Median Graph Shift: A New Clustering Algorithm for Graph Domain

ISSN: 1051-4651 Print ISBN: 978-1-4244-7542-1International audiencen the context of unsupervised clustering, a new algorithm for the domain of graphs is introduced. In this paper, the key idea is to adapt the mean-shift clustering and its variants proposed for the domain of feature vectors to graph clustering. These algorithms have been applied successfully in image analysis and computer vision domains. The proposed algorithm works in an iterative manner by shifting each graph towards the median graph in a neighborhood. Both the set median graph and the generalized median graph are tested for the shifting procedure. In the experiment part, a set of cluster validation indices are used to evaluate our clustering algorithm and a comparison with the well-known Kmeans algorithm is provided

A comparison between two representatives of a set of graphs: median vs barycenter graph

In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.Preprin

A comparison between two representatives of a set of graphs: median vs barycenter graph

Trabajo presentado al Joint IAPR International Workshop on Structural, Syntactic and Statistical Pattern Recognition (SSPR&SPR) celebrado en Esmirna (Turquía) del 18 al 20 de agosto de 2010.In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.This work was supported by projects: 'CONSOLIDER-INGENIO 2010 Multimodal interaction in pattern recognition and computer vision' (V-00069), 'Robotica ubicua para entornos urbanos' (J-01225).Peer Reviewe

Synthesizing species trees from gene trees using the parameterized and graph-theoretic approaches

Gene trees describe how parts of the species have evolved over time, and it is assumed that gene trees have evolved along the branches of the species tree. However, some of gene trees are often discordant with the corresponding species tree due to the complicated evolution history of genes. To overcome this obstacle, median problems have emerged as a major tool for synthesizing species trees by reconciling discordance in a given collection of gene trees. Given a collection of gene trees and a cost function, the median problem seeks a tree, called median tree, that minimizes the overall cost to the gene trees. Median tree problems are typically NP-hard, and there is an increased interest in making such median tree problems available for large-scale species tree construction. In this thesis work, we first show that the gene duplication median tree problem satisfied the weaker version of the Pareto property and propose a parameterized algorithm to solve the gene duplication median tree problem. Second, we design two efficient methods to handle the issues of applying the parameterized algorithm to unrooted gene trees which are sampled from the different species. Third, we introduce the graph-theoretic formulation of the Robinson-Foulds median tree problem and a new tree edit operation. Fourth, we propose a new metric between two phylogenetic trees and examine the statistical properties of the metric. Finally, we propose a new clustering criteria in a bipartite network and propose a new NP-hard problem and its ILP formulation

Interval Query Problem on Cube-Free Median Graphs

In this paper, we introduce the \emph{interval query problem} on cube-free median graphs. Let $G$ be a cube-free median graph and $\mathcal{S}$ be a commutative semigroup. For each vertex $v$ in $G$, we are given an element $p(v)$ in $\mathcal{S}$. For each query, we are given two vertices $u,v$ in $G$ and asked to calculate the sum of $p(z)$ over all vertices $z$ belonging to a $u-v$ shortest path. This is a common generalization of range query problems on trees and grids. In this paper, we provide an algorithm to answer each interval query in $O(\log^2 n)$ time. The required data structure is constructed in $O(n\log^3 n)$ time and $O(n\log^2 n)$ space. To obtain our algorithm, we introduce a new technique, named the \emph{stairs decomposition}, to decompose an interval of cube-free median graphs into simpler substructures.Comment: ISAAC'21, 21 page

Locally Self-Adjusting Skip Graphs

We present a distributed self-adjusting algorithm for skip graphs that minimizes the average routing costs between arbitrary communication pairs by performing topological adaptation to the communication pattern. Our algorithm is fully decentralized, conforms to the $\mathcal{CONGEST}$ model (i.e. uses $O(\log n)$ bit messages), and requires $O(\log n)$ bits of memory for each node, where $n$ is the total number of nodes. Upon each communication request, our algorithm first establishes communication by using the standard skip graph routing, and then locally and partially reconstructs the skip graph topology to perform topological adaptation. We propose a computational model for such algorithms, as well as a yardstick (working set property) to evaluate them. Our working set property can also be used to evaluate self-adjusting algorithms for other graph classes where multiple tree-like subgraphs overlap (e.g. hypercube networks). We derive a lower bound of the amortized routing cost for any algorithm that follows our model and serves an unknown sequence of communication requests. We show that the routing cost of our algorithm is at most a constant factor more than the amortized routing cost of any algorithm conforming to our computational model. We also show that the expected transformation cost for our algorithm is at most a logarithmic factor more than the amortized routing cost of any algorithm conforming to our computational model