Random Number Selection in Self-assembly

We investigate methods for exploiting nondeterminism inherent within the Tile Assembly Model in order to generate uniform random numbers. Namely, given an integer range {0, . . . , n− 1}, we exhibit methods for randomly selecting a number within that range. We present three constructions exhibiting a trade-off between space requirements and closeness to uniformity. The first selector selects a random number with probability Θ(1/n) using O(log n) tiles. The second selector takes a user-specified parameter that guarantees the probabilities are arbitrarily close to uniform, at the cost of additional space. The third selector selects a random number with probability exactly 1/n, and uses no more space than the first selector with high probability, but uses potentially unbounded space.