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首页Nancy Lynch的分布式算法讲座笔记
Nancy Lynch的分布式算法讲座笔记
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这份名为"Distributed Algorithms"的报告是Nancy Lynch教授在秋季学期使用的研究生课程讲义。Nancy Lynch和助教Boaz Patt-Shamir共同编撰了这些笔记,主要包含了详尽的课堂讲座内容。这些笔记旨在提供一个深入理解分布式算法的基础,适用于常规课程的授课活动。 课程的核心部分着重于分布式系统中的关键概念、协议设计、一致性模型、共识问题、分片与容错性、P2P网络结构以及分布式数据结构等主题。读者可能会发现这些笔记经过了精心打磨,但同时也欢迎读者提出关于内容准确性或改进意见,因为作者期待能得到反馈以提高教学质量。 除了常规课堂讲座笔记外,报告还包含了家庭作业任务,这些都是为了帮助学生将理论知识应用到实践中。报告最后附录的部分则是对学期结束后额外讲解的三堂课程的补充材料,这些内容涵盖了未能在预定课程时间范围内涵盖的重要主题。 值得注意的是,这些附加讲座笔记可能相对较粗糙,反映了它们是在学期结束后临时添加的性质。报告的编者对参与课程的学生表示了深深的感谢,他们的学习和反馈对于完善这些资料起到了关键作用。 这份报告是一份宝贵的资源,不仅提供了深入的分布式算法理论知识,也展示了如何在实际教学环境中处理复杂问题和不断完善的教学过程。对于对分布式计算感兴趣的读者,无论是学生还是研究人员,这都是一份值得仔细研读和学习的参考资料。
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Prove rigorously that the algorithms solve the problems.
Analyze the complexity of the algorithms.
Prove corresp onding impossibility results.
Note the emphasis on rigor this is imp ortant in this area, b ecause of the subtle compli-
cations that arise. A rigorous approach seems necessary to be sure that the problems are
meaningful, the algorithms are correct, the impossibility results are true and meaningful,
and the interfaces are suciently well-dened to allow system building.
However, because of the many complications, rigor is hard to achieve. In fact, the devel-
opment of good formal methods for describing and reasoning about distributed algorithms
is the sub ject of a go od deal of recent research. Sp ecically, there has been much serious
work in dening appropriate
formal mathematical models
, b oth for describing the algorithms
and for describing the problems they are supposed to solve a considerable amountof work
has also b een devoted to
proof methods
. One diculty in carrying out a rigorous approach
is that, unlike in many other areas of theoretical computer science, there is no any single
accepted formal model to cover all the settings and problems that arise in this area. This
phenomenon is unavoidable, because there are so manyvery dierent settings to b e studied
(consider the dierence between shared memory and message-passing), and each of them has
its own suitable formal mo dels.
So, rigor is a goal to b e striven for, rather than one that wewillachieveentirely in this
course. Due to time limitations, and (sometimes) the diculty of making formal presenta-
tions intuitively understandable, the presentation in class will b e a mixture of rigorous and
intuitive.
1.1.3 Overview of the Course
There are many dierent orders in which this material could be presented. In this course, we
divide it up rst according to
timing assumptions
, since that seems to b e the most imp ortant
model distinction. The timing mo dels to be considered are the following.
synchronous
: This is the simplest model. We assume that components take steps simulta-
neously, i.e., the execution pro ceeds in synchronous rounds.
asynchronous
: Here we assume that the separate comp onents take steps in arbitrary order.
partial ly synchronous
(timing-based): This is an \in-between" model | there are some
restrictions on relative timing of events, but execution is not completely lock-step.
15
The next division is by the IPC mechanism: shared memory vs. message-passing. Then
we sub divide by the problem studied. And nally, each mo del and problem can be considered
with various failure assumptions.
Wenowgoover the bibliographical list and the tentativeschedule (Handouts 2 and 3).
The bibliographical list do esn't completely corresp ond to the order of topics to be covered
the dierence is that the material on models will b e spread throughout the course as needed.
General references.
These include the previous course notes, and some related bo oks.
There is not really muchin the way of textbo oks on this material.
Intro duction.
The chapter on distributed computing in the handb ook on Theoretical
Computer Science is a sketchyoverview of some of the mo deling and algorithmic ideas.
Mo dels and Pro of Metho ds.
