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Introduction to Distributed AlgorithmPart Two: Fundamental Algorithm

Chapter 6- Wave and Traversal Algorithms

Teacher: Chun-Yuan Lin

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Wave and Traversal Algorithms (1) In the design of distributed algorithms for various applications several

very general problems for process networks appear frequently as subtasks.

broadcasting of information achieving a global synchronization between processes triggering the execution of some event in each process computing a function of which each process holds part of the input

These tasks are so fundamental that solutions to more complicated problems

election (Chapter 7) termination detection (Chapter 8)

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Wave and Traversal Algorithms (2)

The importance of message-passing schemes, called wave algorithms This chapter formally defines wave algorithms (Subsection 6.1.1) proves some general results about them (Subsection 6.1.2) The observation that the same algorithms can be used for all of the

fundamental tasks (Subsections 6.1.3 through 6.1.5) Section 6.2 presents some widely used wave algorithms. Section 6.3 considers traversal algorithms Section 6.4 gives several algorithms for a distributed depth-first search

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Wave and Traversal Algorithms (3)

The treatment of wave algorithms as a separate issue is useful for two reasons. (why study wave algorithm?)

First, the introduction of the concept facilitates the later treatment of more involved algorithms because the properties of their subroutines have already been studied.

Second, certain problems in distributed computing can be solved by generic constructions that yield a specific algorithm when parameterized with a specific wave algorithm.

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Definition and Use of Wave Algorithms

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Definition of Wave Algorithms (1)

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Definition of Wave Algorithms (2)

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Definition of Wave Algorithms (3)

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Definition of Wave Algorithms (4)

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Definition of Wave Algorithms (5)

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Elementary Results about Wave Algorithms (1)

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Elementary Results about Wave Algorithms (2)

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Elementary Results about Wave Algorithms (3)

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Propagation of Information with Feedback (1)

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Propagation of Information with Feedback (2)

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Synchronization

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Computation of Infimum Functions (1)

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Computation of Infimum Functions (2)

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Computation of Infimum Functions (3)

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A Collection of Wave Algorithms (1)

In all cases the algorithm text is given for the process p.

The Ring Algorithm

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A Collection of Wave Algorithms (2)

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A Collection of Wave Algorithms (3) The Tree Algorithm

(when a process receive all messages from its neighbor, decide)

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A Collection of Wave Algorithms (4)

(leave to root, root to leave)

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A Collection of Wave Algorithms (5) The Echo Algorithm

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A Collection of Wave Algorithms (6)

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A Collection of Wave Algorithms (7) The Polling Algorithm

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A Collection of Wave Algorithms (8)

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A Collection of Wave Algorithms (9) The Phase Algorithm

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A Collection of Wave Algorithms (10)

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A Collection of Wave Algorithms (11)

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A Collection of Wave Algorithms (12)

Finn's Algorithm

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A Collection of Wave Algorithms (13)

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A Collection of Wave Algorithms (14)

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Traversal Algorithms (1)

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Traversal Algorithms (2)

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Traversal Algorithms (3) Traversing Cliques

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Traversal Algorithms (4)

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Traversal Algorithms (5) Traversing Tori

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Traversal Algorithms (6) Traversing Hypercubes

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Traversal Algorithms (7) Traversing Connected Networks

A traversal algorithm for arbitrary connected networks was given by Tarry in 1895.

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Time Complexity: Depth-first Search (1)

The processes in Tarry's algorithm are given sufficient freedom, in choosing a neighbor to which to forward the token.

In Section 6.4.1, a minor modification of Tarry's algorithm enables the algorithm to compute depth-first search trees. (classical depth-first search algorithm)

In Subsection 6.4.2 two algorithms will be discussed that compute depth-first search trees in less time than the classical algorithm.

In Subsection 6.4.3 a depth-first search algorithm will be presented for networks with initial knowledge of neighbors' identities.

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Time Complexity: Depth-first Search (2)

The time complexity of distributed algorithms

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Distributed Depth-first Search (1)

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Depth-first Search Algorith'ms Using Linear Time

Awerbuch's solution

Cidon's solution

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Depth-first Search with Neighbor Knowledge

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Remaining Issues (1) Overview of Wave Algorithms

Computing Sums Impossibility of a general construction Computing the sum with a traversal algorithm Computing the sum using a spanning tree Computing the sum using identities

Alternative Definitions of Time Complexity

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Remaining Issues (2)

Definitions based on more restrictive timing assumptions

Definition based on message chains