Locally-iterative Distributed (Δ + 1)-coloring and Applications

We consider graph coloring and related problems in the distributed message-passing model. Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop-neighborhood . In STOC’93 Szegedy and Vishwanathan showed that any locally-iterative Δ + 1-coloring algorithm requires Ω (Δ log Δ + log * n ) rounds, unless there exists “a very special type of coloring that can be very efficiently reduced” [ 44 ]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [ 2 , 3 , 19 , 32 ]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ + 1-coloring algorithm with running time O (Δ + log * n ), i.e., below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results: We obtain self-stabilizing distributed algorithms for Δ + 1-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set, and maximal matching with O (Δ + log * n ) time. This significantly improves previously known results that have O(n) or larger running times [ 23 ]. We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O (Δ + log * n ) time and O (Δ)-edge-coloring in the Bit-Round model with O (Δ + log n ) time. The factors of log * n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. We obtain an arbdefective coloring algorithm with running time O (√ Δ + log * n ). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ε)Δ-coloring within O (√ Δ + log * n ) time and Δ + 1-coloring within O (√ Δ log Δ log * Δ + log * n ) time. This improves the recent state-of-the-art bounds of Barenboim from PODC’15 [ 2 ] and Fraigniaud et al. from FOCS’16 [ 19 ] by polylogarithmic factors. Our algorithms are applicable to the SET-LOCAL model [ 25 ] (also known as the weak LOCAL model). In this model a relatively strong lower bound of Ω (Δ 1/3 ) is known for Δ + 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [ 25 ] only Linial’s O (Δ 2 )-time algorithm and Kuhn-Wattenhofer O (Δ log Δ)-time algorithms are shown to work in it.) We obtain the first linear-in-Δ Δ + 1-coloring algorithms that work also in this model.

Link-delay-aware Reinforcement Scheduling for Data Aggregation in Massive IoT

<div>Over the past few years, the use of wireless sensor networks in a range of Internet of Things (IoT) scenarios has grown in popularity. Since IoT sensor devices have restricted battery power, a proper IoT data aggregation approach is crucial to prolong the network lifetime. To this end, current approaches typically form a virtual aggregation backbone based on a connected dominating set or maximal independent set to utilize independent transmissions of dominators. However, they usually have a fairly long aggregation delay because the dominators become bottlenecks for receiving data from all dominatees. The problem of time-efficient data aggregation in multichannel duty-cycled IoT sensor networks is analyzed in this paper. We propose a novel aggregation approach, named LInk-delay-aware REinforcement (LIRE), leveraging active slots of sensors to explore a routing structure with pipeline links, then scheduling all transmissions in a bottom-up manner. The reinforcement schedule accelerates the aggregation by exploiting unused channels and time slots left off at every scheduling round. LIRE is evaluated in a variety of simulation scenarios through theoretical analysis and performance comparisons with a state-of-the-art scheme. The simulation results show that LIRE reduces more than 80% aggregation delay compared to the existing scheme.</div>

Link-delay-aware Reinforcement Scheduling for Data Aggregation in Massive IoT

<div>Over the past few years, the use of wireless sensor networks in a range of Internet of Things (IoT) scenarios has grown in popularity. Since IoT sensor devices have restricted battery power, a proper IoT data aggregation approach is crucial to prolong the network lifetime. To this end, current approaches typically form a virtual aggregation backbone based on a connected dominating set or maximal independent set to utilize independent transmissions of dominators. However, they usually have a fairly long aggregation delay because the dominators become bottlenecks for receiving data from all dominatees. The problem of time-efficient data aggregation in multichannel duty-cycled IoT sensor networks is analyzed in this paper. We propose a novel aggregation approach, named LInk-delay-aware REinforcement (LIRE), leveraging active slots of sensors to explore a routing structure with pipeline links, then scheduling all transmissions in a bottom-up manner. The reinforcement schedule accelerates the aggregation by exploiting unused channels and time slots left off at every scheduling round. LIRE is evaluated in a variety of simulation scenarios through theoretical analysis and performance comparisons with a state-of-the-art scheme. The simulation results show that LIRE reduces more than 80% aggregation delay compared to the existing scheme.</div>

Lower Bounds for Maximal Matchings and Maximal Independent Sets

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of n .

Massively Parallel Computation via Remote Memory Access

We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms [8, 30] using MapReduce and a distributed hash table service [17]. This extension allows us to give new graph algorithms with much lower round complexities compared to the best-known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in O (1) rounds and connectivity/minimum spanning tree in O (log log m / n n rounds both using O ( n δ ) space per machine for constant δ < 1. In the same memory regime for MPC, the best-known algorithms for these problems require poly log n rounds. Our results imply that the 2-C YCLE conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.

Minimum Partition of an r − Independence System

Graph partitioning has been studied in the discipline between computer science and applied mathematics. It is a technique to distribute the whole graph data as a disjoint subset to a different device. The minimum graph partition problem with respect to an independence system of a graph has been studied in this paper. The considered independence system consists of one of the independent sets defined by Boutin. We solve the minimum partition problem in path graphs, cycle graphs, and wheel graphs. We supply a relation of twin vertices of a graph with its independence system. We see that a maximal independent set is not always a minimal set in some situations. We also provide realizations about the maximum cardinality of a minimum partition of the independence system. Furthermore, we study the comparison of the metric dimension problem of a graph with the minimum partition problem of that graph.

The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, $$G+H$$ G + H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graphs in polynomial time, where a $$P_4$$ P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graph G there is a family $${{\mathcal {S}}}$$ S of subsets of V(G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of $${\mathcal {S}}$$ S . These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain $$S_{i,j,k}$$ S i , j , k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

Sparse Backbone and Optimal Distributed SINR Algorithms

We develop randomized distributed algorithms for many of the most fundamental communication problems in wireless networks under the Signal to Interference and Noise Ratio (SINR) model of communication, including (multi-message) broadcast, local broadcast, coloring, Maximal Independent Set, and aggregation. The complexity of our algorithms is optimal up to polylogarithmic preprocessing time. It shows—contrary to expectation—that the plain vanilla SINR model is just as powerful and fast (modulo the preprocessing) as various extensions studied, including power control, carrier sense, collision detection, free acknowledgements, and geolocation knowledge. Central to these results is an efficient construction of a constant-density backbone structure over the network, which is of independent interest. This is achieved using an indirect sensing technique, where message non-reception is used to deduce information about relative node-distances.

Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

The Massively Parallel Computation (MPC) model is an emerging model that distills core aspects of distributed and parallel computation, developed as a tool to solve combinatorial (typically graph) problems in systems of many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n , the number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding deterministic algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all edges incident to a single node. This poses a considerable obstacle compared to classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier, we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph, with the additional property that solving the problem on this subgraph provides significant progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known algorithm of Luby [SICOMP’86], we obtain O (log Δ + log log n )-round deterministic MPC algorithms for solving the problems of Maximal Matching and Maximal Independent Set with O ( n ɛ ) space on each machine for any constant ɛ > 0. These algorithms also run in O (log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O (log 2 Δ) rounds by Censor-Hillel et al. [DISC’17].