A variant of the arrow distributed directory with low average complexity (extended abstract)

This paper considers an enhancement to the arrow distribu- Ted directory protocol, introduced in [8]. The arrow protocol implements a directory service, allowing nodes to locate mobile objects in a distributed system, while ensuring mutual exclusion in the presence of concurrent re- quests. The arrow protocol makes use of a minimum spanning tree (MST) T_m of the network, selected during system initialization, resulting in a worst-case overhead ratio of (1 + stretch(T_m))=2. However, we observe that the arrow protocol is correct communicating over any spanning tree T of G. We show that the worst-case overhead ratio is minimized by the mi- nimum stretch spanning tree (MSST), and that the problem cannot be approximated within a factor better than (1 +√5)=2, unless P = NP. In contrast, other trees may be more suitable if one is interested in the average-case behavior of the network. We show that in the case where the distribution of the requests is fixed and known in advance, the ex- pected communication is minimized using the minimum communication cost spanning tree (MCT). It is shown that the resulting MCT problem is a restricted case for which one can find a tree T over which the expec- Ted communication cost of the arrow protocol is at most 1:5 times the expected communication cost of an optimal protocol. We also show that even if the distribution of the requests is not fixed, or not known to the algorithm in advance, then if the adversary is ob- livious, one may use probabilistic approximation of metric spaces [2,3] to ensure an expected overhead ratio of O(log n log log n) in general, and an expected overhead ratio of O(log n) in the case of constant dimension Euclidean graphs.