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Introduction to Distributed Algorithms

Gerard Tel

Chapter 9

Anonymous Networks - all with Video Answers

Educators

Chapter Questions

23:32

Problem 1

Let $\psi$ be some postcondition and assume that are given
(1) a process terminating Monte Carlo algorithm $A$ to establish $\psi$; and
(2) a deterministic, process terminating verification algorithm $B$ that tests if $\psi$ holds or not.
Show how to construct a Las Vegas algorithm $C$ to establish $\psi$.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator

Problem 2

Let $\psi$ be some postcondition and assume that a Las Vegas algorithm $A$ to establish $\psi$ is given and the expected number of messages exchanged by $A$ is known, namely $K$. Construct a Monte Carlo algorithm to establish $\psi$ with parameterizable failure probability $\epsilon$ (i.e., the algorithm must terminate and be correct with probability $1-\epsilon$ ).

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08:57

Problem 3

Give a deterministic, centralized name assignment algorithm that uses $2|E|$ messages and $\mathrm{O}(D)$ time units.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 4

[Ang80] Give a deterministic algorithm for election in a clique where communication is by synchronous message passing.

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06:10

Problem 5

Give deterministic, message-terminating algorithms for computing MAX for rings and arbitrary networks of unknown size.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 6

Consider the election problem for anonymous trees of unknown size, where communication is by asynchronous message passing.
(1) Give a randomized algorithm that is partially correct, process-terminates with probability one, and has an expected message complexity of $\mathrm{O}(N)$ messages. (Actually, an expected message complexity of $N+1$ messages is achievable.)
(2) Does a deterministic algorithm exist for this case?

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Problem 7

Prove that there exists no deterministic algorithm that computes the number of processes and that is correct for all anonymous networks of diameter at most two.
Hint: exploit the symmetry in the networks of Figure 6.21.

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Problem 8

Prove that there exists no deterministic algorithm for election in rings of known, even size where communication is by synchronous message passing.
Generalize the proof to show the impossibility of election for all composite ring sizes.

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Problem 9

Define the function $f$ on strings $x=\left(x 1, \ldots, x_k\right)$ of integers, where $f(x)$ is true if all $x_{\mathrm{i}}$ are equal and false otherwise. Is $f$ deterministically computable by a process-terminating algorithm in rings of unknown size? Is $f$ deterministically computable by a message-terminating algorithm in rings of unknown size?

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01:08

Problem 10

The same questions as in Exercise 9.9, but for the case where $g(x)$ is true if $x$ has a nondecreasing cyclic shift, i.e., there exists a cyclic shift $z$ of $x$ such that $i<j \Rightarrow z_1 \leq z_j$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator

Problem 11

Give a process-terminating Monte Carlo algorithm for election on anonymous rings of known size. What is the message complexity and what is the success probability of your algorithm?
Is it possible to achieve a success probability that is arbitrarily close to one?
A pseudo-anonymous network is a network where the processes have identities, but the identities are not necessarily all distinct. A pseudo-anonymous ring (of processes $p_0$ through $p_{N-1}$ say) is periodic if there exists a number $k<N$ such that $i d_i=i d_{i+k}$ for all $i$.

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Problem 12

Show that there exists a deterministic, process terminating algorithm for election on non-periodic pseudo-anonymous rings of known size.

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Problem 13

Show that there exists no probabilistic, process terminating algorithm for election on non-periodic pseudo-anonymous rings of unknown size that is correct with positive probability.

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01:26

Problem 14

Define the function $g$ as in Exercise 9.10. Give a probabilistic, message-terminating algorithm for computing $g$ in a ring of unknown size that is correct with probability $1-\epsilon$.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 15

Is it possible to detect termination of the Itai-Rodeh algorithm (Algorithm 9.5) using Safra's termination-detection algorithm (Algorithm 8.7)?
Can you think of an other termination-detection algorithm discussed earlier that can be used for this purpose?

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