We shall not study this as an isolated topic in the course
{ rather, it is distributed through the various units. The basic mo dels used are
automata-
theoretic
, starting with a basic state-machine mo del with little structure. These state ma-
chines need not necessarily be nite-state.
Invariant assertions
are often proved about
automaton states, by induction. Sometimes we use one automaton to represent the problem
being solved, and another to represent the solution then a correctness proof b oils down
to establishing a corresp ondence that preserves the desired
external behavior
. In this case,
proving the corresp ondence is often done using a
mapping
or
simulation
method. Specially
tailored state machine mo dels have been designed for some special purposes, e.g., for shared
memory models (where the structure consists of processes and shared variables). Another
model is the I/O automaton mo del for
reactive systems
, i.e., systems that interact with an
external environment in an ongoing fashion. This mo del can mo del systems based on shared
variables, but is more appropriate for message-passing systems. One of the key features of
this mo del is that it has goo d
compositionality
properties, e.g., that the correctness of a com-
pound automaton can be proved using the correctness of its comp onents.
Temporal logic
is
an example of a sp ecial set of metho ds (language, logic) mainly designed for proving
liveness
properties (e.g., something eventually happens).
Timedmodels
are mainly newer research
work. Typically, these are specially-tailored models for talking ab out timing-based systems
{ e.g., those whose components have access to system clocks, can use timeouts, etc.
Alge-
braic methods
are an important research subarea (but we will not have time for this). The
algebraic methods describe concurrent processes and systems using algebraic expressions,
then use equations involving these expressions to prove equivalences and implementation
relationships among the processes.
16
Synchronous Message-Passing.
As noted, the material is organized rst by timing
model. The simplest mo del (i.e., the one with the least uncertainty) is the
synchronous
model, in which all the processes take steps in synchronous rounds. The shared-memory
version of this mo del is the PRAM. Since it is studied in other courses, we shall skip this
sub ject, and start with synchronous networks.
We spend the rst twoweeks or so of the course on problems in synchronous networks
that are typical of the distributed setting. In the network setting wehave the processors
at the no des of a graph
G
,communicating with their neighbors via messages in the edges.
We start with a simple toy example, involving
ring computation
. The problem is to elect a
unique leader in a simple network of pro cessors, which are assumed to be identical except for
Unique Identiers (UID's). The uncertainty is that the size of network, and the set of ID's
of processors, are unknown (although it is known that the UID's are indeed unique). The
main application for this problem is a token ring, where there is a single token circulating,
and sometimes it it necessary to regenerate a lost token. We shall see some details of the
modeling, and some typical complexity measures will be studied. For example, we shall
show upper and lower bounds for the time and the amount of communication (i.e., number
of messages) required. We shall also study some other problems in this simple setting.
Next, we'll go through a brief survey of some proto cols in more general networks. We
shall see some proto cols used in unknown synchronous networks of processes to solve some
basic problems like nding shortest paths, dening a minimum spanning tree, computing a
maximal indep endent set, etc.
Then we turn to the problem of
reaching consensus
. This refers to the problem of
reaching agreement on some abstract fact, where there are initial dierences of opinion.
The uncertainty here stems not only from dierent initial opinions, but also from
processor
failures
.We consider failures of dierenttypes: stopping, where a processor suddenly stops
executing its local protocol omission, where messages may be lost en route and Byzantine,
where a faulty pro cessor is completely unrestricted. This has b een an active research area
in the past few years, and there are manyinteresting results, so we shall spend a couple of
lectures on this problem. We shall see some interesting b ounds on the number of tolerable
faults, time, and communication.
Asynchronous Shared Memory.
After \warming up" with synchronous algorithms (in
which there is only a little uncertainty), wemoveinto the more characteristic (and possibly
more interesting) part of the course, on
asynchronous
algorithms. Here, processors are no
longer assumed to take steps in lo ck-step synchrony, but rather can interleave their steps in
arbitrary order, with no b ound on individual pro cess speeds. Typically,the interactions with
the external world (i.e., input/output) are ongoing, rather than just initial input and nal
17
output. The results in this setting have quite a dierentavor from those for synchronous
networks.
The rst problem we deal with is
mutual exclusion
This is one of the fundamental (and
historically rst) problems in this area, and consequently,muchwork has b een dedicated
to exploring it. Essentially, the problem involves arbitrating access to a single, indivisible,
exclusive-use resource. The uncertainty here is ab out who is going to request access and
when. We'll sketch some of the important algorithms, starting with the original algorithm
of Dijkstra. Many important concepts for this eld will be illustrated in this context, in-
cluding progress, fairness, fault-tolerance, and time analysis for asynchronous algorithms.
