Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.Comment: To appear in SODA 201

Simultaneous Coherent Structure Coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity

The clustering of data into physically meaningful subsets often requires assumptions regarding the number, size, or shape of the subgroups. Here, we present a new method, simultaneous coherent structure coloring (sCSC), which accomplishes the task of unsupervised clustering without a priori guidance regarding the underlying structure of the data. sCSC performs a sequence of binary splittings on the dataset such that the most dissimilar data points are required to be in separate clusters. To achieve this, we obtain a set of orthogonal coordinates along which dissimilarity in the dataset is maximized from a generalized eigenvalue problem based on the pairwise dissimilarity between the data points to be clustered. This sequence of bifurcations produces a binary tree representation of the system, from which the number of clusters in the data and their interrelationships naturally emerge. To illustrate the effectiveness of the method in the absence of a priori assumptions, we apply it to three exemplary problems in fluid dynamics. Then, we illustrate its capacity for interpretability using a high-dimensional protein folding simulation dataset. While we restrict our examples to dynamical physical systems in this work, we anticipate straightforward translation to other fields where existing analysis tools require ad hoc assumptions on the data structure, lack the interpretability of the present method, or in which the underlying processes are less accessible, such as genomics and neuroscience

MPI+X: task-based parallelization and dynamic load balance of finite element assembly

The main computing tasks of a finite element code(FE) for solving partial differential equations (PDE's) are the algebraic system assembly and the iterative solver. This work focuses on the first task, in the context of a hybrid MPI+X paradigm. Although we will describe algorithms in the FE context, a similar strategy can be straightforwardly applied to other discretization methods, like the finite volume method. The matrix assembly consists of a loop over the elements of the MPI partition to compute element matrices and right-hand sides and their assemblies in the local system to each MPI partition. In a MPI+X hybrid parallelism context, X has consisted traditionally of loop parallelism using OpenMP. Several strategies have been proposed in the literature to implement this loop parallelism, like coloring or substructuring techniques to circumvent the race condition that appears when assembling the element system into the local system. The main drawback of the first technique is the decrease of the IPC due to bad spatial locality. The second technique avoids this issue but requires extensive changes in the implementation, which can be cumbersome when several element loops should be treated. We propose an alternative, based on the task parallelism of the element loop using some extensions to the OpenMP programming model. The taskification of the assembly solves both aforementioned problems. In addition, dynamic load balance will be applied using the DLB library, especially efficient in the presence of hybrid meshes, where the relative costs of the different elements is impossible to estimate a priori. This paper presents the proposed methodology, its implementation and its validation through the solution of large computational mechanics problems up to 16k cores