$l_{vw}$ and $u_{vw}$ are lower and upper bounds for the distance of the edge $\{v,w\}$.

Here and in the following, $k$ equals $3$. Then DGP is prefixed by an ’M’ for ’molecular’. Note that the (M)DGP is contained in the i(M)DGP by setting the lower and upper bound of each interval to be equal to the actual distance. Saxe [@saxe1980] showed that deciding whether a valid embedding exists (in the DGP sense) is strongly NP-complete for $k=1$ and strongly NP-hard for $k > 1$. Interestingly, the problem becomes solvable in polynomial time if all distances are given [@blumenthal1953dg; @crippen1988distance; @dong2002linear]. Since solving the decision problem (finding a valid embedding) is difficult and even not always possible, we continue by considering the embedding task as an optimization problem, to be solved by heuristics. As a measure of error, one could use the *largest distance mean error* (LDME) defined as: $$\label{eq:ldme}
 \textnormal{LDME}(x) = \sqrt{\frac{1}{|E|} \sum_{ \{v,w\} \in E} \max(l_{vw} - \lVert x_v - x_w \rVert, \lVert x_v - x_w \rVert - u_{vw} , 0)^2}.$$ An embedding $x$ that has an LDME value of $0$ is obviously a solution of the iDGP as each distance constraint is met. One could thus minimize the 