Transversal merge operation: a nondominated coterie construction method for distributed mutual exclusion

A coterie is a set of subsets (called quorums) of the processes in a distributed system such that any two quorums intersect with each other and is mainly used to solve the mutual exclusion problem in a quorum-based algorithm. The choice of a coterie sensitively affects the performance of the algorithm and it is known that nondominated (ND) coteries achieve good performance in terms of criteria such as availability and load. On the other hand, grid coteries have some other attractive features: 1) a quorum size is small, which implies a low message complexity, and 2) a quorum is constructible on the fly, which benefits a low space complexity. However, they are not ND coteries unfortunately. To construct ND coteries having the favorite features of grid coteries, we introduce the transversal merge operation that transforms a dominated coterie into an ND coterie and apply it to grid coteries. We call the constructed ND coteries ND grid coteries. These ND grid coteries have availability higher than the original ones, inheriting the above desirable features from them. To demonstrate this fact, we then investigate their quorum size, load, and availability, and propose a dynamic quorum construction algorithm for an ND grid coterie.     

Recognizing nondominated coteries and wr-coteries by availability

Coterie is a widely accepted concept for solving the mutual exclusion problem. Nondominated coteries are an important class of coteries which have better performance than dominated coteries. The performance of a coterie is usually measured by availability. Higher availability of a coterie exhibits greater ability to tolerate node or communication link failures. In this paper, we demonstrate a way to recognize nondominated coteries using availability. By evaluating the availability of a coterie instead of using a formal proof, the coterie can be recognized as a nondominated coterie or not. Moreover, with regard to wr-coterie, a concept for solving the replica control problem, we also present a similar result for recognizing nondominated wr-coteries. Finally, we apply our results to some well-known coteries and wr-coteries     

A geometric approach for constructing coteries and k-coteries

Quorum-based mutual exclusion algorithms are resilient to node and communication line failures. Recently, some mutual exclusion algorithms successfully use logical structures to construct coteries with small quorums sizes. In this paper, we introduce a geometric approach on dealing with the logical structures and present some useful geometric properties for constructing coteries and k-coteries. Based on those geometric properties, a logical structure named three-sided graph is proposed to provide a new scheme for constructing coteries with small quorums: The smallest quorum size is O(√N) in a homogeneous system with N nodes and O(1) in a heterogeneous system. In addition, we also extend the three-sided graph to the O-sided graph for constructing k-coteries     

Coterie join operation and tree structured k-coteries

The coterie join operation proposed by M.L. Neilsen and M. Mizuno (1994) produces, from a k-coterie and a coterie, a new k-coterie. For the coterie join operation, this paper first shows 1) a necessary and sufficient condition to produce a nondominated k-coterie (more accurately, a nondominated k-semicoterie satisfying nonintersection property) and 2) a sufficient condition to produce a k-coterie with higher availability. By recursively applying the coterie join operation in such a way that the above conditions hold, we define nondominated k-coteries, called tree structured k-coteries, the availabilities of which are thus expected to be very high. This paper then proposes a new k-mutual exclusion algorithm that effectively uses a tree structured k-coterie, by extending Agrawal and El Abbadi's tree algorithm. The number of messages necessary for k processes obeying the algorithm to simultaneously enter the critical section is approximately bounded by k log(n/k) in the best case, where n is the number of processes in the system     

Minimizing the maximum delay for reaching consensus in quorum-basedmutual exclusion schemes

The use of quorums is a well-known approach to achieving mutual exclusion in distributed computing systems. This approach works based on a coterie, a special set of node groups where any pair of the node groups shares at least one common node. Each node group in a coterie is called a quorum. Mutual exclusion is ensured by imposing that a node gets consensus from all nodes in at least one of the quorums before it enters a critical section. In a quorum-based mutual exclusion scheme, the delay for reaching consensus depends critically on the coterie adopted and, thus, it is important to find a coterie with small delay. Fu (1997) introduced two related measures called max-delay and mean-delay. The former measure represents the largest delay among all nodes, while the latter is the arithmetic mean of the delays. She proposed polynomial-time algorithms for finding max-delay and mean-delay optimal coteries when the network topology is a tree or a ring. In this paper, we first propose a polynomial-time algorithm for finding max-delay optimal coteries and, then, modify the algorithm so as to reduce the mean-delay of generated coteries. Unlike the previous algorithms, the proposed algorithms can be applied to systems with arbitrary topology     

