Minimal Expansions in Redundant Number Systems and Shortest Paths in Graphs

Abstract.

We consider digit expansions $n=\sum_{i=0}^l \epsilon_iq^i$ in redundant number systems to base q with $-(q-1)\le \epsilon_i\le q-1$ and consider such an expansion as minimal, if $l + \sum\nolimits_{i = 0}^l {|\varepsilon _i |}$ is minimal. We describe an efficient algorithm for determining a minimal representation and give an explicit characterization of optimal representations for odd q.