${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$, which are just strong enough for solving $k$-set agreement: We introduce the generalized $(n-k)$-loneliness failure detector ${\cal L}(k)$, which we first prove to be sufficient for solving $k$-set agreement, and show that ${\cal L}(k)$ but not ${\cal L}(k-1)$ can be implemented in both models. ${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$ are hence the first message passing models that lie between models where $\Omega$ (and therefore consensus) can be implemented and the purely asynchronous model. We also address $k$-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel $k$ -set agreement algorithm using ${\cal L}(k)$ also works in anonymous systems, it turns out that the loneliness failure detector ${\cal L}={\cal L}(n-1)$ introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between ${\cal L}(k)$ and other failure detectors suitable for solving $k$-set agreement." /> ${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$, which are just strong enough for solving $k$-set agreement: We introduce the generalized $(n-k)$-loneliness failure detector ${\cal L}(k)$, which we first prove to be sufficient for solving $k$-set agreement, and show that ${\cal L}(k)$ but not ${\cal L}(k-1)$ can be implemented in both models. ${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$ are hence the first message passing models that lie between models where $\Omega$ (and therefore consensus) can be implemented and the purely asynchronous model. We also address $k$-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel $k$ -set agreement algorithm using ${\cal L}(k)$ also works in anonymous systems, it turns out that the loneliness failure detector ${\cal L}={\cal L}(n-1)$ introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between ${\cal L}(k)$ and other failure detectors suitable for solving $k$-set agreement." /> ${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$, which are just strong enough for solving $k$-set agreement: We introduce the generalized $(n-k)$-loneliness failure detector ${\cal L}(k)$, which we first prove to be sufficient for solving $k$-set agreement, and show that ${\cal L}(k)$ but not ${\cal L}(k-1)$ can be implemented in both models. ${\cal M}^{{\rm anti}(n-k)}$ and ${\cal M}^{{\rm sink}(n-k)}$ are hence the first message passing models that lie between models where $\Omega$ (and therefore consensus) can be implemented and the purely asynchronous model. We also address $k$-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel $k$ -set agreement algorithm using ${\cal L}(k)$ also works in anonymous systems, it turns out that the loneliness failure detector ${\cal L}={\cal L}(n-1)$ introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between ${\cal L}(k)$ and other failure detectors suitable for solving $k$-set agreement." /> The Generalized Loneliness Detector and Weak System Models for k-Set Agreement

This paper presents two weak partially synchronous system models M^{anti(n-k)} and M^{sink(n-k)}, which are just strong enough for solving k-set agreement: We introduce the generalized (n-k)-loneliness failure detector L(k), which we first prove to be sufficient for solving k-set agreement, and show that L(k) but not L(k-1) can be implemented in both models. M^{anti(n-k)} and M^{sink(n-k)} are hence the first message passing models that lie between models where Ω (and therefore consensus) can be implemented and the purely asynchronous model. We also address k-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel k -set agreement algorithm using L(k) also works in anonymous systems, it turns out that the loneliness failure detector L=L(n-1) introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between L(k) and other failure detectors suitable for solving k-set agreement.

INDEX TERMS

Detectors, Computer crashes, Message passing, Delays, Computational modeling, Biological system modeling, Electronic mail,Distributed systems, models of computation,

CITATION

Ulrich Schmid, "The Generalized Loneliness Detector and Weak System Models for k-Set Agreement", IEEE Transactions on Parallel & Distributed Systems, vol.25, no. 4, pp. 1078-1088, April 2014, doi:10.1109/TPDS.2013.77