Coterie join algorithm

Given a set of nodes in a distributed system, a coterie is a collection of subsets of the set of nodes such that any two subsets have a nonempty intersection and are not properly contained in one another. A subset of nodes in a coterie is called a quorum. An algorithm, called the join algorithm, which takes nonempty coteries as input, and returns a new, larger coterie called a composite coterie is introduced. It is proved that a composite coterie is nondominated if and only if the input coteries are nondominated. Using the algorithm, dominated or nondominated coteries may be easily constructed for a large number of nodes. An efficient method for determining whether a given set of nodes contains a quorum of a composite coterie is presented. As an example, tree coteries are generalized using the join algorithm, and it is proved that tree coteries are nondominated. It is shown that the join algorithm may be used to generate read and write quorums which may be used by a replica control protocol     

Generating and approximating nondominated coteries

A coterie, which is used to realize mutual exclusion in a distributed system is a family C of incomparable subsets such that every pair of subsets in C has at least one element in common. Associate with a family of subsets C a positive (i.e., monotone) Boolean function f_c such that f_c(x)=1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f_c is dual-minor, and is a nondominated (ND) coterie if and only if f_c is self-dual. We introduce an operator ρ, which transforms a positive self-dual function into another positive self-dual function, and the concept of almost-self-duality, which is a close approximation to self-duality and can be checked in polynomial time (the complexity of checking positive self-duality is currently unknown). After proving several interesting properties of them, we propose a simple algorithm to check whether a given positive function is self-dual or not. Although this is not a polynomial algorithm, it is practically efficient in most cases. Finally, we present an incrementally polynomial algorithm that generates all positive self-dual functions (ND coteries) by repeatedly applying p operations. Based on this algorithm, all ND coteries of up to seven variables are computed     

k-组群通信成本的研究

Kakugana在文献[1]引入k-组群（k-Coteries）的概念来解决分布系统中的k-互斥问题。本文研究使用k-组群技术解决k-互斥问题时的通信成本,它是单个组群通信成本的扩 展。在对k-组群通信成本进行精确定义并给出受支配和不受支配k-组群的概念之后,本文证明了不受支配k-组群的通信成本不会大于它所支配的k-组群的通信成本,据此定理 ,给出了获得通信成本最小k-组群的方法。     

A Token-Based Distributed Group Mutual Exclusion Algorithm with Quorums

The group mutual exclusion problem is a generalization of mutual exclusion problem such that a set of processes in the same group can enter critical section simultaneously. In this paper, we propose a distributed algorithm for the group mutual exclusion problem in asynchronous message passing distributed systems. Our algorithm is based on tokens, and a process that obtains a token can enter critical section. For reducing message complexity, it uses coterie as a communication structure when a process sends a request messages. Informally, coterie is a set of quorums, each of which is a subset of the process set, and any two quorums share at least one process. The message complexity of our algorithm is $O(|Q|)$ in the worst case, where $|Q|$ is a quorum size that the algorithm adopts. Performance of the proposed algorithm is presented by analysis and discrete event simulation. Especially, the proposed algorithm achieves high concurrency, which is a performance measure for the number of processes that can be in critical section simultaneously.     

On multilevel voting

Summary In a distributed system mutual exclusion is often used to maintain consistency when restricted operations are performed. Mechanisms guaranteeing mutual exclusions should be both resilient and efficient. Resiliency implies high resource availability in the face of failures, while efficiency implies low overhead incurred by performing restricted operations. In this paper, we propose and study a general paradigm, called multilevel voting, which provides a general framework to assist in the design of resilient and efficient mutual exclusion mechanisms. The proposed method uses multiple level quorum consensus. Unlike another method based on the use of multiple quorum consensus, the proposed model only contains one type of integrity constraints. This has the benefit of being conceptually simple and easy to reason about. The strong resemblance with the traditional quorum consensus makes it easy for the proposed paradigm to embed any technique based on traditional quorum schemes. We show that the proposed model represents the exact class of coteries. This means that not only does it have all the power of coteries, but also all schemes under the model are correct. Thus, should the need arise, we can interchange two schemes freely without using any extra mechanisms to ensure correctness. We study a number of issues that have impact on performance such as the degree of a multilevel scheme and the order of a coterie. We explain how the model can be extended also to model the schemes for the synchronization of read and write of replicated data. We provide algorithms for the design of multilevel schemes in the context of mutual exclusion and that of read and write of replicated data. J. Tang received the M.S. degree in computer science from the University of Iowa in 1983 and the Ph.D degree in computer science from the Pennsylvania State University in 1988. Since 1988, he has been an assistant professor with the Department of Computer Science, Memorial University of Newfoundland, St. John's, Newfoundland, Canada. His current research interests include distributed computing, fault tolerance in distributed systems, database modeling and multidatabase systems.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, individual operating grant OGP0041916     

