Probabilistic Snap-Stabilizing Algorithms for Local Resource Allocation Problems

Scientific Context. Modern networks are very large-scale (about 100 000 nodes). Now, the more a network contains nodes, the greater the probability of failures is. Hence, today fault-tolerance is a main concern for distributed algorithm designers. In particular, since several decades, there is a growing interest for self-repairing methods. For example, (deterministic) self-stabilization [Dij74] is a versatile property, enabling a distributed algorithm to withstand transient faults in a network. Indeed, a self-stabilizing algorithm, after transient faults (e.g., memory corruptions, message losses, etc.) hit and place the network in some arbitrary state, enables the network to recover without external (e.g., human) intervention in finite time. For example, consider a self-stabilizing circulation of a single token. Such an algorithm can be used to implement mutual exclusion in the network: a node can access to the critical section only if it is the token holder. Now, after a transient fault, the token may be duplicated. Consequently, the exclusive access to the critical section is (temporarily) no more guaranteed. The self-stabilizing property ensures in this case that the network recovers a legitimate configuration containing a single token within finite time. However, self-stabilization has a major drawback: after faults, there is a convergence phase during which no safety property can be ensured despite no fault occurs in the network. Recent research has been made to overcome this drawback leading to propose new properties that offer more guarantees, e.g., (deterministic) Snap-stabilization [BDPV99]. After a finite number of transient faults, a snap-stabilizing algorithm immediately operates correctly, without any external (e.g. human) intervention. By contrast, self-stabilization only guarantees that the network eventually recovers to a correct behavior. Snap-stabilization is a powerful technique. But, as for deterministic self-stabilization, many problems have no deterministic snap-stabilizing solution. This is in particular true in anonymous networks, where graph coloring and token passing are known to be impossible to solve under the deterministic (selfor snap-) stabilizing setting. To cope with these impossibility results, we are currently introducing probabilistic snap-stabilization, which basically means that safety properties of each task started after the last fault will be (deterministically) guaranteed, while liveness properties of the task will be only guaranteed with probability 1.