural operation of shift over trees. By means of this characterization a number of results can be easily proved about the structure of this class of terms.     

Approximation algorithms for optimization problems in graphs with superlogarithmic treewidth

We present a generic scheme for approximating NP-hard problems on graphs of treewidth k=ω(logn). When a tree-decomposition of width ? is given, the scheme typically yields an ?/logn-approximation factor; otherwise, an extra logk factor is incurred. Our method applies to several basic subgraph and partitioning problems, including the maximum independent set problem.     

A polynomial-time algorithm for computing K-terminal residual reliability of d-trapezoid graphs

Consider a probabilistic graph G   in which the edges of E(G)$E(G)$ are perfectly reliable, but the vertices of V(G)$V(G)$ may fail with some known probabilities. Given a subset K   of V(G)$V(G)$, the K-terminal residual reliability of G is the probability that all operational vertices in K are connected to each other. This problem can be considered to be a generalization of two well-known reliability problems – the K-terminal reliability problem and the residual connectedness reliability problem.     

BEAMR: An exact and approximate model for the p-median problem

The p-median problem is perhaps one of the most well-known location–allocation models in the location science literature. It was originally defined by Hakimi in 1964 and 1965 and involves the location of p facilities on a network in such a manner that the total weighted distance of serving all demand is minimized. This problem has since been the subject of considerable research involving the development of specialized solution approaches as well as the development of many different types of extended model formats. One element of past research that has remained almost constant is the original ReVelle–Swain formulation [ReVelle CS, Swain R. Central facilities location. Geographical Analysis 1970;2:30–42]. With few exceptions as detailed in the paper, virtually no new formulations have been proposed for general use in solving the classic p-median problem. This paper proposes a new model formulation for the p-median problem that contains both exact and approximate features. This new p-median formulation is called Both Exact and Approximate Model Representation (BEAMR). We show that BEAMR can result in a substantially smaller integer-linear formulation for a given application of the p-median problem and can be used to solve for either an exact optimum or a bounded, close to optimal solution. We also present a methodological framework in which the BEAMR model can be used. Computational results for problems found in the OR_library of Beasley [A note on solving large p-median problems. European Journal of Operational Research 1985;21:270–3] indicate that BEAMR not only extends the application frontier for the p-median problem using general-purpose software, but for many problems represents an efficient, competitive solution approach.     

On special families of morphisms related to δ-matching and don't care symbols

The δ-matching problem is a special version of approximate pattern-matching, motivated by applications in musical information retrieval, where the alphabet Σ is an interval of integers. We investigate relations between δ-matching and pattern-matching with don't care symbol ∗ (a symbol matching every symbol, including itself). We show that the δ-matching is reducible to k instances of pattern-matching with don't cares. We investigate how the numbers δ and k are related by introducing δ-distinguishing families of morphisms. The size of corresponds to k. We show that for minimal families we have .     

Design, optimization and performance evaluation of a content distribution overlay for streaming

As an evolution of peer-to-peer (p2p) file-sharing applications, overlay-based networks are also adopted to efficiently distribute content with real-time constraints to a wide user population. In addition, they can be utilized to exploit application level strategies to overcome limitations imposed by the underlying network infrastructure, e.g., the lack of multicast support.In this perspective, the paper introduces an overlay Content Distribution Network (CDN) able to sustain the real-time delivery of data streams. To better use resources, and to face the churn affecting users, the control and optimization of the CDN are performed through a model predictive control scheme. Simulations of two use cases are provided to show the effectiveness of the proposed solution. In particular, the stream of multimedia and interactive grid data are considered.     

Computing all efficient solutions of the biobjective minimum spanning tree problem

A common way of computing all efficient (Pareto optimal) solutions for a biobjective combinatorial optimisation problem is to compute first the extreme efficient solutions and then the remaining, non-extreme solutions. The second phase, the computation of non-extreme solutions, can be based on a “k-best” algorithm for the single-objective version of the problem or on the branch-and-bound method. A k-best algorithm computes the k-best solutions in order of their objective values. We compare the performance of these two approaches applied to the biobjective minimum spanning tree problem. Our extensive computational experiments indicate the overwhelming superiority of the k-best approach. We propose heuristic enhancements to this approach which further improve its performance.     

An algorithm for computing simple k-factors

A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple k-factors for all values of k?2.     

The impact of IXPs on the AS-level topology structure of the Internet

The AS-level topology of the Internet has been quite a hot research topic in the last few years. However, only a small number of studies have been developed that give a structural interpretation of this graph. Such an interpretation is crucially important in order to test protocols and optimal routing algorithms, to design efficient networks, and for failure detection purposes. Moreover, most research does not highlight the role that IXPs have on the AS-level structure of the Internet, although their role is recognized as fundamental.The initial contribution of this study is an analysis of the most important AS-level topologies that are publicly found on the web and an analysis of the topology obtained when they are merged. We compiled structural information from this topology making considerable use of the k-core decomposition technique to delineate various particular classes of nodes. Next, we associated node properties with a reasonable modus operandi of the ASs on the Internet. The second contribution is a study of the impact that ASs connected to IXPs and BGP connections crossing IXPs have on the AS-level topology. To achieve this, we developed a procedure to gather reliable information related to IXPs and their participants.     

Fast modified global k-means algorithm for incremental cluster construction

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets. Recently, incremental approaches have been developed to resolve difficulties with the choice of starting points. The global k-means and the modified global k-means algorithms are based on such an approach. They iteratively add one cluster center at a time. Numerical experiments show that these algorithms considerably improve the k-means algorithm. However, they require storing the whole affinity matrix or computing this matrix at each iteration. This makes both algorithms time consuming and memory demanding for clustering even moderately large datasets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset. We exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage. Results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k-means algorithms.     

A k-populations algorithm for clustering categorical data

In this paper, the conventional k-modes-type algorithms for clustering categorical data are extended by representing the clusters of categorical data with k-populations instead of the hard-type centroids used in the conventional algorithms. Use of a population-based centroid representation makes it possible to preserve the uncertainty inherent in data sets as long as possible before actual decisions are made. The k-populations algorithm was found to give markedly better clustering results through various experiments.