Given an undirected and unweighted graph $G = (V, E)$, is it possible to find a mapping $f: V \rightarrow \mathbb{R}^k$ for some $k$ such that for every $i, j \in V$, $\|f(i) - f(j)\|_2^2 = \Delta(i, j)$, where $\Delta(i, j)$ is the shortest path length between $i$ and $j$ in $G$?

I have been testing a few counterexamples for which these isometric embeddings in $\ell_2$ fail to exist (for example, the 4-cycle), but in this case they work.

  • $\begingroup$ Notice that the standard metric over the Euclidean space is the $\ell_2$-norm $\|f(i)-f(j)\|_2$, rather than $\|f(i)-f(j)\|_2^2$. Any reason why you consider the squared $\ell_2$-norm? $\endgroup$ – smapers Aug 19 '20 at 14:52
  • $\begingroup$ @smapers Because of the second paragraph? $\endgroup$ – Emil Jeřábek Aug 19 '20 at 14:56
  • $\begingroup$ Oh, ofcourse, thanks! $\endgroup$ – smapers Aug 19 '20 at 14:59
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    $\begingroup$ Metric embeddings is a well-explored topic. See book by Deza-Laurent springer.com/gp/book/9783540616115 or a web search will reveal several lecture notes and papers on this topic. Square distance embeddings are negative type metrics and one can in fact test, via semi-definite-programming, whether one can embed a given finite metric into a Euclidean space (no restriction on dimension). Many problems here are also NP-Hard. $\endgroup$ – Chandra Chekuri Aug 19 '20 at 19:29
  • $\begingroup$ Thanks Chandra, the pointer turned out to be useful. I'm not an expert of the field and the only relevant theorem I was aware of is due to Schoenberg (ams.org/journals/tran/1938-044-03/S0002-9947-1938-1501980-0/…) but it is quite complicate to work with. $\endgroup$ – Andrea Aug 20 '20 at 14:19

This is not possible in general.

The 4-cycle is actually helpful to consider: embedding it in $\mathbb{R}^k$ in the way you describe requires the images of all four vertices to be coplanar, forming a square (since the distances between adjacent vertices must be $1$ and those between the non-adjacent pairs must be $\sqrt{2}$).

Now consider the complete bipartite graph $K_{2,3}$. Let us call its vertices $a_1,a_2,b_1,b_2,b_3$. In any embedding $f$ satisfying your condition, the 4-cycles $(a_1,b_1,a_2,b_2)$, $(a_1,b_1,a_2,b_3)$ and $(a_1,b_2,a_2,b_3)$ would all have to be mapped to squares as above, but the first two of these conditions imply $f(b_2) = f(b_3)$, contradiction.

  • $\begingroup$ Thanks a lot Klaus. $\endgroup$ – Andrea Aug 20 '20 at 14:23

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