Maximal Independent Set Algorithm Research Articles

Eventually lattice-linear algorithms

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice.We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS).Algorithms for SDMDS, MVC and MIS converge in 1 round plus n moves (within 2n moves), GC in n+4m moves, and 2DS in 1 round plus 2n moves (within 3n moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

Loosely-stabilizing maximal independent set algorithms with unreliable communications

Self-stabilization is a promising paradigm for designing highly adaptive distributed systems. However, it cannot be realized when the communication between processes is unreliable with some constant probability. To circumvent such impossibility, this paper adopts the concept of loose-stabilization for the first time, which is a practical alternative of self-stabilization, and proposes three systematic approaches to realize loose-stabilization in the atomic-state model with probabilistically erroneous communications, namely, the redundant-state approach, the step-up approach, and the repetition approach. Further, we apply these approaches to design three corresponding loosely-stabilizing algorithms for the maximal independent set problem.

Breaking the linear-memory barrier in [formula omitted]: Fast [formula omitted] on trees with strongly sublinear memory

Recently, studying fundamental graph problems in the Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow Ω˜(n) memory per machine, where n is the number of nodes in the graph and Ω˜ hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory.In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality.In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in O(log3⁡log⁡n) rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in O˜(log⁡n) rounds in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from Ω˜(n) required by the recent O(log⁡log⁡n)-round MIS algorithm of Ghaffari et al. [PODC'18] to nε for any ε>0, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the LOCAL and PRAM models by Lenzen and Wattenhofer [PODC'11].

Distributed Self-Stabilizing Capacitated Maximal Independent Set Construction in Wireless Sensor Networks

Wireless sensor networks (WSNs) are composed of a large number of wireless self-organized sensor nodes connected through a wireless decentralized distributed network without the aid of a predefined infrastructure. Fault-tolerance and power management are fundamental challenges in WSNs. A WSN is self-stabilizing if it can initially start at any state and obtain a legitimate state in a finite time without any external intervention. Self-stabilization is an important method for providing fault-tolerance in WSNs. Maximal independent set (MIS) is an extensively used structure for many important applications such as clustering (Randhawa and Jain in Wirel Personal Commun 97(3):3355, 2017. https://doi.org/10.1007/s11277-017-4674-5 ) and routing (Attea et al. in Wirel Personal Commun 81(2):819, 2015. https://doi.org/10.1007/s11277-014-2159-3 ; Lipinski in Wirel Personal Commun 101(1):251, 2018. https://doi.org/10.1007/s11277-018-5686-5 ) in WSNs. The capacitated MIS (CapMIS) problem is an extension of MIS in that each node has a capacity that determines the number of nodes it may dominate. In this paper, we propose a distributed self-stabilizing capacitated maximal independent set algorithm (CapMIS) in order to reduce energy consumption and support load balancing in WSNs. To the best of our knowledge, this is the first algorithm in this manner. The algorithm is validated through theoretical analysis as well as testbed implementations and simulations.

Derandomizing local distributed algorithms under bandwidth restrictions

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send $$O(\log n)$$-bit messages in each round of communication. Our framework mostly follows by the derandomization approach of Luby (J Comput Syst Sci 47(2):250–286, 1993) combined with the power of all to all communication. Our key results are as follows: first, we show that in the congested clique model, which allows all-to-all communication, there is a deterministic maximal independent set algorithm that runs in $$O(\log ^2 {\varDelta })$$ rounds, where $${\varDelta }$$ is the maximum degree. When $${\varDelta }=O(n^{1/3})$$, the bound improves to $$O(\log {\varDelta })$$. In addition, we deterministically construct a $$(2k-1)$$-spanner with $$O(kn^{1+1/k}\log n)$$ edges in $$O(k \log n)$$ rounds in the congested clique model.

Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp and Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame and Luby (1990) and Kelsen (1992), running in time roughly (log n ) r ! . We improve the randomized algorithm of Kelsen, reducing the runtime to roughly (log n ) 2 r and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in (log n ) 2 r +3 time and poly ( m , n ) processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank r > 3. Our analysis can also apply when r is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time exp ( O ( log ( mn ) / log log ( mn )).

