Chasing Convex Bodies with Linear Competitive Ratio

We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .

Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.

Cluster analysis for multidimensional objects in fuzzy data conditions

This article presents many different areas of practical applications of multivariate cluster analysis under conditions of fuzzy initial data that are described in the literature. New algorithms and formula expressions are proposed for combining various multi-dimensional objects, the parameters of which are given by fuzzy-sets, into clusters along with calculating the coordinates of the centroids of their membership functions. Various types of clustering criteria are formulated in the form of minimizing the weighted average and the sum of distances between the centroids of objects and clusters presented in different metrics, as well as maximizing the distances between the centroids of different clusters. The formulations and mathematical models of three different NP-hard problems of multidimensional clustering in fuzzy-data conditions are proposed; while solving them any of the considered optimality criteria can be used. Heuristic algorithms for the approximate solution of two formulated problems have been developed. The algorithm for solving the 1st problem is illustrated with a numerical example. The obtained results can serve as a direction for further research and have wide practical applications.

The utility of clusters and a Hungarian clustering algorithm

Implicit in the k–means algorithm is a way to assign a value, or utility, to a cluster of points. It works by taking the centroid of the points and the value of the cluster is the sum of distances from the centroid to each point in the cluster. The aim in this paper is to introduce an alternative way to assign a value to a cluster. Motivation is provided. Moreover, whereas the k–means algorithm does not have a natural way to determine k if it is unknown, we can use our method of evaluating a cluster to find good clusters in a sequential manner. The idea uses optimizations over permutations and clusters are set by the cyclic groups; generated by the Hungarian algorithm.

Wiener Index of k-Connected Graphs

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances of all pairs of vertices in [Formula: see text]. In this paper, we show that for any even positive integer [Formula: see text], and [Formula: see text], if [Formula: see text] is a [Formula: see text]-connected graph of order [Formula: see text], then [Formula: see text], where [Formula: see text] is the [Formula: see text]th power of a graph [Formula: see text]. This partially answers an old problem of Gutman and Zhang.

The asymptotic formula on average weighted path length for scale-free modular network

In this paper, we discuss the average path length for a class of scale-free modular networks with deterministic growth. To facilitate the analysis, we define the sum of distances from all nodes to the nearest hub nodes and the nearest peripheral nodes. For the unweighted network, we find that whether the scale-free modular network is single-hub or multiple-hub, the average path length grows logarithmically with the increase of nodes number. For the weighted network, we deduce that when the network iteration [Formula: see text] tends to infinity, the average weighted shortest path length is bounded, and the result is independent of the connection method of network.

Weighted Consensus Segmentations

The problem of segmenting linearly ordered data is frequently encountered in time-series analysis, computational biology, and natural language processing. Segmentations obtained independently from replicate data sets or from the same data with different methods or parameter settings pose the problem of computing an aggregate or consensus segmentation. This Segmentation Aggregation problem amounts to finding a segmentation that minimizes the sum of distances to the input segmentations. It is again a segmentation problem and can be solved by dynamic programming. The aim of this contribution is (1) to gain a better mathematical understanding of the Segmentation Aggregation problem and its solutions and (2) to demonstrate that consensus segmentations have useful applications. Extending previously known results we show that for a large class of distance functions only breakpoints present in at least one input segmentation appear in the consensus segmentation. Furthermore, we derive a bound on the size of consensus segments. As show-case applications, we investigate a yeast transcriptome and show that consensus segments provide a robust means of identifying transcriptomic units. This approach is particularly suited for dense transcriptomes with polycistronic transcripts, operons, or a lack of separation between transcripts. As a second application, we demonstrate that consensus segmentations can be used to robustly identify growth regimes from sets of replicate growth curves.

Wiener Complexity versus the Eccentric Complexity

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G).

On Fermat-Torricelli Problem in Frechet Spaces

We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.

On the Wiener Index of the Dot Product Graph over Monogenic Semigroups

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the nite dot product of monogenic semigroups has recently been dened and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.