On asymptotic complexity of the optimum Golomb ruler problem: From established stochastic methods to self-avoiding walks

Optimum or near-optimum solutions of the Golomb ruler (gr) problem have applications in information theory, error correction, current transformers, radio frequency selection, placement of antenna arrays in astronomy, among others. In mathematics, there is a well-defined relationship of Golomb rulers and graceful graphs. A massively parallel computing project on gr has been on-going for more than 10 years: the order-24 ruler was claimed as optimum in November 2004 (after a 4-year computational effort), followed by order-25, -26, -27 rulers in 2008, 2009, 2014. The order-28 ruler is work-in-progress. The distribution of waiting times, in years, such as {4, 4, 1, 5} may be impossible to predict under a variable number of processors running simultaneously. This paper proposes a model to experimentally predict the asymptotic runtime complexity of any gr solver that returns the best-known-value (BKV) Golomb ruler defined by the paired list (L = length, M = order). A subset of this list includes {(6,4), (11,5), (17,6), (25,7), (34,8), (44,9),..., (680,30)}. Given the number of processors N and the runtime limit t_lmt, we observe at least N_u ≥ 100 processors reaching the target BKV with the first-passage-time <; t_lmt and say that each such observation is uncensored. In other words, the mean runtime value we measure is based strictly on at least 100 uncensored observations from the experiment. Experiments in this paper focus on a stochastic gr-solver that implements a variation of a self-avoiding walk. In the total number of steps, the solver asymptotic walkLength complexity is 409.2 × 1.0762^L. When measuring the number of CPU seconds on a loaded grid of 100 processors, the solver asymptotic runtime complexity is 0.000206 × 1.08711^L. This solver significantly outperforms the alternative stochastic gr-solvers reported to date.