Optimal deterministic distributed algorithms for maximal independent set in geometric graphs

Finding a maximal independent set (MIS) in a graph is one of the fundamental problems in distributed computing. Researchers in this community are trying to close the time complexity gap of computing a MIS in a graph, way back from Luby’s [STOC’85] O(logn) time (randomized) algorithm and Linial’s [SICOMP’92] Ω(log∗n) lower bound result (n is the number of nodes of the graph). Since then, an extensive research has been done on this problem, however, most of the results are randomized and we are lack of efficient deterministic solution. In fact, no polylogarithmic (in n) time deterministic algorithm is known so far in general graphs — an open question raised by Linial in 1993 [Comb. Probab. Comput.’93]. In this paper, we study the MIS problem in geometric graphs, a class of graphs which has both theoretical and practical importance. We present O(1)-time deterministic algorithms (hence, optimal) for computing MIS in unit interval graphs, unit square graphs and unit disk graphs. The same idea applies to develop optimal MIS algorithm for higher dimensional unit balls and unit hypercubes. We further extend the MIS algorithms to compute approximate maximum independent set (MaxIS) in the above geometric graph classes. The theoretical results are corroborated through extensive experiments to show the effectiveness and efficiency of our algorithms.Our algorithms are fully decentralized, scalable, and robust. In general, nodes exchange only O(logn) bits message through an edge per communication. Moreover, being local, our algorithms are also suitable in failure model, self-stabilizing and appropriate for dynamic environment.

An energy-efficient, self-stabilizing and distributed algorithm for maximal independent set construction in wireless sensor networks

Maximal independent set (MIS) is a very important structure that provides data aggregation, topology control and routing for wireless sensor networks (WSNs). Energy-efficient and fault-tolerant construction of MIS on WSNs is one of the vital tasks. A distributed sensor network is self-stabilizing if it can initially start at any state and regain a legal state in a finite time without any external intervention. Self-stabilization is a considerable method to provide fault tolerance in WSNs. This paper presents a distributed self-stabilizing MIS algorithm which is an improved version of Turau’s algorithm under a fully distributed scheduler for WSNs. The proposed algorithm is theoretically analyzed and evaluated with its counterparts. The proposed algorithm is compared with the other studies through testbed experiments on IRIS nodes and simulations on TOSSIM environment. It is shown that the proposed algorithm outperforms other algorithms in terms of move count and energy consumption.

Deterministic Parallel Algorithms for Bilinear Objective Functions

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial optimization problems, with low time- and processor-complexity. We extend and generalize a technique of Luby for efficiently handling bilinear objective functions. One noteworthy application is an NC algorithm for maximal independent set. On a graph G with m edges and n vertices, this takes $${\tilde{O}}(\log ^2 n)$$ time and $$(m + n) n^{o(1)}$$ processors, nearly matching the best randomized parallel algorithms. Other applications include reduced processor counts for algorithms of Berger (SIAM J Comput 26:1188–1207, 1997) for maximum acyclic subgraph and Gale–Berlekamp switching games. This bilinear factorization also gives better algorithms for problems involving discrepancy. An important application of this is to automata-fooling probability spaces, which are the basis of a notable derandomization technique of Sivakumar (In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp 619–626, 2002). Our method leads to large reduction in processor complexity for a number of derandomization algorithms based on automata-fooling, including set discrepancy and the Johnson–Lindenstrauss Lemma.

Distributed Spanner Construction With Physical Interference: Constant Stretch and Linear Sparseness

This paper presents the first distributed algorithm to construct a spanner for arbitrary ad hoc networks under the physical signal-to-interference-and-noise-ratio (SINR) interference model. Spanner construction is one of the most important techniques for topology control in wireless networks, which intends to find a sparse topology in which only a small number of links need to be maintained, without substantially degrading the path connecting any pair of the nodes in the network. Due to the non-local property of interference, constructing a spanner is challenging under the SINR model, especially when a local distributed algorithm is desired. We meet this challenge by proposing an efficient randomized distributed algorithm that can construct a spanner in $O(\log n\log \Gamma )$ timeslots with a high probability, where $n$ is the total number of nodes and $\Gamma $ describes the ratio of the maximum distance to the minimum distance between nodes. The constructed spanner concurrently satisfies two most desirable properties: constant stretch and linear sparseness. Our algorithm employs a novel maximal independent set (MIS) procedure as a subroutine, which is crucial in achieving the time efficiency of spanner construction. The MIS algorithm improves the best known result of $O(\log ^{2}n)$ [33] to $O(\log n)$ and is of independent interest as the algorithm is applicable also to many other applications. We conduct simulations to verify the proposed spanner construction algorithm, and the results show that our algorithm also performs well in realistic environments.

