Certificate Complexity and Exact Learning
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timestamp T 0 is smaller than the initial value of every clock. Then, the algorithm works as follows. 1. When a process pi requests the resource, it sends a request message (Tm ; p i ; request) to all other processes and puts the message in its request queue. 2. When a process pj receives a message (Tm ; p i ; request), it puts that message in its request queue and sends an acknowledgment (Tm0 ; p j ; ack) to pi . 3. When a process pi releases the resource, it removes all instances of messages (; p i ; request) from its queue, and sends a message (Tm0 ; p i ; release) to all other processes. 4. When a process pj receives a release message from process pi , it removes all instances of messages (; p i ; request) from its queue, and sends a timestamped acknowledgment to pi . 5. Messages in the queue are sorted according to the total order relation ) of Definition 4. A process pi can use the resource when (a) a message (Tm ; p i ; request) appears first in the queue, and (b) pi has received from all other processes a message with a timestamp greater than T m (or equal from any process pj where p i p j ).
can continue to operate even in spite of the failure of some of the processors. State-machine replication ensures that the different replicas remain consistent. The mutual exclusion algorithm proposed by Lamport [5] and described in this entry is actually one of the first known solution to the atomic broadcast problem (see relevant entry). Briefly, in a system with several processes that broadcast messages concurrently, the problem requires that all processes deliver (and process) all message in the same order. Nowadays, there exist several approaches to solving the problem. Surveying many algorithms, Défago et al. [1] have classified Lamport’s algorithm as communication history algorithms, because of the way the ordering is generated.
Applications
Recommended Reading
A brief overview of some applications of the concepts presented in this entry has been provided. First of all, the notion of causality in distributed systems (or lack thereof) leads to a famous problem in which a user may potentially see an answer before she can see the relevant question. The time-independent characterization of causality of Lamport lead to the development of efficient solutions to enforce causal order in communication. In his later work, Lamport [3] gives a more general definition to the “happened before” relation, so that a system can be characterized at various abstraction levels. About a decade after Lamport’s work on logical clock, Fidge [2] and Mattern [6] have developed the notion of vector clocks, with the advantage of a complete characterization of causal order. Indeed, the clock condition enforced by Lamport’s logical clocks is only a one-way implication (see Definition 3). In contrast, vector clocks extend Lamport clocks by ensuring that, for any events a and b, if Chai < Chbi, then a ! b. This is for instance useful for choosing a set of checkpoints after recovery of a distributed system, for distributed debugging, or for dea
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