The Logarithmic Random Bidding for the Parallel Roulette Wheel Selection with Precise Probabilities
Koji Nakano
TL;DR
The paper tackles the problem of exact fitness-proportionate (roulette wheel) selection in parallel settings, where many fitness values may be zero. It introduces a novel technique called logarithmic random bidding, computing $r_i = {\log(\mathrm{rand}())\over f_i}$ for each processor and selecting the maximum to realize $P(i) = F_i = {f_i\over \sum_j f_j}$. A formal proof shows this method yields the correct probabilities for all $i$ with $k$ non-zero fitness values, achieving $O(\log k)$ expected time on a CRCW-PRAM with constant shared memory. The implementation leverages a shared memory cell to aggregate maxima, yielding the same asymptotic time bound and confirming practical efficiency particularly when the fitness vector is sparse. This approach enables fast, precise parallel selection in heuristic algorithms (e.g., ant-colony optimization for the traveling salesman problem) where many fitness entries are zero.
Abstract
The roulette wheel selection is a critical process in heuristic algorithms, enabling the probabilistic choice of items based on assigned fitness values. It selects an item with a probability proportional to its fitness value. This technique is commonly employed in ant-colony algorithms to randomly determine the next city to visit when solving the traveling salesman problem. Our study focuses on parallel algorithms designed to select one of multiple processors, each associated with fitness values, using random wheel selection. We propose a novel approach called logarithmic random bidding, which achieves an expected runtime logarithmic to the number of processors with non-zero fitness values, using the CRCW-PRAM model with a shared memory of constant size. Notably, the logarithmic random bidding technique demonstrates efficient performance, particularly in scenarios where only a few processors are assigned non-zero fitness values.