Dynamic networks of finite state machines

Like distributed systems, biological multicellular processes are subject to dynamic changes and a biological system will not pass the survival-of-the-fittest test unless it exhibits certain features that enable fast recovery from these changes. In most cases, the types of dynamic changes a biological process may experience and its desired recovery features differ from those traditionally studied in the distributed computing literature. In particular, a question seldomly asked in the context of distributed digital systems, and that is crucial in the context of biological cellular networks, is whether the system can keep the changing components confined so that only nodes in their vicinity may be affected by the changes, but nodes sufficiently far away from any changing component remain unaffected.Based on this notion of confinement, we propose a new metric for measuring the dynamic changes recovery performance in distributed network algorithms operating under the Stone Age model (Emek and Wattenhofer, 2013) [1], where the class of dynamic topology changes we consider includes inserting/deleting an edge, deleting a node together with its incident edges, and inserting a new isolated node. Our main technical contribution is a distributed algorithm for maximal independent set (MIS) in synchronous networks subject to these topology changes that performs well in terms of the aforementioned new metric. Specifically, our algorithm guarantees that nodes which do not experience a topology change in their immediate vicinity are not affected and that all surviving nodes (including the affected ones) perform O((C+1)log2⁡n) computationally-meaningful steps, where C is the number of topology changes; in other words, each surviving node performs O(log2⁡n) steps when amortized over the number of topology changes. This is accompanied by a simple example demonstrating that the linear dependency on C cannot be avoided.

Designing Self-Stabilizing Systems Using Game Theory

Self-stabilizing systems tolerate transient faults by always returning to a legitimate system state within a finite time. This goal is challenged by several system features such as arbitrary system states after faults, various process execution models, and constrained process communication means. This work designs self-stabilizing distributed algorithms from the perspective of game theory, achieving an intended system goal through private goals of processes. We propose a generic game design for identifying a maximal independent set (MIS) or a maximal weighted independent set (MWIS) among all processes in a distributed system. From the generic game several specific games can be defined which differ in whether and how neighboring players influence each other. Turning the game designs into self-stabilizing algorithms, we obtain the first algorithms for the MWIS problem and also the first self-stabilizing MIS algorithm that considers node degree (including an analysis of its performance ratio). We also show how to handle simultaneous moves of processes in some process execution models. Simulation results indicate that, for various representative network topologies, the new algorithm outperforms existing methods in terms of MIS size and convergence rate. For the MWIS problem, the new algorithms performed only slightly worse than centralized greedy counterparts.

Symmetry breaking depending on the chromatic number or the neighborhood growth

We deterministically compute a Δ+1 coloring and a maximal independent set(MIS) in time O(Δ1/2+Θ(1/h)+log∗n) for Δ1+i≤Δ1+i/h, where Δj is defined as the maximal number of nodes within distance j for a node and Δ:=Δ1. Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(Δ+log∗n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h≥36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O(logχ+log∗n) for Δ∈Ω(log1+1/log∗nn) and χ∈O(Δ/log1+1/log∗nn). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest Δ+1 coloring algorithm running in time O(logΔ+logn). The algorithm works without knowledge of χ and uses less than Δ colors, i.e., (1−1/O(χ))Δ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account. We also improve on the state of the art deterministic computation of (2,c)-ruling sets.

An optimal bit complexity randomized distributed MIS algorithm

We present a randomized distributed maximal independent set (MIS) algorithm for arbitrary graphs of size n that halts in time O(log n) with probability 1 − o(n −1 ), and only needs messages containing 1 bit. Thus, its bit complexity par channel is O(log n). We assume that the graph is anonymous: unique identities are not available to distinguish between the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multi- plicative constant) for the bit complexity and improves the best previous randomized distributed MIS algorithms (deducedfromtherandomizedPRAMalgorithmduetoLuby (SIAM J. Comput. 15:1036-1053, 1986)) for general graphs which is O(log 2 n) per channel (it halts in time O(log n)

An application of the maximal independent set algorithm to course allocation

In this paper, we demonstrated one of the many applications of the Maximal Independent Set Algorithm in the area of course allocation. A program was developed in Pascal and used in implementing a modified version of the algorithm to assign teaching courses to available lecturers in any academic environment and it proved to be very effective. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 97-106