Università degli Studi dell’AquilaAcademic Year 2009/2010

Course: Algorithms for Distributed SystemsInstructor: Prof. Guido Proietti

Time: Monday: 10.15 – 11.45 – Room 0.6 Wednesday: 11.45 – 13.30 – Room 0.6Questions?: Wednesday 16.30-17.30

Slides plus other infos:http://www.di.univaq.it/~proietti/didattica.html

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 :

is autonomous communicates by messages needs to synchronize with others to achieve a

common goal (load balancing, fault tolerance, an application..)

Modern Distributed Applications

Collaborative computing Military command and control Online strategy games Massive computation

Distributed Real-time Systems

Process Control 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 structureFIRST PART: Algorithms for COOPERATIVE DS

1. Leader Election2. Minimum spanning tree3. Maximal independent set

SECOND PART: Algorithms for UNRELIABLE DS1. Benign failures: consensus problem2. Byzantin failures: consensus problem

THIRD PART: Algorithms for CONCURRENT DS1. Mutual exclusion

Mid-term Written Examination: First (?) week of December

FOURTH PART: DS SECURITY1. Elements of cryptography

FIFTH PART: Algorithms for NON COOPERATIVE (STRATEGIC) DS1. Strategic equilbria theory2. 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 Cooperative distributed algorithms: algorithms:

Message Passing SystemMessage Passing System

Cooperative distributed Cooperative distributed algorithms: algorithms:

Message Passing SystemMessage Passing System

A Formal ModelA 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: p0, p1, … , pn-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 outbufi[k], inbufi[k], i=1,…,r.

qi : the state set for pi, 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 [q0,q1,…,qn-1] where qi is the state of pi

Events: Computation events (internal computations plus sending of messages), and message delivering events

ExecutionC0 1 C1 2 C2 3 … where

Ci : A configuration

i : An event

C0 : An initial configuration

Asynchronous SystemsNo upper bound on delivering

timesAdmissible 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 buffer2. Makes some internal computations3. 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, r0

– 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 = r0

– DFS defines a tree, with r0 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)