Delay optimal coteries on regular symmetric networks

Coteries, introduced by Garcia-Molina and Barbara [Journal for the Association for Computing Machinery, 32 (4) (1985) 841], are an important and effective tool for enforcing mutual exclusion in distributed systems. Communication delay is an important performance measure for a coterie. Fu et al. [IEEE Transactions on Parallel and Distributed Systems, 8 (1) (1997) 59] emphasize that while calculating communication delay, the actual distances between different sites in a network must be taken into account and using this idea, obtain delay optimal coteries for trees, rings and hypercubes. Also, topology of an interconnection network plays an important role in the performance of a distributed system. For certain applications, it is desirable that the degree of each node in the interconnection network is the same. Constant Degree Four Cayley Graphs introduced by Vadapalli and Srimani [IEEE Transactions on Parallel and Distributed Systems 7(2) (1996) 26] provide an ideal topology for such applications. They are regular, have a logarithmic diameter and a node connectivity of four. In this paper, we prove that no coterie on an arbitrary network can have a delay of less than half the diameter of the network and use this result to obtain delay optimal coteries on regular symmetric networks with special reference to constant degree four Cayley interconnection-network.     

Nondominated coteries on graphs

Let C and D be two distinct coteries under the vertex set V of a graph G=(V,E) that models a distributed system. Coterie C is said to G-dominate D (with respect to G) if the following condition holds: For any connected subgraph H of G that contains a quorum in D (as a subset of its vertex set), there exists a connected subgraph H' of H that contains a quorum in C. A coterie C on a graph G is said to be G-nondominated (G-ND) (with respect to G) if no coterie D(≠C) on G G-dominates C. Intuitively, a G-ND coterie consists of irreducible quorums. This paper characterizes G-ND coteries in graph theoretical terms, and presents a procedure for deciding whether or not a given coterie C is G-ND with respect to a given graph G, based on this characterization. We then improve the time complexity of the decision procedure, provided that the given coterie C is nondominated in the sense of Garcia-Molina and Barbara (1985). Finally, we characterize the class of graphs G on which the majority coterie is G-ND     

k-Coteries for tolerating network 2-partition

A network partition, which makes it impossible for some pairs of processes to communicate with each other, is one of the most serious network failures. Although the notion of k-coterie is introduced to design a k-mutual exclusion algorithm that is robust against network failures, the number of processes allowed to simultaneously access the critical section may fatally decrease once network partition occurs. We discuss how to construct a k-coterie such that the k-mutual exclusion algorithm adopting it is robust against a network 2-partition. To this end, we introduce the notion of complemental k-coterie, and show that complemental k-coteries meet our requirements. We then give methods for constructing complemental k-coteries, and show a necessary and sufficient condition for a k-coterie to be complemental.     

Nondominated local coteries for resource allocation in grids and clouds

The resource allocation problem is a fundamental problem in Grid and Cloud computing environments. This paper focuses on constructing nondominated (ND) local coteries to solve the problem in a distributed way. Distributed algorithms using coteries usually incur low communication overheads and have high degrees of fault-tolerance, and ND coteries are candidates for the algorithms to achieve the highest degree of fault-tolerance. A new type of coteries, called p-coteries, is defined to aid the construction of local coteries. Theorems about the nondomination of p-coteries are then developed, and an operation, called pairwise-union (p-union), is proposed to help generate ND p-coteries, which in turn can be used to generate ND local coteries for solving the resource allocation problem.     

