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Introduction to Distributed Algorithms

Gerard Tel

Chapter 7

Election Algorithms - all with Video Answers

Educators

Chapter Questions

Problem 1

Prove that a comparison election algorithm for arbitrary networks is a wave algorithm if the event in which a process becones leader is regarded as a decision event.

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Problem 2

Show that the time complexity of Algorithm 7.1 is $2 D$.

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01:00

Problem 3

Prove the identity $\sum_{1=1}^m \mathrm{H}_4=(m+1) \mathrm{H}_m-m$ used in Sub. section 7.2.1.

Rukhmani Jain
Rukhmani Jain
Numerade Educator
05:16

Problem 4

Show that $\ln (N+1)<\mathrm{H}_N<\operatorname{in}(N)+1$. (In denotes the natural logarithm.)

Bryan Lynn
Bryan Lynn
Numerade Educator

Problem 5

Consider the Cisiny-Roberts algorithm under the assumption that every process ss an initzaivor. For what distribution of identities over the rung is the message complerrty minimal and exactly how many mesiages cre exchanged in this case?

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01:10

Problem 6

What is the aw rage case complexity of the Chang-Roberts algornthrn of there are exactiy $S$ inituators, where each choice of $S$ processes is equally likely to be the sel os nutiators?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:19

Problem 7

Gine an invtz:: onfiguration for Algorathm 7.7 for which the algorithm actually requares !lov $\mathrm{V}\rfloor+1$ rounds. Also give an initial configurat,in for which the algorib: $\%$ requires only two rounds, regardless of the number of mutiators. Is it pusible for the algorithm to terminate in one round?

James Chok
James Chok
Numerade Educator
01:21

Problem 8

Determine the et $E_{\mathrm{CR}}$ (as defined before Lemma 7.10) for the Chang-Roberts algorithrt

Raj Bala
Raj Bala
Numerade Educator
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Problem 9

Apply extinctrois to the ring algorithm and compare the algoruthm with the Chang-Roberts algorthm. What is the difference and what is the effect of this difference?

James Kiss
James Kiss
Numerade Educator
02:06

Problem 10

Determine, for each of the seven message types used in the Gallager/Humblet/Spira algorithm, whether a message of this type can be serit to a node in the sleep state.

Thomas Emment
Thomas Emment
Numerade Educator
04:16

Problem 11

Assume that the GHS algorithm uses an additional wake-up procedure that guarantes thai each node starts the algorithm within $N$ time units.
Prove by induction that after at most $5 \mathrm{Nl}-3 \mathrm{~N}$ time units each node is at level $l$.
Prove that the algorithm terminates within $5 N \log N$ time units.

Ibrahima Barry
Ibrahima Barry
Numerade Educator
05:24

Problem 12

Show that an $\mathrm{O}(N \log N)$ algorithm for election in planar networks exists.

Chris Trentman
Chris Trentman
Numerade Educator
05:16

Problem 13

Show that there sursts an $\mathrm{O}(N \log N)$ rlection alyorithm for tort without a sense of directron.

Bryan Lynn
Bryan Lynn
Numerade Educator
05:16

Problem 14

Show that there extsts an $O(N \log N)$ elrction alyorythen for hypercubes without a sense of drection.

Bryan Lynn
Bryan Lynn
Numerade Educator
05:24

Problem 15

Show that there exsts an $O(N(\log N+h))$ electon algurtihn for networks wnth bounded degree $k$ (i.e.. networks where each noje has at most $k$ neighbors).

Chris Trentman
Chris Trentman
Numerade Educator