Università degli Studi dell’Aquila Academic Year 2009/2010 Course: Algorithms for Distributed Systems Instructor: Prof. Guido Proietti Time: Monday: – – Room 0.6 Wednesday: – – Room 0.6 Questions?: Wednesday Slides plus other infos:

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..)

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 …

The Internet

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?

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

Cooperative distributed algorithms: Message Passing System A Formal Model

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

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

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

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

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

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.

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

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

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|)

DFS: an example (1/2)

DFS: an example (2/2)

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

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)