Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

We show an $\Omega\big(\Delta^{\frac{1}{3}-\frac{\eta}{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(\Delta^{1+\eta}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In particular, given a network graph \mbox{$G=(V,E)$}, in the weak \LOCAL\ model nodes communicate in synchronous rounds and they can use unbounded local computation. We assume that the nodes have no identifiers, but that instead, the computation starts with an initial valid vertex coloring. A node can \textbf{broadcast} a \textbf{single} message of \textbf{unbounded} size to its neighbors and receives the \textbf{set of messages} sent to it by its neighbors. That is, if two neighbors of a node $v\in V$ send the same message to $v$, $v$ will receive this message only a single time; without any further knowledge, $v$ cannot know whether a received message was sent by only one or more than one neighbor. Neighborhood graphs have been essential in the proof of lower bounds for distributed coloring algorithms, e.g., \cite{linial92,Kuhn2006On}. Our proof analyzes the recursive structure of the neighborhood graph of the respective model to devise an $\Omega\big(\Delta^{\frac{1}{3}-\frac{\eta}{3}}\big)$ lower bound on the runtime for any deterministic distributed $\mathcal{O}\big(\Delta^{1+\eta}\big)$-graph coloring algorithm. Furthermore, we hope that the proof technique improves the understanding of neighborhood graphs in general and that it will help towards finding a lower (runtime) bound for distributed graph coloring in the standard \LOCAL\ model. Our proof technique works for one-round algorithms in the standard \LOCAL\ model and provides a simpler and more intuitive proof for an existing $\Omega(\Delta^2)$ lower bound.


Similar Publications

Grids allow users flexible on-demand usage of computing resources through remote communication networks. A remarkable example of a Grid in High Energy Physics (HEP) research is used in the ALICE experiment at European Organization for Nuclear Research CERN. Physicists can submit jobs used to process the huge amount of particle collision data produced by the Large Hadron Collider (LHC). Read More


The $CONGEST$ model for distributed network computing is well suited for analyzing the impact of limiting the throughput of a network on its capacity to solve tasks efficiently. For many "global" problems there exists a lower bound of $\Omega(D + \sqrt{n/B})$, where $B$ is the amount of bits that can be exchanged between two nodes in one round of communication, $n$ is the number of nodes and $D$ is the diameter of the graph. Typically, upper bounds are given only for the case $B=O(\log n)$, or for the case $B = +\infty$. Read More


We consider the problem of routing in presence of faults in undirected weighted graphs. More specifically, we focus on the design of compact name-independent fault-tolerant routing schemes, where the designer of the scheme is not allowed to assign names to nodes, i.e. Read More


On the one hand, the correctness of routing protocols in networks is an issue of utmost importance for guaranteeing the delivery of messages from any source to any target. On the other hand, a large collection of routing schemes have been proposed during the last two decades, with the objective of transmitting messages along short routes, while keeping the routing tables small. Regrettably, all these schemes share the property that an adversary may modify the content of the routing tables with the objective of, e. Read More


The paper is devoted to an analytical study of the "master-worker" framework scalability on multiprocessors with distributed memory. A new model of parallel computations called BSF is proposed. The BSF model is based on BSP and SPMD models. Read More


In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. Read More


The main goal for this article is to compare performance penalties when using KVM virtualization and Docker containers for creating isolated environments for HPC applications. The article provides both data obtained using commonly accepted synthetic tests (High Performance Linpack) and real life applications (OpenFOAM). The article highlights the influence on resulting application performance of major infrastructure configuration options: CPU type presented to VM, networking connection type used. Read More


In this paper, we propose a vital data analysis platform which resolves existing problems to utilize vital data for real-time actions. Recently, IoT technologies have been progressed but in the healthcare area, real-time actions based on analyzed vital data are not considered sufficiently yet. The causes are proper use of analyzing methods of stream / micro batch processing and network cost. Read More


The paper addresses the problem of emulating a regular register in a synchronous distributed system where clients invoking ${\sf read}()$ and ${\sf write}()$ operations are anonymous while server processes maintaining the state of the register may be compromised by rational adversaries (i.e., a server might behave as \emph{rational malicious Byzantine} process). Read More


Many modern parallel computing systems are heterogeneous at their node level. Such nodes may comprise general purpose CPUs and accelerators (such as, GPU, or Intel Xeon Phi) that provide high performance with suitable energy-consumption characteristics. However, exploiting the available performance of heterogeneous architectures may be challenging. Read More