Anonymous Processors with Synchronous Shared Memory

We consider synchronous distributed systems in which anonymous processors communicate by shared read-write variables. The goal is to have all the processors assign unique names to themselves. We consider the instances of this problem determined by whether the number $n$ is known or not, and whether concurrently attempting to write distinct values into the same memory cell is allowed or not, and whether the number of shared variables is a constant independent of $n$ or it is unbounded. For known $n$, we give Las Vegas algorithms that operate in the optimum expected time, as determined by the amount of available shared memory, and use the optimum $O(n\log n)$ expected number of random bits. For unknown $n$, we give Monte Carlo algorithms that produce correct output upon termination with probabilities that are $1-n^{{-\Omega(1)}$,} which is best possible when terminating almost surely and using $O(n\log n)$ random bits.