### Abstract

In an "anonymous" network the processors have no identity numbers. We investigate the problem of computing a given function f on an asynchronous anonymous network in the sense that each processor computes f (I) for any input I = (I(υ_{1}), . . . , I(υ_{n})), where I(υ_{i}) is the input to processor υ_{i}, i = 1, 2, . . . , n. We address the following three questions: (1) What functions are computable on a given network? (2) Is there a "universal" algorithm which, given any network G and any function f computable on G as inputs, computes f on G? (3) How can one find lower bounds on the message complexity of computing various functions on anonymous networks? We give a necessary and sufficient condition for a function to be computable on an asynchronous anonymous network, and present a universal algorithm for computing f(I) on any network G, which accepts G and f computable on G, as well as {I(υ_{i})}, as inputs. The universal algorithm requires O(mn) messages in the worst case, where n and m are the numbers of processors and links in the network, respectively. We also propose a method for deriving a lower bound on the number of messages necessary to solve the above problem on asynchronous anonymous networks.

Original language | English |
---|---|

Pages (from-to) | 331-356 |

Number of pages | 26 |

Journal | Mathematical Systems Theory |

Volume | 29 |

Issue number | 4 |

Publication status | Published - Jul 1 1996 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Mathematics(all)
- Computational Theory and Mathematics

### Cite this

*Mathematical Systems Theory*,

*29*(4), 331-356.

**Computing functions on asynchronous anonymous networks.** / Yamashita, Masafumi; Kameda, T.

Research output: Contribution to journal › Article

*Mathematical Systems Theory*, vol. 29, no. 4, pp. 331-356.

}

TY - JOUR

T1 - Computing functions on asynchronous anonymous networks

AU - Yamashita, Masafumi

AU - Kameda, T.

PY - 1996/7/1

Y1 - 1996/7/1

N2 - In an "anonymous" network the processors have no identity numbers. We investigate the problem of computing a given function f on an asynchronous anonymous network in the sense that each processor computes f (I) for any input I = (I(υ1), . . . , I(υn)), where I(υi) is the input to processor υi, i = 1, 2, . . . , n. We address the following three questions: (1) What functions are computable on a given network? (2) Is there a "universal" algorithm which, given any network G and any function f computable on G as inputs, computes f on G? (3) How can one find lower bounds on the message complexity of computing various functions on anonymous networks? We give a necessary and sufficient condition for a function to be computable on an asynchronous anonymous network, and present a universal algorithm for computing f(I) on any network G, which accepts G and f computable on G, as well as {I(υi)}, as inputs. The universal algorithm requires O(mn) messages in the worst case, where n and m are the numbers of processors and links in the network, respectively. We also propose a method for deriving a lower bound on the number of messages necessary to solve the above problem on asynchronous anonymous networks.

AB - In an "anonymous" network the processors have no identity numbers. We investigate the problem of computing a given function f on an asynchronous anonymous network in the sense that each processor computes f (I) for any input I = (I(υ1), . . . , I(υn)), where I(υi) is the input to processor υi, i = 1, 2, . . . , n. We address the following three questions: (1) What functions are computable on a given network? (2) Is there a "universal" algorithm which, given any network G and any function f computable on G as inputs, computes f on G? (3) How can one find lower bounds on the message complexity of computing various functions on anonymous networks? We give a necessary and sufficient condition for a function to be computable on an asynchronous anonymous network, and present a universal algorithm for computing f(I) on any network G, which accepts G and f computable on G, as well as {I(υi)}, as inputs. The universal algorithm requires O(mn) messages in the worst case, where n and m are the numbers of processors and links in the network, respectively. We also propose a method for deriving a lower bound on the number of messages necessary to solve the above problem on asynchronous anonymous networks.

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UR - http://www.scopus.com/inward/citedby.url?scp=33748851013&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33748851013

VL - 29

SP - 331

EP - 356

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 4

ER -