Chapter 13, Fault Tolerance in Asynchronous Systems Video Solutions, Introduction to Distributed Algorithms

Problem 1

Omission of any of the three requirements of Definition 13.3 (termination, agreement, non-triviality) for the consensus problem allows a very simple solution. Show this by presenting the three simple solutions.

Andrew John De Los Santos

Numerade Educator

12:07

Problem 2

In the proof of Lemma 19.6 it is assumed that each of the $2^N$ assignments of a bit to the $N$ processes produces a possible input configuration.
Give deterministic, 1-crash robust consensus protocols for each of the following restrictions on the input values.
(1) It is given that the parity of the input is even (i.e., there are an even number of processes with input 1) in each initial configuration.
(2) There are two (known) processes $r_1$ and $r_2$, and each initial configuration satisfies $x_{r_1}=x_{r_2}$.
(3) In each initial configuration there are at least $\lceil(N / 2)+1\rceil$ processes with the same input.

Chris Trentman

Numerade Educator

Problem 3

Show that there is no $t$-initially-dead-robust election algorithm for $t \geq N / 2$.

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Problem 4

Show that no algorithm for e-approximate agreement can tolerate $t \geq N / 2$ crashes.

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02:10

Problem 5

Give a bijection from the set
$$
\{(s, r): N-t \leq s \leq N \text { and } 1 \leq r \leq s\}
$$
to integers in the range $[1, \ldots, K]$.

Goutam Chand

Numerade Educator

01:15

Problem 6

Is Algorithm 13.2 non-trivial?

Nick Johnson

Numerade Educator

01:01

Problem 7

Adapt the proof of Theorem 13.15 for the case that $G_T$ consusts of $k$ connected components.

Raj Bala

Numerade Educator

Problem 8

In this exercise we consider the problem of $[k, l]$-election, which generalizes the usual election problem. The problem requires that all correct processes decide on either 0 ("defeated") or $s 11$ ("elected"), and that the number of processes that decide 1 is between $k$ and $l$ (inclusive).
(1) What are the uses of $[k, l]$-election?
(2) Demonstrate that no deterministic 1-crash robust algorithm for $[k, k]$ election exists.(if $0<k<N$ ).
(3) Give a deterministic $t$-crash robust algorithm for $[k, k+2 t]$-election.

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01:29

Problem 9

Does the convergence requirement imply that the expected number of steps is bounded?
Is the expected number of steps bounded in all algorithms of this section?

Bobby Barnes

University of North Texas

00:04

Problem 10

Show that if all correct processes start round $k$ of the crashrobust consensus algorithm (Algorithm 13.3), then all correct processes will also finish round $k$.

Sanchit Gogia

Numerade Educator

Problem 11

(1) Prove, that if more that $(N+t) / 2$ processes start the crash-robust consensus algorithm (Algorithm 13.3) with input $v$, then a decision for $v$ is taken in three rounds.
(2) Prove, that if more than $(N-t) / 2$ processes start the algorithm with input $v$, then a decision for $v$ is possible.
(3) Is a decision for $v$ possible if exactly $(N-t) / 2$ processes start the algorithm with input $v$ ?
(4) What are the bivalent input configurations of the algorithm?

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06:34

Problem 12

(1) Prove that, if more than $(N+t) / 2$ correct processes start Algorithm 19.5 with input $v$, a $v$-decision is eventually taken.
(2) Prove that, if more than $(N+t) / 2$ correct processes start Algorithm 13.5 with input $v$ and $t<N / 5$, a $v$-decision is taken within two rounds.

Lucas Gagne

Numerade Educator

Problem 13

Prove that no asynchronous $t$-Byzantine-robust broadcast algorithm exists for $t \geq N / 3$.

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06:34

Problem 14

Prove that during the execution of Algorithm 19.6 at most $N(3 N+1)$ messages are sent by correct processes.

Lucas Gagne

Numerade Educator