Global Optimization vs Local Optimization in the grid model.

(Panel a) Dependence of 〈log t〉 on the travel length l (red points). The solid blue line is the best fit logarithmic function (${\chi }_2^{}\simeq 2.1$). (Panel b) Dependence of σ_l as a function of l. Dots corresponds to paths sampled by using (blue) or not using (red) intermediate optimization steps between the origin and destination nodes of each path. (Panel c) Dependence of 〈logt〉 upon travel length l (blue points) with paths sampled by picking intermediate stops between the origin and the destination nodes. The solid blue line is the best fit logarithmic function. Both optimization processes yield the correct dependence of μ_l on l, but the local piecewise optimization also reproduces the steady behavior of σ_l.