Suppose I have a sequence of non-negative integers $J=\{j_1,j_2,\ldots,j_n\}$ and want to find (if possible) a set of integers $I=\{0=i_1<i_2< \cdots < i_m\}$ such that $j_t$ counts the number of pairs $(i_k, i_\ell)$ with $i_\ell>i_k$ and $i_\ell-i_k=t$.

E.g., if $J=\{3,2,2,2,1\}$, then we can take $I=\{0,1, 2, 4, 5\}$. Of course, $\sum_{k=1}^{n} j_{k}=m(m-1)/2$, $i_m=n$, $j_n=1$.

My question is what is known about this problem (citations to the literature,...) and whether there exists an efficient algorithm for finding such $I$ for a given $J$, or determining that no such $I$ exists for a given $J$.