# Spiral Galaxy

This study strengthens the links between Mean Payoff Games (MPGs) and Energy Games (EGs). Firstly, we offer a faster $O(|V|^2|E|W)$ pseudo-polynomial time and $\Theta(|V|+|E|)$ space deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in MPGs. This improves the best previously known estimates on the pseudo-polynomial time complexity to:
$O(|E|\log |V|) + \Theta\Big(\sum_{v\in V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\Big) = O(|V|^2|E|W)$,
where $\ell_{\Gamma}(v)$ counts the number of times that a certain energy-lifting operator $\delta(\cdot, v)$ is applied to any $v\in V$, along a certain sequence of Value-Iterations on reweighted EGs; and $\texttt{deg}_{\Gamma}(v)$ is the degree of $v$. This improves significantly over a previously known pseudo-polynomial time estimate, i.e. $\Theta\big(|V|^2|E|W + \sum_{v\in V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\big)$, as the pseudo-polynomiality is now confined to depend solely on $\ell_\Gamma$. Secondly, we further explore on the relationship between Optimal Positional Strategies (OPSs) in MPGs and Small Energy-Progress Measures (SEPMs) in reweighted EGs. It is observed that the space of all OPSs, $\texttt{opt}_{\Gamma}\Sigma^M_0$, admits a unique complete decomposition in terms of extremal-SEPMs in reweighted EGs. This points out what we called the “Energy-Lattice $\mathcal{X}^*_{\Gamma}$ associated to $\texttt{opt}_{\Gamma}\Sigma^M_0$“. Finally, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of $\mathcal{X}^*_{\Gamma}$, and for computing the corresponding partitioning of $\texttt{opt}_{\Gamma}\Sigma^M_0$.