We shall see upp er bounds on the amount of shared memory, corresponding lower b ounds,
and imp ossibility results. We shall also discuss generalizations of mutual exclusion to more
general resource allocation problems. For example, we will consider the
Dining Philosophers
problem { a prototypical resource allocation problem.
Next, we shall study the concept of
atomic registers
:sofar,wehave been assuming indi-
visible access to shared memory.Buthow can one implement this on simpler architectures?
We shall look at several algorithms that solve this problem in terms of weaker primitives. An
interesting new prop erty that app ears here is
wait-freeness
,which means that any op eration
on the register must complete regardless of the failure of other concurrent op erations.
An
atomic snapshot
is a convenient primitive for shared read-write memory. Roughly
speaking, the ob jectiveis to take an instantaneous snapshot of all the memory locations at
once. An atomic snapshot is a useful primitiveto have for building more powerful systems.
We shall see how to implement it.
A
concurrent timestamp system
is another nice primitive. This is a system that issues,
upon request, timestamps that can b e used by programs to establish a consistent order
among their operations. The twist here is how to implementsuch systems with bounded-
memory. It turns out out such bounded timestamp systems can b e built in terms of atomic
snapshots. Also, a concurrent timestamp system can b e used to build more p owerful forms
of shared memory,suchasmulti-writer multi-reader memory.
We shall also reconsider the
consensus
problem in the asynchronous shared memory
model, and provetheinteresting fact it is impossible to solve in this setting.
A p ossible new topic this time is
shared memory for multiprocessors
.Much recent research
is aimed at identifying dierenttypes of memory abstractions that are used, or mightbe
useful, for real systems. Systems architects who develop suchtyp es of memory frequently do
not give precise statements of what their memory guarantees (esp ecially in the presence of
concurrent accesses and failures). So recently, theoreticians have started trying to do this.
18
Asynchronous Message-Passing Systems.
This section deals with algorithms that op-
erate in asynchronous networks. Again, the system is modeled as a graph with pro cessors at
nodes, and communication links are represented by the edges, but now the system do es not
operate in rounds. In particular, messages can arrive at arbitrary times, and the pro cessors
can take steps at arbitrary sp eeds. One mightsaythat wenowhave \looser coupling" of
the components of the system: wehave more independence and uncertainty.
Computing in static graphs
. The simplest type of setting here is computation in a xed
(unknown) graph in which the inputs arrive at the beginning, and there is a single output
to be produced. Some examples are leader election in ring, and minimum spanning tree
computation.
Network synchronization
.At this point, we could plunge into a study the many sp ecial-
purpose algorithms designed expressly for asynchronous distributed networks. But instead,
we shall rst try to imp ose some structure on such algorithms by considering \algorithm
transformations" that can be used to run algorithms designed for a simpler computation
model on a a complex asynchronous network.
The rst example here arises in the very imp ortant pap er by Lamp ort, where he shows
a simple metho d of assigning consistent
logical times
to events in a distributed network.
This can be used to allow an asynchronous network to simulate one in which the nodes
have access to p erfectly synchronized real-time clocks. The second example is Awerbuch's
synchronizer
, which allows an asynchronous network to simulate the lock-step synchronous
networks discussed in the rst part of the course (at least, those without failures), and to
do so eciently.We shall contrast this simulation result with an interesting lower bound
that seems to say that any such simulation must b e inecient (the apparentcontradiction
turns out to depend on the kind of problem being solved). Third, we shall see that an
synchronous network can simulate a centralized (non-distributed) state machine. And fourth,
an asynchronous network can be used to simulate asynchronous shared memory.Any of these
simulations can b e used to run algorithms from simpler mo dels in the general asynchronous
network model.
Next, we shall lo ok at some specic problems, such as resource allocation. We shall see
how to solvemutual exclusion, dining philosophers etc. in networks.
Detection of stable properties
refers to a class of problems with a similar avor and a
common solution. Suppose that there is a separate algorithm running, and wewantto
design another algorithm to \monitor" the rst. To monitors here might mean, for instance,
to detect when it terminates or deadlo cks, or to take a \consistent snapshot" of its state.
We shall also revisit the
consensus
problem in the context of networks. The problem is
easy without faults, but with faults, it is mostly impossible (even for very simple types of
19
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