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Local mutual exclusion guarantees that no two neighboring processes enter acritical section at the same time while satisfying both mutual exclusion and nostarvation properties. On the other hand, processes may want to execute someoperation simultaneously with the neighbors. Of course, we can use a globallysynchronized clock to achieve the task but it is very expensive to realize itin a distributed system in general.  In this paper, we define a new concept neighborhood mutual remainder. Adistributed algorithm that satisfies the neighborhood mutual remainderrequirement should satisfy global fairness, l-exclusion and repeated localrendezvous requirements. Global fairness is satisfied when each process (thatrequests to enter the critical section infinitely often) executes the criticalsection infinitely often, l-exclusion is satisfied when at most l neighboringprocesses enter the critical section at the same time, and repeated localrendezvous is satisfied when for each process infinitely often no process inthe closed neighborhood is in the critical or trying section.  We first formalize the concept of neighborhood mutual remainder, and give asimple self-stabilizing algorithm to demonstrate the design paradigm to achieveneighborhood mutual remainder. We also present two applications of neighborhoodmutual remainder to a Look-Compute-Move robot system. One is for implementing amove-atomic property and the other is for implementing FSYNC scheduler, whererobots possess an independent clock that is advanced in the same speed. Theseare the first self-stabilizing implementations of the LCM synchronization.

Neighborhood Mutual Remainder: Self-Stabilizing Implementation of Look-Compute-Move Robots (Extended Abstract)