There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In Distributed Algorithms, Nancy Lynch provides a blueprint for designing, implementing, and analyzing distributed algorithms. She directs her book at a wide audience, including students, programmers, system designers, and researchers.



Distributed Algorithms contains the most significant algorithms and impossibility results in the area, all in a simple automata-theoretic setting. The algorithms are proved correct, and their complexity is analyzed according to precisely defined complexity measures. The problems covered include resource allocation, communication, consensus among distributed processes, data consistency, deadlock detection, leader election, global snapshots, and many others.



The material is organized according to the system model-first by the timing model and then by the interprocess communication mechanism. The material on system models is isolated in separate chapters for easy reference.



The presentation is completely rigorous, yet is intuitive enough for immediate comprehension. This book familiarizes readers with important problems, algorithms, and impossibility results in the area: readers can then recognize the problems when they arise in practice, apply the algorithms to solve them, and use the impossibility results to determine whether problems are unsolvable. The book also provides readers with the basic mathematical tools for designing new algorithms and proving new impossibility results. In addition, it teaches readers how to reason carefully about distributed algorithms-to model them formally, devise precise specifications for their required behavior, prove their correctness, and evaluate their performance with realistic measures.


Table of Contents

1 Introduction 
2 Modelling I; Synchronous Network Model 
3 Leader Election in a Synchronous Ring 
4 Algorithms in General Synchronous Networks 
5 Distributed Consensus with Link Failures 
6 Distributed Consensus with Process Failures 
7 More Consensus Problems 
8 Modelling II: Asynchronous System Model 
9 Modelling III: Asynchronous Shared Memory Model 
10 Mutual Exclusion 
11 Resource Allocation 
12 Consensus 
13 Atomic Objects 
14 Modelling IV: Asynchronous Network Model 
15 Basic Asynchronous Network Algorithms 
16 Synchronizers 
17 Shared Memory versus Networks 
18 Logical Time 
19 Global Snapshots and Stable Properties 
20 Network Resource Allocation 
21 Asynchronous Networks with Process Failures 
22 Data Link Protocols 
23 Partially Synchronous System Models 
24 Mutual Exclusion with Partial Synchrony 
25 Consensus with Partial Synchrony

Distributed algorithms

An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

Parameterized Complexity

We survey the current techniques to cope with the problem of string matching that allows errors. This is becoming a more and more relevant issue for many fast growing areas such as information retrieval and computational biology. We focus on online searching and mostly on edit distance, explaining the problem and its relevance, its statistical behavior, its history and current developments, and the central ideas of the algorithms and their complexities. We present a number of experiments to compare the performance of the different algorithms and show which are the best choices. We conclude with some directions for future work and open problems.

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A guided tour to approximate string matching

We consider the problem of combining ranking results from various sources. In the context of the Web, the main applications include building meta-search engines, combining ranking functions, selecting documents based on multiple criteria, and improving search precision through word associations. We develop a set of techniques for the rank aggregation problem and compare their performance to that of well-known methods. A primary goal of our work is to design rank aggregation techniques that can e ectively combat \spam," a serious problem in Web searches. Experiments show that our methods are simple, e cient, and e ective.

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Rank aggregation methods for the Web

We consider the following problem. Suppose a rooted tree T is available for preprocessing. Answer on-line queries requesting the lowest common ancestor for any pair of vertices in T. We present a linear time and space preprocessing algorithm that enables us to answer each query in $O(1)$ time, as in Harel and Tarjan [SIAM J. Comput., 13 (1984), pp. 338–355]. Our algorithm has the advantage of being simple and easily parallelizable. The resulting parallel preprocessing algorithm runs in logarithmic time using an optimal number of processors on an EREW PRAM. Each query is then answered in $O(1)$ time using a single processor.

On Finding Lowest Common Ancestors: Simplification and Parallelization

We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the local-ratio technique. We note that they can equivalently be interpreted within the primal-dual schema.

/pdf/a-unified-approach-to-approximating-resource-allocation-and-z6teux7tet.pdf

A unified approach to approximating resource allocation and scheduling

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log

 2
 
|X|) . The second algorithm achieves an approximation factor of O(min{log τ

 *
 log log τ

 *
 
, log n log log n)} , where τ

 *
 is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

We study the problem of scheduling activities of several types under the constraint that, at most, a fixed number of activities can be scheduled in any single time slot. Any given activity type is associated with a service cost and an operating cost that increases linearly with the number of time slots since the last service of this type. The problem is to find an optimal schedule that minimizes the long-run average cost per time slot. Applications of such a model are the scheduling of maintenance service to machines, multi-item replenishment of stock, and minimizing the mean response time in Broadcast Disks. Broadcast Disks recently gained a lot of attention because they were used to model backbone communications in wireless systems, Teletext systems, and Web caching in satellite systems.

The first contribution of this paper is the definition of a general model that combines into one several important previous models. We prove that an optimal cyclic schedule for the general problem exists, and we establish the NP-hardness of the problem. Next, we formulate a nonlinear program that relaxes the optimal schedule and serves as a lower bound on the cost of an optimal schedule. We present an efficient algorithm for finding a near-optimal solution to the nonlinear program. We use this solution to obtain several approximation algorithms.

1 A 9/8 approximation for a variant of the problem that models the Broadcast Disks application. The algorithm uses some properties of “Fibonacci sequences.” Using this sequence, we present a 1.57-approximation algorithm for the general problem.

2 A simple randomized algorithm and a simple deterministic greedy algorithm for the problem. We prove that both achieve approximation factor of 2. To the best of our knowledge this is the first worst-case analysis of a widely used greedy heuristic for this problem.

Minimizing service and operation costs of periodic scheduling

We consider two types of buffering policies that are used in network switches supporting Quality of Service (QoS). In the FIFO type, packets must be transmitted in the order in which they arrive; the constraint in this case is the limited buffer space. In the bounded-delay type, each packet has a maximum delay time by which it must be transmitted, or otherwise it is lost. We study the case of overloads resulting in packet loss. In our model, each packet has an intrinsic value, and the goal is to maximize the total value of transmitted packets.
Our main contribution is a thorough investigation of some natural greedy algorithms in various models. For the FIFO model we prove tight bounds on the competitive ratio of the greedy algorithm that discards packets with the lowest value when an overflow occurs. We also prove that the greedy algorithm that drops the earliest packets among all low-value packets is the best greedy algorithm. This algorithm can be as much as 1.5 times better than the tail-drop greedy policy, which drops the latest lowest-value packets.
In the bounded-delay model we show that the competitive ratio of any on-line algorithm for a uniform bounded-delay buffer is bounded away from 1, independent of the delay size. We analyze the greedy algorithm in the general case and in three special cases: delay bound 2, link bandwidth 1, and only two possible packet values.
Finally, we consider the off-line scenario. We give efficient optimal algorithms and study the relation between the bounded-delay and FIFO models in this case.

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Baruch Schieber

Papers

On Finding Lowest Common Ancestors: Simplification and Parallelization

A unified approach to approximating resource allocation and scheduling

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

Minimizing service and operation costs of periodic scheduling

Buffer Overflow Management in QoS Switches