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After a failure occurs in a distributed computing system, it is often necessary to reorganize the active nodes so that they can continue to perform a useful task. The first step in such a reorganization or reconfiguration is to elect a coordinator node to manage the operation. This paper discusses such elections and reorganizations. Two types of reasonable failure environments are studied. For each environment assertions which define the meaning of an election are presented. An election algorithm which satisfies the assertions is presented for each environment.