Chapter 8, Termination Detection Video Solutions, Introduction to Distributed Algorithms

Problem 1

Characterize the active and passive states of Algorithm A.2. Where are these states found in Algorithm A.1?
The time complexity of a termination-detection algorithm is defined as the worst-case number of time units (under the idealized assumptions of Definition 6.31) between the termination of the basic computation and the call to Announce.

Chris Trentman

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01:15

Problem 2

What is the time complexity of the Dijkstra-Scholten algorithm?

Nick Johnson

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Problem 3

The Shavit-Francez algorithm is applied in an arbitrary network with unique identities, and to keep control-message overheads low the Gallager-Humblet-Spira algorithm is used as the wave algorithm. The time complexity of the detection is $\Omega(N \log N)$.
Can you improve the time complexity to $\mathrm{O}(N)$ at the cost of exchanging $\mathrm{O}(N)$ additional control messages?
Exercise 8.4 Why is the predicate $P_0$ in the derivation of the DijkstraFeijen-Van Gasteren algorithm not falsified if $p_j$ is activated by $p_i$, where $j \leq t$ or $i>t$ ?

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Problem 4

Why is the predicate $P_0$ in the derivation of the DijkstraFeijen-Van Gasteren algorithm not falsified if $p_j$ is activated by $p_i$, where $j \leq t$ or $i>t$ ?

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Problem 5

Show that for each $m$ there exists a basic computation that exchanges $m$ messages and forces the Dijkstra-Feijen-Van Gasteren algorithm to exchange $m \cdot(N-1)$ control messages.

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Problem 6

What modifications must be made in Algorithm 8.9 to implement rule $5 a$ of the credit-recovery algorithm, instead of rule $5 b$ ?

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Problem 7

It is assumed in Rana's algorithm that processes have identities. Now assume instead that processes are anonymous, but have a means of sending messages to their successors in the ring, and that the number of processes is known. Modify Algorithm 8.10 to work under this assumption.

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00:04

Problem 8

Show the correctness of Rana's algorithm (Algorithm 8.10) from an invariant of the algorithm.

Sanchit Gogia

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