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Evan Conway

Publications and source records attributed to Evan Conway.

2 recordsLinked to original sources

POrTAL: Plan-Orchestrated Tree Assembly for Lookahead

When tasking robots in partially observable environments, these robots must efficiently and robustly plan to achieve task goals under uncertainty. Although many probabilistic planning algorithms exist for this purpose, these algorithms can be inefficient if executed with the robot's limited computational resources, or may produce policies that take more steps than expected to achieve the goal. We therefore created a new, lightweight, probabilistic planning algorithm, Plan-Orchestrated Tree Assembly for Lookahead (POrTAL), that combines the strengths of two baseline planning algorithms, FF-Replan and POMCP. We demonstrate that POrTAL is an anytime algorithm that generally outperforms these baselines in terms of the final executed plan length given bounded computation time, especially for problems with only moderate levels of uncertainty.

cs.RO

Decomposing a factorial into large factors

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erd\H{o}s and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

math.NT