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Distributed colouring and communication in rings with local knowledge

Dessmark, Anders LU and Pelc, A (2004) In Combinatorics, Probability & Computing 13(2). p.123-136
Abstract
We consider two interrelated tasks in a synchronous n-node ring: distributed constant colouring and local communication. We investigate the impact of the amount of knowledge available to nodes on the time of completing these tasks. Every node knows the labels of nodes up to a distance r from it, called the knowledge radius. In distributed constant colouring every node has to assign itself one out of a constant number of colours, so that adjacent nodes get different colours. In local communication every node has to communicate a message to both of its neighbours. We study these problems in two popular communication models: the one-way model, in which each node can only either transmit to one neighbour or receive from one neighbour, in any... (More)
We consider two interrelated tasks in a synchronous n-node ring: distributed constant colouring and local communication. We investigate the impact of the amount of knowledge available to nodes on the time of completing these tasks. Every node knows the labels of nodes up to a distance r from it, called the knowledge radius. In distributed constant colouring every node has to assign itself one out of a constant number of colours, so that adjacent nodes get different colours. In local communication every node has to communicate a message to both of its neighbours. We study these problems in two popular communication models: the one-way model, in which each node can only either transmit to one neighbour or receive from one neighbour, in any round, and the radio model, in which simultaneous receiving from two neighbours results in interference noise. Hence the main problem in fast execution of the above tasks is breaking symmetry with restricted knowledge of the ring. We show that distributed constant colouring and local communication are tightly related and one can be used to accomplish the other. Also, in most situations the optimal time is the same for both of them, and it strongly depends on the knowledge radius. For knowledge radius r = 0, i.e., when each node knows only its own label, our bounds on time for both tasks are tight in both models: the optimal time in the one-way model is Theta(n), while in the radio model it is Theta(log n). For knowledge radius r = 1 both tasks can be accomplished in time 0(log log n) in the one-way model, if the ring is oriented. For 2 less than or equal to r less than or equal to clog(*) n, where c < 1/2 the upper bounds on time are 0(log ((2r)) n) in the one-way model and 0(log((2[r/2])) n) in the radio model; the lower bound is Omega(log(*) n), in both models. For r greater than or equal to (log(*) n)/2 both tasks can be completed in constant time, in the one-way model, and distributed constant colouring also in the radio model. Finally, if r greater than or equal to log(*) n then constant time is also enough for local communication in the radio model. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Combinatorics, Probability & Computing
volume
13
issue
2
pages
123 - 136
publisher
Cambridge University Press
external identifiers
  • wos:000220908000001
  • scopus:3142539294
ISSN
1469-2163
DOI
10.1109/IPDPS.2001.925025
language
English
LU publication?
yes
id
1c8e794c-d8f1-4696-9112-0f53ae5c14cf (old id 281036)
date added to LUP
2007-10-23 13:43:30
date last changed
2017-01-01 05:12:20
@article{1c8e794c-d8f1-4696-9112-0f53ae5c14cf,
  abstract     = {We consider two interrelated tasks in a synchronous n-node ring: distributed constant colouring and local communication. We investigate the impact of the amount of knowledge available to nodes on the time of completing these tasks. Every node knows the labels of nodes up to a distance r from it, called the knowledge radius. In distributed constant colouring every node has to assign itself one out of a constant number of colours, so that adjacent nodes get different colours. In local communication every node has to communicate a message to both of its neighbours. We study these problems in two popular communication models: the one-way model, in which each node can only either transmit to one neighbour or receive from one neighbour, in any round, and the radio model, in which simultaneous receiving from two neighbours results in interference noise. Hence the main problem in fast execution of the above tasks is breaking symmetry with restricted knowledge of the ring. We show that distributed constant colouring and local communication are tightly related and one can be used to accomplish the other. Also, in most situations the optimal time is the same for both of them, and it strongly depends on the knowledge radius. For knowledge radius r = 0, i.e., when each node knows only its own label, our bounds on time for both tasks are tight in both models: the optimal time in the one-way model is Theta(n), while in the radio model it is Theta(log n). For knowledge radius r = 1 both tasks can be accomplished in time 0(log log n) in the one-way model, if the ring is oriented. For 2 less than or equal to r less than or equal to clog(*) n, where c &lt; 1/2 the upper bounds on time are 0(log ((2r)) n) in the one-way model and 0(log((2[r/2])) n) in the radio model; the lower bound is Omega(log(*) n), in both models. For r greater than or equal to (log(*) n)/2 both tasks can be completed in constant time, in the one-way model, and distributed constant colouring also in the radio model. Finally, if r greater than or equal to log(*) n then constant time is also enough for local communication in the radio model.},
  author       = {Dessmark, Anders and Pelc, A},
  issn         = {1469-2163},
  language     = {eng},
  number       = {2},
  pages        = {123--136},
  publisher    = {Cambridge University Press},
  series       = {Combinatorics, Probability & Computing},
  title        = {Distributed colouring and communication in rings with local knowledge},
  url          = {http://dx.doi.org/10.1109/IPDPS.2001.925025},
  volume       = {13},
  year         = {2004},
}