On distributed dynamic channel allocation in mobile cellularnetworks

Distributed dynamic channel allocation (DDCA) is a fundamental resource management problem in mobile cellular networks. It has a flavor of distributed mutual exclusion but is not exactly a mutual exclusion problem. We establish the exact relationship between the two problems. Specifically, we introduce the problem of relaxed mutual exclusion to model one important aspect of the DDCA problem. We develop a general algorithm that guarantees relaxed mutual exclusion for a single resource and prove necessary and sufficient conditions for the information structure. Considering distributed dynamic channel allocation as a special case of relaxed mutual exclusion, we apply and extend the algorithm to further address the issues that arise in distributed channel allocation such as deadlock resolution, dealing with multiple channels, design of efficient information structures, and channel selection strategies. Based on these results, we propose an example distributed channel allocation scheme using one of the information structures proposed. Analysis and simulation results are provided and show that the results of this research can be used to design more efficient distributed channel allocation algorithms     

Obtaining coteries that optimize the availability of replicateddatabases

Techniques for implementing the coterie scheme and for obtaining optimal coteries for a system are presented. Central to the techniques is the notion of an acceptance set, which is an alternative representation of the information contained in a coterie. Using this concept, the coterie scheme can be implemented efficiently, and an optimal coterie for a system can be obtained more directly. The problem of determining an optimal acceptance set is formulated as a sparse zero-one linear programming problem. Hence, the optimization problem can be handled using the very rich class of existing techniques for solving such problems. Experimental results indicate that the optimization approach is feasible for up to eight nodes at least. The ways in which the scheme and the optimization approach can be used for systems that distinguish between read and write operations are indicated     

(h,k)-Arbiters for h-out-of-k mutual exclusion problem

h-Out-of-k mutual exclusion is a generalization of the 1-mutual exclusion problem, where there are k units of shared resources and each process requests h (1hk) units at the same time. Though k-arbiter has been shown to be a quorum-based solution to this problem, quorums in k-arbiter are much larger than those in the 1-coterie for 1-mutual exclusion. Thus, the algorithm based on k-arbiter needs many messages. This paper introduces the new notion that each request uses different quorums depending on the number of units of its request. Based on the notion, this paper defines two (h,k)-arbiters for h-out-of-k mutual exclusion: a uniform (h,k)-arbiter and a (k+1)-cube (h,k)-arbiter. The quorums in each (h,k)-arbiter are not larger than the ones in the corresponding k-arbiter; consequently, it is more efficient to use (h,k)-arbiters than the k-arbiters. A uniform (h,k)-arbiter is a generalization of the majority coterie for 1-mutual exclusion. A (k+1)-cube (h,k)-arbiter is a generalization of square grid coterie for 1-mutual exclusion.     

A delay optimal coterie on the k-dimensional folded Petersen graph

Coteries are an effective tool for enforcing mutual exclusion in distributed systems. Communication delay is an important metric to measure the performance of a coterie. The topology of the interconnection network in a distributed system also has an impact on its performance. The k-dimensional folded Petersen graph, a graph with 10^k nodes and diameter 2k, qualifies as a good network topology for large distributed systems. In this paper, we present a delay optimal coterie on the k-dimensional folded Petersen graph, FP_k. For any positive integer k, the coterie has message complexity 4^k and delay k. Moreover, this coterie is not vote-assignable.     

Quorum-based algorithms for group mutual exclusion

We propose a quorum system, which we referred to as the surficial quorum system, for group mutual exclusion. The surficial quorum system is geometrically evident and is easy to construct. It also has a nice structure based on which a truly distributed algorithm for group mutual exclusion can be obtained and processed loads can be minimized. When used with Maekawa's algorithm, the surficial quorum system allows up to /spl radic/2n/m(m-l) processes to access a resource simultaneously, where n is the total number of processes and m is the total number of groups. We also present two modifications of Maekawa's algorithm so that the number of processes that can access a resource at a time is not limited to the structure of the underlying quorum system, but to the number that the problem definition allows.     

Asynchronous group mutual exclusion

Abstract. Mutual exclusion and concurrency are two fundamental and essentially opposite features in distributed systems. However, in some applications such as Computer Supported Cooperative Work (CSCW) we have found it necessary to impose mutual exclusion on different groups of processes in accessing a resource, while allowing processes of the same group to share the resource. To our knowledge, no such design issue has been previously raised in the literature. In this paper we address this issue by presenting a new problem, called Congenial Talking Philosophers, to model group mutual exclusion. We also propose several criteria to evaluate solutions of the problem and to measure their performance. Finally, we provide an efficient and highly concurrent distributed algorithm for the problem in a shared-memory model where processes communicate by reading from and writing to shared variables. The distributed algorithm meets the proposed criteria, and has performance similar to some naive but centralized solutions to the problem. Received: November 1998 / Accepted: April 2000