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Università degli Studi dell’Aquila Academic Year 2009/2010 Course: Algorithms for Distributed Systems Instructor: Prof. Guido Proietti Time: Monday: 10.15 – 11.45 – Room 0.6 Wednesday: 11.45 – 13.30 – Room 0.6 Questions?: Wednesday 16.30-17.30 Slides plus other infos: http://www.di.univaq.it/~proietti/didattica.html

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Distributed System Set of computational devices connected by a communication network. Old platform : Usually a number of WSs over a LAN Now, ranges from a LAN to a sensor network to a mobile network Each node in a DS : n is autonomous n communicates by messages n needs to synchronize with others to achieve a common goal (load balancing, fault tolerance, an application..)

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Modern Distributed Applications Collaborative computing n Military command and control n Online strategy games n Massive computation Distributed Real-time Systems n Process Control n Navigation systems, Airline Traffic Monitoring (ATM) Mobile Ad hoc Networks Rescue Operations, emergency operations, robotics Wireless Sensor Networks Habitat monitoring, intelligent farming Grid Stock market …

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The Internet

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Some Issues in Building Distributed Applications Reliability (connectivity) Security (cryptography) Consistency (mutual exclusion) Cooperativeness (game theory) Fault-tolerance (failures, recoveries…) Scalability: How is the performance affected as the number of nodes increase ? Performance: What is the complexity of the designed algorithm?

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Course structure FIRST PART: Algorithms for COOPERATIVE DS 1.Leader Election 2.Minimum spanning tree 3.Maximal independent set SECOND PART: Algorithms for UNRELIABLE DS 1.Benign failures: consensus problem 2.Byzantin failures: consensus problem THIRD PART: Algorithms for CONCURRENT DS 1.Mutual exclusion Mid-term Written Examination: First (?) week of December FOURTH PART: DS SECURITY 1.Elements of cryptography FIFTH PART: Algorithms for NON COOPERATIVE (STRATEGIC) DS 1.Strategic equilbria theory 2.Algorithmic mechanism design (AMD) 3.AMD for Graph optimization problems SIXTH PART (???): Algorithms for WIRELESS DS Final Oral Examination: depending of the mid-term rate

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Cooperative distributed algorithms: Message Passing System A Formal Model

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The System Topology: a network (connected undirected graph) Processors (nodes) Communication channels (edges) Degree of synchrony: asynchronous versus synchronous (universal clock) Degree of symmetry: anonymous (processors are indistinguishable) versus non-anonymous Degree of Uniformity: uniform (number of processors is unknown) versus non-uniform Local algorithm: the algorithm associated to a single processor Distributed algorithm: the “composition” of local algorithms

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Notation n processors: p 0, p 1, …, p n-1. Each processor knows nothing about the network topology, except for its neighbors, numbered from from 1 to r Communication takes place only through message exchanges, using buffers associated with each neighbor, namely outbuf i [k], inbuf i [k], i=1,…,r. q i : the state set for p i, containing a distinguished initial state; each state describes the internal status of the processor and the status of the buffers

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Configuration and events System configuration: A vector [q 0,q 1,…,q n-1 ] where q i is the state of p i Events: Computation events (internal computations plus sending of messages), and message delivering events

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Execution C 0 1 C 1 2 C 2 3 … where C i : A configuration i : An event C 0 : An initial configuration

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Asynchronous Systems No upper bound on delivering times Admissible execution: each message sent is eventually delivered

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Synchronous Systems Each processor has a (common) clock, and computation takes place in rounds. At each round each processor: 1. Reads the incoming messages buffer 2. Makes some internal computations 3. Sends messages which will be read in the next round.

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Message Complexity The total number of messages sent during any admissible execution of the algorithm. In other words, the number of delivery events.

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Time Complexity Synchronous: The number of rounds until termination. Asynchronous: not really meaningful

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Example: Distributed Depth-First Search –General overview of a sequential algorithm: –Begin at some source vertex, r 0 –when reaching any vertex v »if v has an unvisited neighbor, then visit it and proceed from it »otherwise, return to parent(v) –when we reach the parent of some vertex v such that parent(v) = NULL, then we terminate since v = r 0 –DFS defines a tree, with r 0 as the root, which reaches all vertices in the graph –“back edges” = graph edges not in tree –sequential time complexity = O(|edges|+|nodes|)

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DFS: an example (1/2)

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DFS: an example (2/2)

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Distributed DFS (cont’d.) –Distributed version (token-based): the token traverses the graph in a depth-first manner using the algorithm described above 1.Start exploration (visit) at root r. 2.When v is visited for the first time: 2.1 Inform all neighbors of v that v has been visited. 2.2 Wait for acknowledgment from all neighbors. 2.3 Resume the DFS process. –Message complexity is O(|E|) (optimal, because of the lower bound of (|edges|) to explore every edge) »note that edges are not examined from both endpoints; when edges (v,w) is examined by v, w then knows that v has been visited

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Distributed DFS (cont’d.) Time complexity analysis (sync. DS) We ensure that vertices visited for the first time know which of their neighbors have/have not been visited; thus we make no unnecessary vertex explorations: »algorithm: freeze the DFS process; inform all neighbors of v that v has been visited; get Ack messages from those neighbors; restart DFS process constant number of rounds for each new discovered node »only O(n) nodes are discovered time complexity = O(n)

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