Sparse ruler

Summary

A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length $L$ with $m$ marks is a sequence of integers $a_1, a_2, ..., a_m$ where $0 = a_1 < a_2 < ... < a_m = L$. The marks $a_1$ and $a_m$ correspond to the ends of the ruler. In order to measure the distance $K$, with $0\le K\le L$ there must be marks $a_i$ and $a_j$ such that $a_j-a_i=K$.