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Introduction to Distributed Algorithms

Gerard Tel

Chapter 11

Synchrony in Networks - all with Video Answers

Educators

Chapter Questions

Problem 1

Improve the protocol for sending information by "coding in time" so that message $m$ is transmitted in $\mathrm{O}(\sqrt{m})$ time units using a constant number of bits.

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Problem 2

Theorem 9.8 states that:
There exists no deterministic process-terminating algorithm for compute a non-constant function $f$ if the ring size is not known.
Does this theorem hold for synchronous networks as well?

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Problem 3

Prove that in Algorithm 11.1 a process can receive during the execution of pulse $i$ (the interval $(2 i \mu, 2(i+1) \mu)$ on its local clock) only messages of pulses $i$ and $i+1$.
(As a consequence, the pulse number of a message can be determined using its time of receipt and the parity of the pulse number.)

Victor Salazar
Victor Salazar
Numerade Educator

Problem 4

Assume the synchronization phase of Algorithm 11.1 is extended in such a way that process $p$ records the clock time $\delta_{p q}=C L O C K_p^{(r)}$ at the time $\tau$ at which $p$ receives the (start) message of $q . \quad\left(\delta_{p q}=0\right.$ if this message caused the execution of init in p.) Prove thatwhen $p$ receives a pulse-i message of $q$, the value of $p$ 's clock is between $\delta_{p q}+2 i \mu-\mu$ and $\delta_{p q}+2 i \mu+\mu$.
(As a consequence, the pulse number of a message received from $q$ at clock time $c$ is found as $\left\lfloor\left(c-\delta_{p q}+\mu\right) / 2\right\rfloor$; so no information concerning the message's pulse number need be included in the message.)

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Problem 5

Give algorithms for flooding messages in cliques, tori, and hypercubes that require $N-1$ messages and $\mathrm{O}(D)$ time units. (It must be assumed that the tori and hypercubes are labeled.)

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02:47

Problem 6

Give synchronous election algorithms for networks of known size, in the case where processes do not necessarily start the election in the same pulse, but may initiate the algorithm in different pulses.

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 7

Determine the message complexity of the asynchronous election algorithms obtained by combining the synchronous algorithms of Section 11.2 with the synchronizers of Section 11.3.

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01:21

Problem 8

Prove that a neighbor of a node of level $f$ in the BFS tree has level $f-1, f$, or $f+1$.

Prashant Bana
Prashant Bana
Numerade Educator
01:15

Problem 9

Analyze the complexity of the breadth-first search algorithms in terms of $N,|E|$, and $D$ (the network diameter).

Nick Johnson
Nick Johnson
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Problem 10

[Vit85] The aim of this exercise is to investigate the effect of selective delays in Archimedean networks. Let a network be given with Archimedean ratio $s$.
Election is performed by applying extinction to a traversal algorithm with message complexity $W$. Each message of process $p$ is delayed by $f(p)-1$ clock ticks in each process. A separate wake-up procedure ensures that each process launches its traversal within Du time units after the start of the algorithm.
(1) Prove that the algorithm terminates within D.u+W.u.f(p $\left.p_0\right)$ time units, where $p_0$ is the process with the smallest identity.
(2) Prove that the token of process $p$ is sent at most $1+t /(f(p)-2)$ times during a time interval of length $t$.
(3) Derive a formula for the worst-case message complexity of the algorithm.
(4) Show, by varying $f$, that a linear time complexity can be obtained.

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